Identifying demand shocks from production data Carlos D. Santos Nova School of Business and Economics March 23, 2017
Abstract Standard productivity estimates contain a mixture of cost e¢ ciency and demand conditions. In this article, I propose a method to identify the distribution of the demand shock using production data. Identi…cation does not depend on functional form restrictions. It is also robust to dynamic demand considerations and ‡exible labor. In the parametric case the ratio of intermediate inputs to the wage bill (input ratio) contains information about the magnitude of the demand shock. The method is tested using data from Spain which contains information on prices and demand conditions. Finally, we generate Montecarlo simulations to evaluate the method’s performance and sensitivity. Keywords: Demand, Production, Identi…cation, Input choices, Productivity JEL classification: C18, C23, C33, C51, D24
1
Introduction
The traditional productivity literature is centered in the estimation of total factor productivity (TFP). However, standard productivity estimates often measure a mixture of cost e¢ ciency (TFPQ) and demand conditions. Such mismeasurement is due to the inexistence of data on physical output (and prices) and the presence of imperfectly competitive Nova School of Business and Economics, UNL, 1099-032 Lisboa, Portugal.
1
markets, a problem studied by Klette and Griliches (1996). Recent research has shown that demand conditions, in particular persistence, are important determinants of performance. For example, …rms exploit demand accumulation by lowering margins earlier in their life and adjusting them upwards as they age (Foster et al., 2008, 2016) and …rms’ responses to demand shocks are much larger than the reaction to e¢ ciency shocks (Pozzi and Schivardi, 2016). The importance of demand is far reaching. For example, if we apply these …ndings to the trade literature, most of the empirical evidence on the productivity advantage of exporters could be due to demand and brand building, not cost advantages. It also rationalizes the stylized facts on the relation between number of countries and number of products exported (Eaton, Kortum and Kramarz, 2011) and explains why international trade is so highly concentrated in just a few …rms. Escalation strategies due to demand accumulation are known since at least Sutton’s (1991, 1998) seminal work. Demand is also a fundamental component of the business cycle literature. Santos, Costa and Brito (2016), using price data, investigate the response of markups to demand and supply shocks and …nd that markups are countercyclical and negatively correlated with the demand shocks while procyclical and positively correlated with productivity shocks. Gilchrist et al. (forthcoming) show how dynamic demand considerations, together with …nancial constraints, modify the pricing strategy and rationalize markup countercyclicality. The generalized applicability and study of demand e¤ects is prevented by the lack of widely available price information, which restricts generalized cross industry studies. This article tries to tackle this limitation by constructing a measure of the demand shock from the optimality conditions used in the productivity literature (e.g. De Loecker (2011) and Gandhi et al (2016)). By revealed preferences, observed decisions can potentially convey information on exact demand conditions faced by the …rms when making such decisions. In this article, I use observed input choices to recover the distribution of shocks to demand, when prices (and physical output) are not directly observed by the econometrician. The idea is to exploit adjustment costs and the di¤erences in timing for the various input choices. When one input is more ‡exible to adjust (say materials)
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and another is less ‡exible (say labour), the asymmetry of responses provides information about the unexpected ’news’. The use of optimal input choices has a long tradition in the productivity literature (see for example Hall (1988) or Klette (1999)). De Loecker (2011) and De Loecker and Warzynski (2012) propose using input shares to recover markups. Under optimality, the input share (ratio of intermediate inputs to revenues) to the output elasticity of the intermediate input (for example the input coe¢ cient of a standard Cobb Douglas production function) is equal to the markup. When physical output is observed, the input elasticities can be estimated via production function and the markup directly constructed by dividing the input share by the input elasticity. This idea nicely avoids any demand speci…cation while the introduction of persistent input price di¤erences addresses the unidenti…cation problems raised in Bond and Soderbom, (2005) and Ackerberg, Caves and Frazer (2006). Unidenti…cation in the estimation of production function arises because all ‡exible inputs are optimally chosen conditional on the same set of variables. There is thus no meaningful source of variation in the variable inputs, conditional on the …xed inputs and productivity (collinearity). This means the output elasticity of the ‡exible inputs cannot be separately identi…ed from the e¤ects of the state variables, capital and e¢ ciency. Gandhi et al (2016) propose using the same input shares to address the collinearity problems, and non-parametrically identify the production function. From the optimality conditions, the optimal input share is a function of the output elasticity. We can then non-parametrically estimate it in a …rst step regression. I build on ideas from these two approaches to obtain meaningful variation in optimal input choices. De Loecker (2011) allows for ‡exible demand with persistent idiosyncratic markups while it requires data on physical output to estimate the input elasticity. On the other hand, Gandhi et al (2016) do not require data on physical output but impose some restrictions to the demand speci…cation. For example, markups must be common across …rms to prevent endogeneity of the input share regression. One important distinction from previous work is that while Gandhi et al (2016) and De Loecker (2011) use a transforma-
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tion of the …rm’s …rst order condition (the input share) to estimate the coe¢ cients of the ‡exible inputs and the markups, I use an alternative transformation of the …rm’s …rst order condition (the input ratio) to identify the distribution of the shocks to demand. The two approaches are complementary. The main advantage of De Loecker (2011) is the ‡exibility of "allowing for various price-setting models and puts no restrictions on underlying consumer demand" or the markup. It proposes that markups are estimated using the share of intermediate inputs. On the other hand, the main disadvantage is that it requires price level information to estimate the output elasticities and does not address the unidenti…cation problem raised by Ackerberg, Caves and Frazer (2006) and Bond and Soderbom (2005). The unidenti…cation problem is addressed by Ghandi et al. (2016) which uses the same input share as De Loecker (2011) but to estimate the output elasticities, instead of estimating the markups. However, contrary to De Loecker (2011), it needs to restrict markups to be constant across …rms. This di¤erence is analyzed in more detail when we introduce the input ratio in Section (2.2). The current article allows for two shocks, to e¢ ciency and to demand. Compared to the literature, it does not require data on physical output (i.e. the case most commonly faced by researchers). As shown by Santos et al. (2016), the introduction of persistent demand shocks and the use of price data solves the unidenti…cation problem. On the other hand, I do not attempt to estimate the input elasticities.1 Instead, attention is focused into obtaining a measure of the demand shock. We can identify the distribution of demand shocks when we specify the full input optimization decisions and add a timing of decisions where materials are chosen after labour. This choice is consistent with recent …ndings. For example, regarding the relevance of demand shocks, Pozzi and Schivardi (2016) …nd evidence that …rms react more to demand shocks than to productivity shocks. It also generates testable implications that we will explore. The system of equations obtained from the …rst order conditions for materials and labour, together with the output equation is however not su¢ cient to identify the demand shocks. We need to introduce 1
Markups are identi…ed from the input share up to a constant so that variations in markup are given directly by the variations in the input share, as discussed by De Loecker and Warzynski (2012). The level of the markup is tied down by the estimate of the input elasticity. Thus, the problem of obtaining the markup is identical to the correct estimation of the input elasticity. Once we know the elasticity, obtaining the markup is straightforward.
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auxiliary assumptions similar to the ones that have been used to address the input endogeneity problem in production functions (e.g. Olley and Pakes, 1996). In particular, I use a separability restriction on the demand shocks. This extra equation allows us to solve the system of equations (under monotonicity and invertibility conditions) for the optimal inputs and output as a function of all the observables and the demand shock. In the parametric case under CES demand and Cobb Douglas technology, the demand shock is directly obtained from the input ratio (up to a constant and a monotone transformation). Finally, I compare the estimated measure, against self-reported measures of market conditions on a long panel of …rms across all industrial sectors. I also provide empirical evidence on testable implications and a series of robustness checks. I conclude with a series of Montecarlo exercises to evaluate the model’s performance and sensitivity.
2
Framework
The distinctive feature of this paper is the introduction of two exogenous shocks, to demand and supply. We thus have two unobservable components and, for the moment, both the production and demand functions are left unparameterized. We require at least two inputs used in production. Input choices satisfy the following conditions: Assumption 1 (A.1) Sequential adjustment: Inputs are set sequentially. For example, materials are chosen conditional on labour (L).
Assumption 2 (A.2) Information: (i) Firms observe their productivity before any decision is made. (ii) Demand conditions are observed after labour is chosen (adjustment costs to labour prevent its instantaneous adjustment to match demand conditions). For the remainder of the article I will consider three inputs: capital, labour and materials. Assumptions (A.1) and (A.2) generate an exclusion restriction since the demand shock shifts one of the input choices but not the other. Persistence is allowed in the demand component. At any given period the demand component can be divided into a predictable and an unpredictable (or ’news’) component. The ’news’component moves 5
materials independently from labour. Labour is thus allowed to be correlated with the predictable part of demand. Firms with larger levels of demand have a larger workforce. Assumptions (A.1) and (A.2) generate testable predictions. Note that (A.1) is not the main cause of concern. Its validity might vary by industry/country/time but overall it is sensible to assume that materials are more responsive to demand shocks than labour. For example, in some countries, labour regulations will imply that labour is almost as di¢ cult to adjust as physical capital. The second Assumption (A.2), allows us to "control" for unobserved heterogeneity in productivity and therefore separate the demand shock from the productivity shock. It is implicit that productivity is a predictable process known to the …rm, while demand is much more unpredictable. This assumption can be more controversial than (A.1) and I discuss the e¤ect of departures from it in Section (2.5). Furthermore, I will empirically test how productivity and demand shocks shift the two inputs. Under correct model speci…cation, we expect productivity shocks to shift both input choices while the demand shock shifts materials, but not labour. We can now introduce the timing, the demand and the production functions. Timing Given the above assumptions the timing of the model is the following: 1. Productivity is observed: The …rm decides on (i) Investment plan where capital will take one year to be delivered. (ii) Its labour hiring/…ring decisions where labour becomes immediately available. 2. Demand shock is observed: Conditional on the capital existing in the beginning of the period and the labour set after observing productivity, the …rms buys materials to execute production and deliver its …nal products. For expositional reasons, we start with a simple model in this section and show, in the coming sections, how results are equivalent in the more complex settings with (i) dynamic demand and (ii) non-…xed labour subject to adjustment costs. Demand Inverse demand for the product of …rm i in period t is pit = p(qit ; Zit ; "dit ) 6
where qit is the quantity, pit the price and Zit a set of observed variables that in‡uence demand. Zit can include either exogenous or endogenous market structure variables (e.g. the number of competitors or the vector of competitors’quantities/prices). The structure of the pricing game and its equilibrium are left uncharacterized. The function p (:) is the inverse demand function conditional on consumer tastes and the equilibrium strategies of the opponent …rms. In the endogenous case Zit will be correlated with "dit . Finally, "dit is the demand component (unobserved to the econometrician) that we now specify. Unobserved demand satis…es the following property: E "dit j
it
= g1 [
it ]
where
it
is a vector containing the information known to …rm i when it makes its labour hiring decision. This information set contains the demand level from last period ("di;t 1 ), together with all other state variables, like the capital stock or productivity. One can also include the whole (or a restricted) history of signals. For example, if we restrict unobserved demand to be an exogenous Markovian process, "dit
1
becomes a su¢ cient statistic.
Supply Let ! it denote productivity/e¢ ciency. While we leave it general for now, in the parametric speci…cation of Section (2.4) we will restrict attention to the Hicks neutral case. In particular, in the Hicks neutral case the marginal rate of technical substitution does not depend on the productivity term, which is unobserved by the econometrician. Output can be produced using several inputs with technology qit = q(Kit ; Lit ; Mit ; ! it ) where q (:) is strictly increasing in ! for any (K; L; M ).
2.1
Solution
From the setting speci…ed above we can obtain the optimality conditions for the two inputs.
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By the end of the period the …rm chooses the level of materials which maximizes the following objective function max p(qit ; Zit ; "dit )qit Mit
pM it Mit
s.t. qit = q(Kit ; Li;t ; Mit ; ! i;t ) where pM it is the price of materials. From the …rst order condition [ where
p;q
p;q
+ 1] p(qit ; Zit ; "dit )qm = pM it
(1)
@q(:) . @M
We can rewrite the reduced
< 0 is the elasticity of demand and qm =
form optimal solution as Mit = m(! it ; Lit ; Kit ; Zit ; "dit ) In the beginning of the period the …rm chooses the level of employment that maximizes the expected returns max E"d p(qit ; Zit ; "dit )qit Li;t
pM it Mit j
it
pLit Lit
s.t. qit = q(Kit ; Lit ; Mit ; ! it ) which gives2 E"d
@p(qit ; Zit ; "dit )qit @qit j @qit @Lit
where ql =
@q(:) . @L
it
= E"d [
p;q
+ 1] p(qit ; Zit ; "dit )ql j
it
= pLit
(2)
In the Appendix we consider the extension to the case with two
types of labour and quadratic adjustment costs to labour. In this case there will be a further wedge in equation (2). For the remainder of the article, I will assume standard economic restrictions in the production and inverse demand functions such that there is a unique optimal choice of intermediate inputs and labour. 2
From equation (1), h it @qit L pit = E"d @p(:)q @qit @Lit + we obtain equation (2).
h i M @p(qit ;Zit ;"d it )qit pit Mit = pM it . The total derivative is E"d @Lit i @Mit pM pL it it (by the envelope theorem). Replacing the above @Lit
@p(qit ;Zit ;"d it )qit @qit @qit @Mit @p(:)qit @qit @qit @Mit
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2.2
The input ratio
First notice that combining the two …rst order conditions for Mit and Lit from equations (1) and (2) we obtain pM [ ql Mit it Mit = L pi;t Li;t F OC ql Lit E"d [ =
[ qm Mit ql Lit E"d [ 1
=
m:l |{z}
Supply (ETS)
where
m:l
+ 1] p(qit ; Zit ; "dit )qm d p;q + 1] p(qit ; Zit ; "it )ql j
p;q
+ 1] p(qit ; Zit ; "dit )ql d p;q + 1] p(qit ; Zit ; "it )ql j
(3) it
p;q
E"d | it
d p;q (qit ; Zi;t ; "it ) + 1 d p;q (qit ; Zi;t ; "it ) + 1
{z
it
p(qit ; Zi;t ; "dit )ql p(qit ; Zi;t ; "dit )ql j
Demand
it
}
is the elasticity of technical substitution (ETS).3 The ratio in expression (3)
depends on the ETS and the ratio of marginal return to labour to expected marginal return to labour (demand side). In case of no uncertainty, the denominator and numerator of the second term simply cancel out and the input ratio equals the ETS. In the case of Hicks neutral productivity, the ETS is a deterministic function of the inputs. Stochastic variation in the input ratio must thus come from some other stochastic element. Input price variation does not alter the ratio either (unless input prices are unknown before choosing inputs). Input prices are stochastic and their variation, per …rm and over time, is incorporated in the ratio. For example, if pLit increases, …rms substitute away from Lit towards Mit (e.g. by outsourcing parts of production) such that equation (3) is satis…ed. The optimality condition implies that inputs are chosen so that the expected marginal rate of technical substitution equals the input price ratio. Because Lit cannot be automatically adjusted to the demand shock, in practice the marginal rate will depart from its ideal level, i.e. equal to the input price ratio and,
ql qm
6=
pL it . pM it
The wedge
ql pL = it qm pM it
will be informative about the size of the demand shock "d . This is the reason why the ratio identi…es (a monotonic transformation of) the distribution of shocks to demand. 3
The ETS,
ml
=
ql M qm L
where
ql qm
is the marginal rate of technical substitution.
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Alternatively pM M it Mit h it i h = = L pit Lit E"d qqml j it Lit E"d =
Mit E"d [ m:l Mit j
Mit Mit m:l Lit j
it
i
(4) Lit
it ]
The last step is possible because Lit is known at the time Mit is being chosen. When the ETS is constant, as with Cobb Douglas technology, the input ratio is nothing more than a constant times the wedge between optimal choice of materials and its ex-ante expected value (strictly speaking we only require the ETS not to depend on M so it can be taken out of the expectation). This wedge, which causes the input ratio to depart from its optimal value (the ETS), is a function of the demand shock. Thus, the input ratio contains valuable information about the demand shock. The input share Instead of the input ratio, the literature has used the input share to either estimate the markup (De Loecker, 2011) or the output elasticity (Ghandi et al., 2016). To illustrate how this is done rearrange equation (1)
where ([
p;q
q;m (Kit ; Lit ; Mit )
ln
pM it Mit = ln pit qit
=
@q(:) M @M q
q;m
(:)
ln
it
is the output elasticity and
it
is the markup
+ 1] 1 ). This equation connects De Loecker (2011) and Ghandi et al
(2016) and shows how it is di¢ cult to use the input share to separately identify the output elasticity (
q;m )
et al (2016) from the markup (
which is the approach followed in Ghandi it )
which is the approach followed by De
Loecker (2011) without more data and/or more assumptions. To tackle this, De Loecker (2011) uses price level information to estimate the output elasticity (
q;m )
elsewhere and uses this equation to recover the markup directly from
the data. Santos, et al (2016) show that this approach does not su¤er from the unidenti…cation problem as long as there are serially correlated demand shocks which make past values of the variable input valid instruments. On the other hand, Ghandi et al. (2016) make progress by restricting the markup 10
to vary only across time (
it
=
t)
and this way recover the output elasticity
without requiring information on …rm level prices (they also include an ex-post shock). Finally, note that we face two problems when estimating equation (4). The …rst is that qit , ql and
m:l
are functions of !, which is unobserved. The second is that even if ! was
observed, M and all other inputs are correlated with "dit and therefore endogenous. The structure of the model gives us conditions to use a control function approach to control for ! and "d by inverting the optimality condition for labour and materials. Furthermore, lagged values of intermediate inputs (Mi;t 1 ) are valid instruments for the current level of intermediate inputs. The nonparametric case with instrumental variables is discussed in the Appendix. However, we prefer to avoid the estimation complications of using nonparametric IV, so we present the parametric case.
2.3 2.3.1
Extensions Dynamic demand
For convenience, we have considered the case with static demand in the previous section. The results can be extended to the case of dynamic demand. Here we consider the case using a demand speci…cation similar to Foster et al (2016). We also consider in Appendix A.1 a demand speci…cation similar to Gourio and Rudanko (2014). For example, in Foster et al. (2016), demand takes a similar form qit =
ln(pit ) +
a
ln(agei;t ) +
z
ln(Zi;t;1 ) + ln("dit )
(5)
where Zit is the demand "stock" of the …rm which depends on the previous …rm sales. The stock evolves as Zit = (1
) Zi;t
1
+ pit qit . Consider the more general, non-parametric
case with pit = p(qit ; Zi;t 1 ; "dit ). For convenience let’s abstract from age as this adds nothing substantial to the analysis and we would need to keep track of one extra state
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variable. As shown in Appendix A.1, the input ratio is pM it Mit = pLi;t Li;t
d p;q (qit ; Zi;t ; "it ) + 1 d p;q (qit ; Zi;t ; "it ) + 1
1 m:l |{z}
Supply (ETS)
E"d | it
p(qit ; Zi;t ; "dit ) (1 + EVZ0 ) ql p(qit ; Zi;t ; "dit ) (1 + EVZ0 ) ql j {z
Demand
it
}
where EVZ0 = E"di;t+1 ;!i;t+1 VZ0 it ("di;t+1 ; ! i;t+1 ; Zit j"dit ; ! it ; Zi;t 1 ) is the e¤ect sales today on the demand stock tomorrow. The main di¤erence from before is the continuation term EVZ0 . So, the marginal returns of increasing Mit consists of the increase in short-term revenues and future revenues due to the increase in the demand stock Z. Put in another way, …rms are willing to produce more and receive a lower price because every unit they sell also increases future sales. This generates the exact same input ratio as equation (4). 2.3.2
Non-…xed employment (overtime/temporary work)
One restriction implied by Assumption (A.1) is the impossibility of short term labour adjustments, an extreme form of adjustment costs. In practice, …rms adjust its workforce by regularly using overtime or temporary workers to meet ‡uctuations in production. Allowing for more general adjustment costs in equation (3) implies that the wedge from the input ratio and the ETS is now a function of the adjustment costs as explained in the Appendix A.2. The equations for the input ratio and the temporary employment (LT ) to permanent employment (LP = L
LT ) are the following
pM 1 it Mit = P pitL Lit m:l
1
+2
LTit LPit
(6)
P
pLit LTit LTit = P 2 E"d [LTit j pLit LPit where 0 <
it ]
< 1 is the cost saving (…scal) advantage of temporary work and
(7) is the
measure of the quadratic adjustment costs. Equation 6 nicely shows how a demand shock is transmitted to the input ratio via temporary employment. A positive shock will lead to an increase in the ratio of temporary to permanent employment which leads companies to increase the materials relative to the wage costs. This is the optimality condition when LTit > 0 (interior solution). When LTit = 0 (binding constraint) employment is quasi…xed and we are back in the original case. Note that the ratio of temporary employment
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to permanent employment is a function of the demand shock. The demand shock will be transmitted to the input ratio via temporary employment since companies will hire temporary employment to adjust production and at the margin temporary employment and materials should be equated.
2.4
Parametric case
We now present the parametric case for the static demand model with employment …xed in the short run. This model will be used for estimation. Nonparametric identi…cation is contained in the Appendix. In the static case with Cobb Douglas technology (qit = ! it Mit Lit Kit ) the ETS is = and the ratio simpli…es to pM it Mit = pLit Lit
Mit E"d [Mit j
(8)
it ]
so the wedge between the input ratio and the ETS is the di¤erence between optimal input choice and the expected optimal input choice. This wedge is a monotone transformation of the demand shock. This is because Mit will depart from E"d [Mit j
it ]
due to the
unexpected component. If we also parametrize the demand function we obtain an analytic solution for optimal input choice. If we let demand be of the CES type p(qit ; Zit ; "dit ) = qit h(Zit )"dit where the demand shifter (h(Zit )) can be a function of any subset of the observables, Zit and
< 0. The optimal input choice is ( +1)
Mit =
( + 1) hit ! it Lit Kit
"dit
1 pM it
1
1 ( +1)
(9)
where for short, the demand level shifter is hit = h(Zit ). This parametric speci…cation guarantees separability of the input demand function in the demand shock. The input ratio is simpli…ed since the terms in the numerator and the denominator of the input ratio cancel out, except for the component related with the demand shock. Replacing equation
13
(9) in the input ratio (8) pM it Mit = pLi;t Li;t
E
h
"dit "dit
1
1
1 ( +1) 1 ( +1)
j
it
i
The input ratio is a sole function of the ETS and the demand component. For simpli…cation, let the demand shock be a separable process, "dit = g( where
it
it ;
d it )
= g(
it )e
d it
(10)
is a vector which includes all the history observed by …rm up the moment
it makes its labor choices. The important restriction here is separability. Write ~dit = 1
1 d ( +1) it
and normalize E
d it
~dit = 0 (these are standard normalization and location
restrictions, see Matzkin (2007)), ln
pM it Mit = ln ( = ) + ~dit pLi;t Li;t
(11)
Equation (11) can be estimated by ordinary least squares. In fact, the ’news’to demand will be identical to the input ratio up to a constant where this constant is the ETS, = . In summary, when the two inputs are not fully ‡exible, the only input shifted by the demand shock is materials. Once we condition on all the other inputs we "control" for productivity and past demand shocks and the remaining residual is the news to demand. Heterogeneity and serial correlation Cross sectional heterogeneity in input usage could re‡ect heterogeneity in production technology. Some …rms use more labour while others use more materials. Such heterogeneity in production can be allowed in the previous case with the …xed e¤ects component ( = = ( i = i )) - technology is allowed to vary by …rm. The results presented below either use the input ratio directly or an estimate of equation (11) individual …rm speci…c e¤ects and lagged dependent variable (see next for serial correlation concerns). These account for the fact that in the general case, = is not constant across …rms (cross sectional heterogeneity) and time (serial correlation). Note that in this case the estimated "news" shock contains a …rm speci…c component, ~dit =
d i it .
14
Furthermore, we can have di¤erences in the quality of inputs or their usage (e.g. number of workers being a bad proxy for labour inputs due to the di¤erences in hours usage). The input ratio is calculated with the value of inputs rather than physical units. This allows us to account for both di¤erences in input usage (e.g. more hours) since using more of an input will lead to a larger expenditure. It also allows us to account for di¤erences in quality since higher quality inputs will be more expensive. On the margin, the optimal conditions already account for such di¤erences, and the input ratio is obtained from the …rst order conditions. A second concern is due to serial dependence in the elasticity of technical substitution. This can arise when there is a slight departure from the Cobb-Douglas speci…cation, and the ETS is not constant but a function of inputs. This generates serial dependence in the input ratio generated by serial dependence in the ETS. We will account for this by using a simple dynamic panel data model to estimate the demand shock.
2.5
Discussion
I now discuss departures from Assumptions A.1 and A.2. Under the null that both assumptions are true, the news to demand is (positively) correlated with materials and uncorrelated with employment. On the other hand, if we depart from these assumptions and let the shock to demand also be correlated with the employment, the input ratio would simply equal the inverse of the ETS 1 pM it Mit = L pit Lit m:l The ETS depends only on production inputs (plus productivity in the non-Hicks neutral case) and not on demand side variables. Thus, due to misspeci…cation, what we recover as the "demand shock" is a function of supply side variables, which can for example generate a positive correlation with materials and negative correlation with employment. By …nding a non-zero correlation with employment we reject the null hypothesis. A second set of tests can be performed when we observe output prices. In particular, we expect to observe a positive correlation between the demand shocks with both output
15
and price, while supply shocks should be positively correlated with output and negatively correlated with prices. However, we should emphasize that if prices and quantities are observed, we can estimate demand shocks directly via demand function rather than indirectly via the input ratio (as done in Foster, Haltiwanger and Syverson, 2016 or Santos, Costa and Brito, 2016). In the empirical section, I perform both sets of tests and also compare how the shocks to demand recovered via input ratio compare with the shocks recovered via demand function.
3
Empirical results - Industrial sector in Spain
This section evaluates how the proposed method performs in the data. The sample is part of the ESEE (Encuesta sobre Estrategias Empresariales) collected by the Fundacion Empresa Publica. It consists of an unbalanced sample of 30,466 observations for 4,094 …rms over the period 1990-2006 for the whole manufacturing sector in Spain with an average 1,800 observations per year (for a description of the data see Doraszelski and Jaumandreu, 2013). The survey collects information on the variables we require for the analysis, namely, the input choices (employment, wage bill and expenditures in intermediate inputs), the capital stock, revenues, output price variation and some measures of market structure). Besides these standard measures, the dataset is particularly attractive for our purposes since …rms directly report demand conditions. The exact questions are: "Classify the evolution of your market in the year: expansive, stable and recessive" and "Classify the evolution of your market share in the year: increased, constant and decreased". Most datasets do not include the measures we observe, namely, the self reported demand conditions and price variations both for output and inputs. This makes it ideal to evaluate how shocks to demand recovered via input ratio compare with (i) actual demand conditions and (ii) shocks to demand obtained as the residual from estimating a simple demand function. For a …rst glance, Figure (1) reports how the input ratio (expenses in materials over 16
the wage bill) exhibits a strong correlation with self reported demand shocks (average per year for all the manufacturing sector). Next, we recover the demand shocks as the residual from estimating equation (11) with lagged dependent variable and individual …rm speci…c e¤ects. This accounts for cross sectional heterogeneity and serial correlation in the ETS. Figure (1) plots the time series of the estimated average and compares this to the reported demand shocks, for the whole industrial sector. Figure (A.1) in the Appendix repeats the same exercise separately for each individual industry. Overall, the estimated demand shocks closely match the self reported …gures with the correlations ranging from 0.18 for "Food and Tobacco" to 0.89 for "Basic Metals". Table (1) reports the moments of the demand shock, namely, the standard deviation, serial correlation up to order 5 and cross sectional dependence. Notice that for the cross sectional dependence we need to restrict the sample to …rms that have more than 9 observations to guarantee that there are at least two observations overlapping between each …rm. We also report Pesaran’s test of cross sectional independence. The standard deviation is about 34.5% which results in about 95% of the demand shocks being contained in the
69% interval. Demand shocks
exhibit a signi…cant cross-sectional dependence. However, this dependence is explained by macro shocks and becomes insigni…cant once we account for the year e¤ects (dummies). Finally, in Table (2) we evaluate the correlations between estimated and reported demand conditions at a more disaggregate (…rm) level. Again, estimated shocks are signi…cantly correlated with both the self reported demand shocks (Panel A and C) and changes in market shares (Panel B and C). Changes in demand conditions are more important that changes in market share (Panel C) as the market share might increase without changes to overall demand, and vice versa. The coe¢ cients of these regressions can be interpreted as semi-elasticities. A move from stable to expansionary demand conditions is equivalent to a 5% (Food) to 25% (Other transportation) estimated increase in demand, depending on the industry. In summary, the results show that the demand shocks are signi…cantly correlated with self reported market conditions both at the industry and the more disaggregated …rm level.
17
.1
.8
0
.7
.8
.3
-.3
.4
-.2
.5
-.1 all_ind
.6
Input ratio (detrended) .6 .5 .7
1990
1995
2000
2005
.4
YEAR 1.6
1.8
2 2.2 Self-reported demand shocks
2.4
Reported changes in Market Conditions Estimated Demand Shocks
2.6
Figure 1: Input ratio vs self reported demand conditions:whole manufacturing sector. Demand shock (moments) Standard deviation 0.345 Serial correlation of order 1 2 3 4 5 -0.046 -0.005 -0.038 -0.084 -0.121 Cross sectional dependence Average absolute o¤ diagonal 0.287 Pesaran’s p-value1 0.000 Av. abs. o¤ diag. (net of Year e¤ ects) 0.274 Pesaran’s p-value1 0.955 1 Pesaran’s test of cross sectional dependence
Table 1: Moments of the estimated demand shocks: standard deviation, serial correlation and cross sectional dependence.
3.1
Serial dependence
By Assumption in equation (10), the news to demand (~dit ) is serially uncorrelated, and cannot be forecasted from period t. Table (3) tests the serial correlation using a dynamic panel data estimator. The coe¢ cient on the lagged dependent variable is small in magnitude and not statistically signi…cant. Furthermore, the serial correlation speci…cation tests reject the hypothesis of serial correlation of order two in the di¤erenced residuals. We do not …nd any evidence to reject the assumption of serial uncorrelation.
3.2
Testable implications
As discussed in Section 2.5, the model delivers several testable implications. According to the exclusion restriction used in the identi…cation, we expect materials to be more ‡exible than labour. We should thus observe (i) that labour is more volatile than materials. Since we actually observe price variation for each individual …rm, we can further (ii) compare the 18
19
0.012 0.062*
Self-reported mkt. Mkt. share
0.094** 0.004
0.032*
0.098***
Non-metal Minerals
0.138*** 0.009
0.043**
0.144***
Basic Metal
2,583
0.033* 0.012
0.021**
0.046***
Food
0.118*** 0.023*
0.059***
0.142***
Fabricat. Metal
522
760
O¢ c./data Precision Panel A 0.162*** 0.07 Panel B 0.079*** 0.044 Panel C 0.128*** 0.043 0.041** 0.033
Equip.
2,559
0.180*** 0.006
0.060***
0.186***
Electric.
689
Dep. Variable: Estimated Demand Shocks Drinks Text. Foot. Timber Panel A 0.127* 0.131*** 0.113** 0.077* Panel B 0.053 0.086*** 0.069*** 0.025 Panel C 0.103* 0.058* 0.048 0.072 0.030 0.067*** 0.052* 0.005
0.145*** -0.006
0.032*
0.140***
Vehicles
749
0.094* 0.005
0.030*
0.100**
Paper
555
0.176* 0.106**
0.158***
0.262***
Other Transp.
1,339
0.058* 0.031
0.049**
0.094***
Publish.
Table 2: Correlation of estimated demand shocks with market conditions.
Observations 1,808 858 2,417 1,959 405 1,776 1,218 Notes: Results for linear regression of the estimated demand shocks, by industry. Each panel reports separate results. Standard errors clustered at the …rm level. * p<0.1; ** p<0.05; *** p<0.01
Self-reported mkt. Mkt. share
Mkt. share
Self-reported mkt.
Sector
720
0.065**
Mkt. share
Observations
0.076*
Meat
Self-reported mkt.
Sector
1,261
0.028 0.074*
0.083***
0.114***
Furniture
1,808
0.067** 0.020
0.032*
0.084***
Chemic.
572
0.070* 0.066**
0.090***
0.155***
Misc.
1,342
0.056** 0.013
0.030*
0.070***
Plastics
Dep. Var:
ln( t-stat -1.465 3.627
d i;t )
Coef. ln( di;t 1 ) -0.026 Constant 0.003 Serial correlation tests AR(1) -10.676 AR(2) 1.592 Firms 2,808 Observation 22,259 Notes: GMM-type instruments for the di¤erenced equation: ln( di;t 2 ), ln( di;t
p-value 0.143 0.000 0.000 0.112
3 ),
ln(
d i;t 4 )
Table 3: Dynamic panel data (GMM) estimates: test for serial correlation. proxy for demand and supply shocks with prices and physical output and, (iii) evaluate the sensitiveness of employment and materials to the demand and supply shocks. Using price variation, a price index is constructed (base year: 2000) for each …rm. A proxy for physical output is constructed by de‡ating …rm revenues with the …rm level price index. This proxy is then used to estimate a standard Cobb Douglas production function and recover total factor productivity (TFP) as the residual. The simplest case is to let e¢ ciency be divided into three components: ! it = ! i +! t +
it .
The estimated coe¢ cients
are reported in Table (A.5.2) in the Appendix. TFP is the residual from the production function equation, including the …xed e¤ects. We start by analyzing the volatility of the di¤erent variables, to see if they are consistent with the model’s assumptions. 3.2.1
Volatility
According to the exclusion restriction used in the identi…cation, we expect materials to be more ‡exible than labour. Furthermore, the identi…cation conditions can vary with the characteristics of the di¤erent industries, namely, the working of its speci…c labour markets. Table (4) reports the standard deviation of the growth rates of the di¤erent variables: revenues, employment (labour, hours and wages, temporary workers) and materials. It con…rms that materials are the most volatile of all components. This is consistent across all industries. Hours, number of workers and total wage bill all exhibit similar (smaller) volatilities, in most cases, also smaller than the variation in revenues (the exceptions being the Food and Footwear industries). The Food and Tobacco and the Footwear industries are particularly interesting because they exhibit a volatility of 20
wages which is much smaller than the volatility in employees/hours. This suggests that employment is much easier to adjust in these two industries when compared to the others. For any positive shock, more workers are hired but since wages adjust by less, it means that the "new" workers are relatively cheap when compared to the ones already employed. (i) (ii) (iii) (iv) (v) Standard deviation of Growth rates ( Revenues Employment Nom. De‡at. Workers Hours Wages Industry Meat 0.19 Food 0.22 Drinks 0.18 Text. 0.24 Foot. 0.25 Timber 0.23 Paper 0.23 Publish. 0.20 Chemic. 0.22 Plastics 0.23 Nonmetal Minerals 0.24 Basic metal 0.26 Fabricated metal 0.24 Equipment 0.27 O¢ ce/data/precision 0.29 Electricals 0.27 Vehicles 0.24 Other transport 0.44 Furniture 0.23 Misc. 0.23 Cross industry correlations
0.18 0.22 0.19 0.23 0.25 0.23 0.22 0.21 0.22 0.23 0.23 0.24 0.24 0.27 0.30 0.27 0.24 0.44 0.23 0.23
0.16 0.24 0.16 0.19 0.33 0.18 0.14 0.16 0.18 0.19 0.19 0.17 0.18 0.17 0.24 0.21 0.18 0.23 0.20 0.19
0.17 0.24 0.16 0.19 0.33 0.18 0.14 0.17 0.19 0.19 0.19 0.18 0.19 0.17 0.24 0.21 0.18 0.23 0.20 0.19
0.17 0.18 0.16 0.18 0.25 0.18 0.16 0.16 0.20 0.19 0.20 0.18 0.21 0.18 0.22 0.22 0.17 0.25 0.19 0.19
(vi) log) of: Materials
(vii) Share 1
(iv)/(ii) Hours to Revs
(vi)/(ii) Mater. to Revs
0.28 0.29 0.31 0.48 0.37 0.34 0.27 0.33 0.31 0.32 0.35 0.33 0.34 0.36 0.42 0.38 0.33 0.55 0.36 0.36
0.27 0.26 0.14 0.21 0.32 0.28 0.14 0.13 0.09 0.21 0.20 0.15 0.23 0.15 0.17 0.21 0.16 0.19 0.25 0.20 1.00
0.91 1.10 0.88 0.83 1.33 0.79 0.66 0.82 0.84 0.81 0.83 0.73 0.77 0.64 0.79 0.78 0.76 0.52 0.87 0.85 0.59
1.55 1.32 1.64 2.03 1.51 1.48 1.26 1.61 1.40 1.38 1.49 1.34 1.43 1.37 1.41 1.42 1.38 1.25 1.56 1.57 0.17
Notes: Subsample of …rms with more than 10 employees and growth rates under 300% due to sensitivity of standard deviations to extreme values. 1 Share of temporary workers
Table 4: Volatility of employment and materials across industries. Labour market conditions vary across industries. The last three columns of Table (4) report how the share of temporary workers in total employment varies across industries and how the volatility of hours and materials compare to the volatility of revenues. The last row present the cross industry correlations between the last three columns. The share of temporary work is larger in industries with more volatile employment while the correlation with the volatility of materials is much smaller. The two correlations are consistent with temporary employment being used to circumvent rigid labour legislation and allow an easier adjustment of employment to shocks. For overtime, besides being used infrequently and in small amounts, we did not …nd any evidence of being a relevant margin of adjustment. In principle because it is easier and cheaper to hire temporary workers instead of using overtime which is typically very costly and highly regulated.
21
3.2.2
E¤ects on prices and output
The second set of implications is about the correlation of demand and supply shocks with prices and output. Demand shocks are predicted to be positively correlated with both prices and quantities while supply shocks are predicted to exhibit a positive correlation with quantities and negative with prices. Overall, we con…rm that output is positively correlated with the shocks to demand and TFP, while prices have a negative correlation with shocks to TFP and either positive or insigni…cant correlation with shocks to demand (Table 5).4 Output is more responsive to both demand and supply shock when compared to prices. This is consistent with models of sticky prices (see also Santos, Costa and Brito 2016). The results are internally consistent with the model. In particular, the ratio of the output elasticity of TFP to the price elasticity of TFP is a proxy for the elasticity of demand. The ratio of the same elasticities with respect to demand shocks is a proxy for the elasticity of supply. Overall, all demand elasticities have the correct sign. We …nd either a positive or insigni…cant elasticity of supply across industries. Positive elasticity is consistent with either short run increasing marginal costs or capacity constraints. 3.2.3
E¤ects on inputs
The …nal set of implications compares the predicted correlations of the shocks to demand and supply with the input choices. As explained in Section 2.5, the shock to productivity is expected to be correlated with both inputs, while the shock to demand should only be correlated with materials. Table (6) reports the results from estimating a log-linear approximations to the optimal input choices: d ln pM ^ it + "m it Mit = m0 + m1 ^ it + m2 ln Kit + m3 ! it
ln pLit Lit = l0 + l1 ^dit + l2 ln Kit + l3 ! ^ it + "lit The estimated shock to demand is insigni…cantly correlated with employment, except for Food, Textiles, Timber, Paper, Non-Metal Minerals and Electricals. In these sectors it is 4
A zero correlation of prices and the shocks to demand, implies a ‡at inverse supply schedule (e.g. constant marginal costs).
22
Dep. Var.: Indep. Var.: Industry Meat Food Drinks Text. Foot. Timber Paper Publish. Chemic. Plastics Non-metal Minerals Basic Metal Fabricat. Metal Equip. O¢ c./data Precision Electric. Vehicles Other Transp. Furniture Misc.
ln(Output) Demand TFP shock 0.284*** 1.409** 0.263*** 0.555** 0.114* 0.290 0.0981*** 0.373** 0.262*** 0.448** 0.327*** 0.785*** 0.0746 0.0427 0.227*** 0.817*** 0.663*** 0.845*** 0.348*** 0.732** 0.282*** 0.495*** 0.623*** 0.421 0.341*** 0.336*** 0.455*** 0.452** 0.537*** 1.014* 0.353*** 0.538** 0.610*** 0.692** 0.683*** 0.734** 0.130 0.309* 0.325*** 0.515
ln(Prices) Demand TFP shock 0.0126 -0.204*** -0.00314 -0.136*** 0.0253* -0.0967** -0.00281 -0.0518*** -0.0141 -0.0339* 0.00701 -0.0260 0.0912* -0.217*** 0.0214* -0.0825 0.0583*** -0.117** 0.0257 -0.0392 0.0193* -0.107** 0.140*** -0.170*** 0.0276*** -0.0838*** 0.0365*** -0.0313** -0.00956 -0.108 0.0121 -0.0869*** 0.0183 -0.0401** 0.0453*** -0.0392* -0.0179 -0.0702*** 0.00336 -0.0680*
Notes: Results for linear regression of the log of de‡ated revenues and the log of the price index on the growth of TFP and on the estimated news to demand. S.e.’s clustered at the …rm level
Table 5: Correlation of estimated demand and TFP with prices and quantities, by industry. statistically signi…cant but exhibits a negative correlation, which could signal misspeci…cation. The shock is signi…cantly correlated with materials across all sectors. As for TFP, it is positively and signi…cantly correlated with both employment and materials across most sectors. Table (A.2) in the Appendix repeats the same exercise while using the self-reported changes to demand instead of the estimated measure. Results are similar if we use the estimated demand shock or the self-reported demand measure. Overall, for the majority of sectors the results match the predictions and we do not have evidence to reject correct speci…cation. The di¤erence in magnitudes is also striking. TFP exhibits a large correlation with both inputs. This compares with the small correlation of the demand shock with employment, but not with materials.
3.3 3.3.1
Robustness Demand estimates
We can repeat the previous exercise replacing the estimated shocks to demand obtained via the input ratio with the estimated shocks to demand obtained from directly estimating the demand function. This is possible because our data contains information on …rm level
23
24
1667
0.823*** 0.738*** 0.438 3.975***
-0.153*** 0.719*** -0.130 3.728***
Non-metal Minerals
607
0.911*** 0.639*** 0.810 6.528**
-0.119 0.617*** 1.076*** 5.221***
Meat
739
0.918*** 0.757*** 0.162 3.632***
-0.040 0.710*** -0.430* 3.674***
Basic Metal
2237
0.901*** 0.889*** 0.455 2.110***
-0.113** 0.703*** 0.325* 3.903***
Food
2234
0.887*** 0.710*** 0.159 4.020***
-0.115 0.617*** 0.166 4.959***
Fabricat. Metal
434
0.785*** 0.588*** 1.826*** 6.818***
-0.005 0.811*** 0.646* 1.901
Drinks
1836
0.862*** 0.810*** -0.129 3.416***
-0.081 0.658*** 0.376* 5.183***
Equip.
2398
0.950*** 0.667*** 1.671*** 4.906***
-0.102** 0.570*** 0.449*** 5.938***
Text.
0.765*** 0.672*** -0.186 5.355**
0.901*** 0.665*** 0.613 5.361 369
bill) -0.125** 0.565*** 0.748*** 6.623***
Electric.
602
0.565*** 0.606*** 1.465*** 5.965***
1669
0.868*** 0.640*** 1.044** 6.298***
ln(materials)
O¢ c./data Precision ln(wage 0.011 0.567*** 0.660** 6.785***
724
bill) -0.291*** 0.492*** 0.927*** 6.378***
Timber
ln(materials)
ln(wage -0.164 0.427*** 0.527** 7.707***
Foot.
1.045*** 0.767*** -0.179 4.293*** 514
1101
0.050 0.696*** 0.318 4.498***
Other Transp.
1226
1.024*** 0.610*** 1.010*** 5.818***
-0.051 0.527*** 1.046*** 6.820***
Publish.
0.731*** 0.848*** 0.965 2.398*
-0.023 0.724*** 0.448* 3.875***
Vehicles
697
0.815*** 0.768*** 0.233 3.813***
-0.237** 0.693*** -0.127 4.008***
Paper
Table 6: Input demand equations (with estimated demand shocks).
Notes: OLS estimates of the wage bill (top panel) and materials (bottom panel) on the estimated demand shock, TFP and capital stock, by industry. Year dummies included. S.e.’s clustered at the …rm level. All variables in logs. * p<0.05; ** p<0.01; *** p<0.001
N
Demand shock ln(Capital) TFP Cons.
Dep. Var:
Dep. Var: Demand shock ln(Capital) TFP Cons.
Industry
N
Demand shock ln(Capital) TFP Cons.
Dep. Var:
Dep. Var: Demand shock ln(Capital) TFP Cons.
Industry
1148
0.899*** 0.436*** 1.674*** 8.279***
-0.104 0.424*** 1.365*** 7.894***
Furniture
1683
1.132*** 0.723*** 0.300 4.980***
0.094 0.701*** 0.260 4.348***
Chemic.
558
1.025*** 0.539*** 1.576*** 6.948***
-0.107 0.601*** 0.458 5.412***
Misc.
1244
0.906*** 0.760*** 0.764* 3.964***
0.018 0.664*** 0.487*** 4.668***
Plastics
price indices. First we need to de…ne the demand model. We follow the speci…cation of Foster et al. (2016). Let demand be of the CES type as in equation (5) with an AR1 unobserved demand component ("dit = ("di;t 1 )g1 ln (qit ) =
ln(pit ) +
a
ln(agei;t ) +
z
d it ).
(12)
ln(Zi;t )
+g1 [ln (qi;t 1 ) + ln (pi;t 1 )
a
ln(agei;t 1 )
z
ln(Zi;t 1 )] + ln
d it
where Zit is the demand stock. The stock is the amount of cumulative (past) sales, P Zit = ts=0 Yis (1 )t s where is the rate at which consumers are lost. I.e., for each
sale, next period only (1
) sales will occur. To construct the demand stock, we need
a measure of the initial demand stock for the …rst year we observe the …rm in sample, because we do not observe …rms from birth. If we assume revenues grow at an expected rate of , the revenue of a company with age a is Yia = Yi0 a and the demand stock is P P a s . In the …rst year the company enters the )a s = Yia as=0 1 Zia = as=0 Yis (1
sample the demand stock at that period is a function of the revenues in that year multiplied a 1 a 1 (1 ) by a term that depends on the age of the …rm and growth rate, Zia = Yia 1 . 1 (1 ) We use a growth rate of = 1:063 since 6.3% is the average growth rate of sales for companies in our sample over the …rst 10 years of life (age). For simplicity we set
= 0.
One problem with estimating equation (12) is price (pit ) endogeneity due to the correlation with the shock to demand (
d it ).
We use a supply side instrument, TFP. Equation (12) is
then estimated with a GMM estimator where the set of instruments is current and lagged TFP, age and capital stock and lagged output, prices and demand stock. We now analyze the results, starting with the estimates for the dynamic demand model, followed by a comparison of the residuals obtained with this method with the residuals obtained from the input ratio and conclude with an evaluation of the e¤ect of the demand shock on inputs identical to the one conducted in Section 3.2.3. Estimated elasticities in Table (A.3) in the Appendix range from -0.4 for O¢ ce/data/Precision to -14.6 for Furniture. Elasticities are in absolute value much smaller (closer to zero) than if we ignore the demand stock component (i.e., setting
z
= 0). This is because in the
dynamic case, due to the persistent demand component, sales respond less to variations
25
d in price. The shocks to demand are obtained as the residual (ln\ it ) from equation (12).
In an exercise similar to the comparison with the self-reported demand shock in Table (2), we compare the shock obtained via demand function with the shock obtained via input ratio (full set of results reported in Table (A.4) in the Appendix). As reported, the shocks obtained via demand function are positively correlated with the shock obtained via input ratio. They are also both positively correlated with the self-reported demand shock, which signals the three measures have strong collinearity. This is con…rmed with a principal component analysis of the three shocks, which reveals that 1 factor can explain 45% of the total variation. Finally, we can test the same set of predictions that was conducted in Table (6): shocks to productivity are expected to have an e¤ect on both inputs while shocks to demand should only be correlated with materials. The di¤erence is the demand shock estimated via the input ratio is now replaced with the measure obtained as the residual from estimating the demand equation. The results in Table (A.5) again show an overall positive correlation of materials with TFP and shocks to demand and a positive correlation of labour with TFP but not with the shocks to demand. Overall, the evidence is consistent with Assumptions (A.1) and (A.2) and the model’s predictions. It is reassuring to verify that materials do respond to unexpected demand shocks while employment remains irresponsive. 3.3.2
Temporary employment
Dep. Var.: ln(1+LT =LP ) ln(1+LT =LP ) (hours) Shock to demand via demand function Self-reported Constant Observations Firms
(i) Whole sample
(ii) LT > 0
0.0453*
0.0548*
(iii) Whole sample
(iv) LT > 0
(v) Whole sample Materials to wage bill 0.0398 0.926**
0.0197
0.612***
0.619***
0.124*** 0.250*** 0.452***
30,034 4,079
23,267 3,675
24,734 3,460
(vi) LT > 0 (hours)
(vii) Whole sample
(viii) LT > 0 (hours)
1.369***
0.522
1.091*** 0.106*** 0.197*** 0.466*** 8,401 1,477
0.108*** 0.238*** 0.489***
0.542***
0.615***
0.126*** 0.250*** 0.454***
19,138 3,091
28,805 4,078
10,239 1,773
24,665 3,455
Notes: Results for linear regression with …xed e¤ects and year dummies. S.e.’s clustered at the …rm level. *** p<0.001, ** p<0.01, * p<0.05
Table 7: Regression estimates for the input ratio on adjustable employment and shocks to demand.
26
Our …nal robustness exercise consists of evaluating how temporary labour responds to demand shocks. Equations (6) and (7) obtained from Section (2.3.2) can be estimated using two measures of ‡exible labour: the fraction of overtime hours to total hours and the fraction of temporary employees to total employees.5 Table (7) reports the results for equation (6) and shows that the temporary employment is positively correlated with the input ratio. This is true for the fraction of temporary employees in column (i) or the fraction of overtime hours in column (v). When the adjustment cost constraint binds, temporary employment is set to zero. Columns (ii) and (vi) show that restricting to the sample with positive adjustment delivers a stronger correlation, as expected. In the remaining columns we introduce the measures of shocks to demand (self -reported and estimated as the residual of the demand function). We conclude that the correlation between the input ratio and temporary employment is weakened and for the case of temporary employees, it becomes insigni…cant. This means that the ratio of temporary to permanent employment is "transmitting" the demand shock to the input ratio. We further investigate this "transmission" by estimating equation (7) for how the news about demand correlate with the ratio of temporary employment to permanent employment in Table (8). Both measures of temporary employment are positively correlated with the demand shock. Conditioning on the other shocks to demand (recovered from the demand function or self-reported), the shock to demand obtained from the input ratio loses signi…cance. Overall, the evidence is consistent with the optimal equations for temporary employment, (6) and (7).
4
Montecarlo simulations
In the last section we evaluate model performance, by generating simulated data from several parametric models, implement our method to recover the demand shocks, and compare with the true demand shocks. As baseline model we use the Cobb Douglas 5
Temporary work is more frequently observed in the data than overtime (77% of the observations report temporary workers and 38% report overtime hours). This is due to the labor legislation that created a well known dual labour market in Spain. Temporary work is used as a way to avoid entering long term contracts that have relatively large …ring costs.
27
(i)
(ii)
(iii) (iv) ln(LT =LP )
Dep. Var.:
(v)
workers Shock to demand input ratio via demand function Self-reported Constant
(vi)
hours
0.0812**
0.0342 0.165***
0.0101 0.139*** 0.229***
0.171***
0.150** 0.0609
0.123* 0.0405 0.182***
-1.556***
-1.534***
-1.658***
-4.391***
-4.380***
-4.482***
8,391 1,474
8,385 1,474
Observations 19,992 19,105 19,092 8,952 Firms 3,192 3,082 3,082 1,542 Notes: Results for linear regression with …xed e¤ects. S.e.’s clustered at the …rm level. *** p<0.001, ** p<0.01, * p<0.05
Table 8: Regression estimates for adjustable employment on demand shocks. Technology and CES demand functions as speci…ed above qit = ea0 ! it Mit Lit Kit p(qit ; Zit ; "dit ) = qit Zi;tz 1 "dit where the demand stock is Zit = (1
) Zi;t
1
+ Yit and Yit = pit qit . Both demand and
supply shocks follow an AR1 (in logs). ln "dit = g" ln "dit ln ! it = g! ln ! it
1
+
1
+
d it it
For computational simplicity, we abstract from dynamic decisions and solve the static pricing problem. In particular, we ignore the investment problem and let capital evolve as ln Kit = ln Ki;t
1
+ iit where i s N (0; 0:15) allowing for a 15% standard deviation to
investment. We also ignore the pricing e¤ect on future Zit . Demand and supply shocks follow a normal distribution
d it ;
it
s N (0; 0:35) and the parameters are set to g"
z
g!
pl pm a0
2 0:25 0:33 0:6 0:3 0:1 0:95 0:95 1
1
10
The model is simulated for a panel of 500 …rms over 20 periods (the …rst …ve periods are dropped). To calculate the expectation we use 1,000 draws. The initial period values for the persistent variables are drawn ln Ki0 ; ln ! i0 ; ln "di0 s N (x; 1) where the mean x is [10; 2; 2], respectively. Initial demand stock is nonstochastic, set to ln Zi0 = 0. 28
In Figure (2) we report the …t (scatter plot and correlation) between the true simulated d it )
demand shock (
and the estimate. We present 4 di¤erent cases. Three with N =
100; 500; 1; 000 and T = 15. And one with N = 100 and T = 5. We can see that even with a small sample of 100 companies over 5 years, the correlation between the two shocks is over 99.8%. The …t increases with sample size (both N and T ). Figure (2)reports how the method performs to departures from the baseline case. We use N = 500 and T = 15. Case (i) is the baseline. Again, model …t is very large. Case (ii) uses a translog model qit = ea0 ! it Mit
+
l
ln(Lit )+
The translog parameters ( l ;
m
m
ln(Mit )
and
+
Lit
l)
l
ln(Lit )
Kit
+
k
ln(Kit )+
l
ln(Lit )+
l
ln(Mit )
make the ETS non-constant which adds "noise"
to the input ratio. The ETS (and consequently the input ratio) is now a function of the inputs, L and M . The parameters are set to m
=
0:025,
l
=
0:025,
k
=
l
=
m
= 0:3,
= 0:6,
= 0:1,
l
= 0:025,
= 0. The parameters which interact with
capital are set to zero since they have no meaningful e¤ect on the marginal returns to M or L, or to the ETS. Figure (2) shows that the correlation between the true and estimated demand shock is reduced while still remaining large at 93%. Case (iii) introduces quadratic ( Lit )2 . Li;t 1
adjustment costs to labour of the form
Every time the company wants to adjust
its labour up or down, it has to pay this extra cost. We set
= 20. Quadratic adjustment
costs have a very marginal e¤ect on the correlation between the two shocks which remain above 99.4%. Finally, case (iv) considers cubic demand ln (pit ) =
1
ln(qit ) +
2
ln(qit )2 +
3
ln(qit )3
1
z
ln("dit )
ln(Zi;t )
Cubic demand generates non-constant markups. This creates a further non-constant wedge between marginal returns and marginal cost. Taking the (input) ratio should cancel the e¤ect in the denominator and the numerator but only in the case where both inputs are fully ‡exible. Otherwise, the input ratio becomes a function of all the inputs, demand components, the shock and the productivity term. We set and
3
1
=
20,
2
=
0:05
= 0:005 to generate sensible elasticities over the range of prices/quantities. For
example, demand elasticity (
1
(1 + 2
2
ln(qit ) + 3
29
1
3
ln(qit )2 ) ) is -13 at ln q = 10; -4 at
0 −1 −2
−1
0
1
2
True demand shock
N=1,000, T=15
1
Corr = 99.95
0 −1 −2 −2
−1
0
1
True demand shock
2
N=100, T=15 1
Corr = 99.87
0 −1 −2
−1
0
1
2
True demand shock
N=100, T=5
1 Corr = 99.81 0 −1 −2
−1
0
1
True demand shock
(i) Baseline case 1
(ii) Translog production function
Corr = 99.94
0
−1 −2
−1
0
1
2
True demand shock
(iii) Labour quadratic adj. costs 1
Corr = 99.45
0
−1 −2
−1
0
1
True demand shock
2
Estimated demand shock Estimated demand shock (via input ratio) (via input ratio)
Corr = 99.94
Estimated demand shock Estimated demand shock (via input ratio) (via input ratio)
N=500, T=15 1
Estimated demand shockEstimated demand shock (via input ratio) (via input ratio)
Estimated demand shockEstimated demand shock (via input ratio) (via input ratio)
ln q = 20 and -1.7 at ln q = 30. 1 Corr = 93.01 0
−1 −2
−1
0
1
2
True demand shock
0.1
(iv) Cubic demand Corr = 99.04
0
−0.1 −2
−1
0
1
2
True demand shock
Figure 2: Scatterplot and correlation between simulated and estimated demand shocks using (a) four di¤erent sample sizes (N and T) (b) four di¤erent parametrizations. Two important parameters are related with the demand stock component ( z , ). In the Appendix we evaluate the sensitivity of the results to varying these parameters. In particular, in Figure (A.2)
z
is increased from 0:25 to 0:75, in Figure (A.3)
is increased
from 0:33 to 0:66 and …nally in Figure (A.4) the production function parameters are set to
= 0:35 and
= 0:7 allowing for increasing returns in the two inputs. Overall the
results remain unchanged.6
5
Final Remarks
Understanding the source of economic shocks is important to guide economic policy. Unfortunately, it is very di¢ cult to disentangle demand from supply using standard …rm level microeconomic datasets. In this paper I have proposed a simple and intuitive way to do so by using the input ratio. Shocks to demand create a wedge between the input ratio and the elasticity of technical substitution, allowing us to infer the demand shock from its variation. This result is robust to dynamic demand or temporary employment. I have presented evidence of how the input ratio compare with other measures of demand shocks, including self reported and estimated residual from demand functions. I have also provided a series of testable implications. Overall, the evidence supports the use of the input ratio to recover shocks to demand. 6
We have also evaluated (not reported) the sensitivity to the AR1 components, namely setting g" = g! = 0:5 and g" = g! = 0:99. Results remain virtually unchanged.
30
Acknowledgments Financial support by FCT (Fundação para a Ciência e a Tecnologia), Portugal, is gratefully acknowledged. This article is part of the Strategic Project UID/ECO/00124/2013, and it is also …nanced by POR Lisboa under the project LISBOA-01-0145-FEDER-007722. I would like to thank Steve Bond, Yossi Spiegel and seminar participants in Alicante, Tilburg, ESEM2013 ad EARIE2013 for useful comments and suggestions.
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[15] Gale,D. and H. Nikaido (1965) "The Jacobian Matrix and Global Univalence of Mappings", Mathematische Annalen, 159, 81-93. [16] Gandhi, A., Navarro, S. and Rivers, D. (2016) "On the Identi…cation of Production Functions: How Heterogeneous is Productivity", manuscript [17] Gilchrist, S., Schoenle, R., Sim, J. and Zakrajsek, (forthcoming) "In‡ation Dynamics During the Financial Crisis", American Economic Review [18] Griliches, Z. and Mairesse, J. (1998) "Production Functions: The Search for Identi…cation", in Econometrics and Economic Theory in the Twentieth Century: The Ragnar Frisch Centennial Simposium, 169-203 [19] Gourio, F. and Rudanko, L. (2014) "Customer Capital", The Review of Economic Studies, 81(3), pp.1102-1136. [20] Hall, R. (1988) "The Relation between Price and Marginal Cost in U.S. Industry", Journal of Political Economy, 96(5), 921-947 [21] Klette, T. (1999) "Market Power, Scale Economies and Productivity: Estimates from a Panel of Establishment Data", The Journal of Industrial Economics, 47(4), 451-476 [22] Klette, T. and Griliches, Z. (1996) "The Inconsistency of Common Scale Estimators When Output Prices are Unobserved and Endogenous", Journal of Applied Econometrics, 11(4), 343-361 [23] Levinshon, J. and Petrin, A. (2003) "Estimating Production Functions Using Inputs to Control for Unobservables", The Review of Economic Studies, 70(2), pp.317-342. [24] Marschak, J. and W.H. Andrews (1944), "Random Simultaneous Equations and the Theory of Production", Econometrica, 12 (3 and 4), 143-205 [25] Matzkin, R. (2007) "Nonparametric Identi…cation" in Handbook of Econometrics, Vol. 6b, edited by J.J. Heckman and E.E. Leamer [26] Melitz, M. (2003) "The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity", Econometrica, 71, 1695-1725 [27] Newey, W. and Powell, J. (2003) “Instrumental Variables Estimation of Nonparametric Models,”Econometrica [28] Olley, and Pakes A. (1996) "The Dynamics of Productivity in the Telecommunications Equipment Industry", Econometrica, 64(6), 1263-1297 [29] Pozzi, A. and Schivardi, F. (2016) "Demand or productivity: What determines …rm growth?", Rand Journal of Economics, 47(3), 608–630 [30] Santos, C., Costa, L. and Brito, P. (2016) "Markup and productivity estimation for single-product …rms: The Portuguese case 2004-2010", INOVA WP 609 [31] Sutton, J. (1991): Sunk Costs and Market Structure, MIT Press. [32] Sutton, J. (1998): Technology and Market Structure, MIT Press.
32
A
Appendix [for online publication]
A.1
Dynamic Demand (extension)
A.1.1
Foster et al (2016)
Let Zit be the demand "stock" of the …rm which depends on the previous …rm sales. The stock evolves as Zit = (1
) Zi;t
1
+ pit qit . We consider the non-parametric case here
where pit = p(qit ; Zi;t 1 ; "dit ) with Zit as speci…ed above. Foster et al. (2016) also consider age as a state variable. For convenience let’s abstract from age as this adds nothing substantial to the analysis and we would need to keep track of one extra state variable. By the end of the period the …rm chooses the level of materials which maximizes the following objective function V ("dit ; ! it ; Zi;t 1 ) = max p(qit ; Zi;t 1 ; "dit )qit Mit
pM it Mit
+ E"di;t+1 ;!i;t+1 V ("di;t+1 ; ! i;t+1 ; Zit j"dit ; ! it ; Zi;t 1 ) s.t. qit = q(Kit ; Li;t ; Mit ; ! i;t ) Zit = (1 where
) Zi;t
1
+ pit qit
< 1 is the discount factor and we now have an extra term which includes
the continuation value. Notice that if we let yit = p(qit ; Zi;t 1 ; "dit )qit , and since Zit = (1
) Zi;t
1
+ yit , the sum of static returns and continuation value can be written as
yit + E"di;t+1 ;!i;t+1 V (:::j"dit ; ! it ; Zi;t 1 ; yit ). This is a fundamental property and greatly simpli…es the analysis since we can use the chain rule to …rst di¤erentiate with respect to yit (by construction the derivative of Zit with respect to yit is 1) and then we di¤erentiate with respect to M . The …rst order condition is [
p;q
+ 1] p(qit ; Zit ; "dit )qm (1 + EVZ0 ) = pM it
(A.1)
where EVZ0 = E"di;t+1 ;!i;t+1 VZ0 it ("di;t+1 ; ! i;t+1 ; Zit j"dit ; ! it ; Zi;t 1 ) is the e¤ect sales today on the demand stock tomorrow. The main di¤erence from before is the continuation term EVZ0 . So, the marginal returns of increasing Mit (LHS) consists of the increase in short-term 1
revenues and future revenues due to the increase in the demand stock Z. Put in another way, …rms are willing to produce more and receive a lower price because every unit they sell also increases future sales. We can rewrite the reduced form optimal solution as Mit = m(! it ; Lit ; Kit ; Zi;t 1 ; "dit ) In the beginning of the period the …rm chooses the level of employment that maximizes the expected returns 2 6 max E"dit 4 Li;t
E"di;t+1 ;!i;t+1 V
("di;t+1 ; ! i;t+1 ; Zit j"dit ; ! it ; Zi;t 1 )j
s.t. qit = q(Kit ; Lit ; Mit ; ! it ) Zit = (1
) Zi;t
1
3
pM it Mit +
p(qit ; Zit ; "dit )qit
it
7 5
pLit Lit
+ pit qit
which gives7 E"dit
@p(qit ; Zit ; "dit )qit @qit j @qit @Lit
it
= E"dit [
p;q
+ 1] p(qit ; Zit ; "dit ) (1 + EVZ0 ) ql j
it
= pLit (A.2)
Again the same dynamic element is at place. Increasing revenues also brings in future additional sales. Using the previous two we obtain the input ratio pM it Mit = pLi;t Li;t
m:l |{z}
Supply (ETS)
and again
d p;q (qit ; Zi;t ; "it ) + 1 d p;q (qit ; Zi;t ; "it ) + 1
1 E"d | it
p(qit ; Zi;t ; "dit ) (1 + EVZ0 ) ql p(qit ; Zi;t ; "dit ) (1 + EVZ0 ) ql j {z
Demand
pM Mit it Mit = L E"d [ m:l Mit j pit Lit A.1.2
it
}
it ]
Gourio and Rudanko, 2014
We now consider the extension to the dynamic demand model formulated by Gourio and Rudanko (2014) and show how the analysis is similar in a model with endogenous dynamic 7
From
The
equation total
h
derivative
(A.1), is
@p(qit ;Zit ;"d it )qit @qit @qit @Mit
E"d
@p(qit ;Zit ;"d it )qit
(1 + EVZ0 ) (1+ EVZ0 ) pM it Mit
@Lit
@p(:)qit @qit it @qit 0 0 E"d @p(:)q @qit @Lit (1 + EVZ ) + @qit @Mit (1 + EVZ ) rem). Replacing the above we obtain equation (A.2).
2
pM it
@Mit @Lit
i
pM it .
= pL it
=
pL it (by the envelope theo-
demand considerations. Contrary to Foster et al (2016) this model avoids dynamic pricing decisions. Instead, dynamic demand considerations are addressed with sales force e¤ort decisions. Since sales force is an expense and does not build up this allows us to split the dynamic considerations about future demand (stock building) from the optimal input allocations (labour and materials). Consider the previous model but let’s assume prices are given, as in Gourio and Rudanko (2014). Let’s also ignore capital decisions to simplify the analysis, as everything holds if we add the extra state variable, capital. In this case the production function is
qit = q0 Lit Mit ! it where comparing to the previous model, q0 = Kit so we are assuming capital is …xed. We adapt the speci…cation in Gourio and Rudanko (2014) so that the number of units sold is given by
qit = nit + sit (
bit ) sit
where nit is the number of consumers carried from the last period, bit is the number of buyers in the market (given) and s is the number of sellers (sales force). The function b s
=
b s
is the (Mortensen-Pissarides) matching function with
2 (0; 1). In this
sense, s (:) is the number of new customers attracted when there are b buyers and s sellers. Also, assume that employing these sellers generates a cost which is increasing in the number of sellers. In particular let the cost be quadratic (s) = s2 =2. Notice that given this structure, the number of units sold is determined by the number of customers (customer base) the number of buyers and the number of sellers. In this way, the …rms’ input decision is how to best choose the intermediate inputs to minimize the cost (dual). We can show this by maximizing the …rms’ pro…t (value) function. We …rst need to introduce two more elements …rst the customer base n decays at rate that every period a fraction
n
n.
This means
of the customers leave the company. That means that
the customer base available for next period is ni;t+1 = (1
n ) qit .
On the other hand,
let the number of potential buyers in the market follow a …rst order Markov process, 3
bit = h(bit 1 )e"it for some smooth function h (:). This introduces a shock to demand "it which is absent in the original model of Gourio and Rudanko (2014). Again, this shock is observed after Lit has been determined. In the last period after the demand shock ("it ) is revealed, the company wants to maximize its long term value (note that choosing sit is equivalent to choosing Mit since Mit =
qit e!it Lit
1=
nit +sit (bit =sit ) e!it Lit
=
1=
and we have to account for the fact that increasing
sit will make the company sell more and also incur larger costs as it needs to purchase more intermediate inputs, Mit )
pM it Mit
max pqit sit
(sit ) + E
i;t+1 ;"i;t+1
[V (ni;t+1 ; ! i;t+1 ; bi;t+1 )jbit ; nit ; ! it ]
s.t. qit = Lit Mit e!it qit = nit + sit (bit =sit ) bit = h(bit 1 )e"it ni;t+1 = (1
n ) qit
! i;t+1 = g(! it )e
it+1
where E [V (ni;t+1 ; ! i;t+1 ; bi;t+1 )jbit ; nit ; ! it ] is the expected continuation value from a company with customer base nit , productivity ! it and a stock of potential buyers of bit . Or equivalently
max sit
8 > < p nit + s1 b it it > : + E
s.t. bit = h(bit 1 )e"it ni;t+1 = (1
i;t+1 ;"i;t+1
nit + s1it e!it Lit
bit
1=
s2it =2
9 > =
> [V (ni;t+1 ; ! i;t+1 ; bi;t+1 )jbit ; nit ; ! it ] ;
n ) qit
! i;t+1 = g(! it )e
pM it
it+1
The …rst order condition is
4
p (1 = pM it
)
1
bit sit
+ (1 1=
e!it Lit
where E [Vn (:)] = E
n ) (1
nit + s1it bit h
@V (:) @n0
i
bit sit
) (1
E [Vn (ni;t+1 ; ! i;t+1 ; bi;t+1 )jbit ; nit ; ! it ] bit sit
)=
(1
)
. The …rst term, p (1
(A.3)
+ sit bit sit
)
, is the marginal short
term bene…t from increasing the sales force while the second two terms are the marginal cost of doing so. First the cost in terms of increasing intermediate input pur1=
1 e!it Lit chases, pM it
nit + s1it bit
(1
)=
(1
bit sit
)
and the second is simply
the marginal cost of more sellers, sit . The …nal component is the marginal net bene…t for the future increase in the customer base,
(1
n ) (1
)
bit sit
E [Vn (ni;t+1 ; ! i;t+1 ; "i;t+1 )].
In the beginning of the period the …rms sets its labour before knowing the shock to demand,"it and maximizes the expected future value 0
B max E"it @ Lit
pM it Mit
pqit + E
i;t+1 ;"i;t+1
(sit ))
pLit Lit
[V (ni;t+1 ; ! i;t+1 ; bi;t+1 )jbit ; nit ; ! it ]
s.t. qit = Lit Mit e!it
1 C A
qit = nit + sit (bit =sit ) bit = bit 1 e"it ni;t+1 = (1
n ) qit
! i;t+1 = g(! it )e
it+1
Or equivalently 0
B max E"it @ Lit
p nit + s2it =2
sit1
n ) qit
! i;t+1 = g(! it )e
nit + s1it
bit
1=
e!it Lit
pLit Lit + E [V (ni;t+1 ; ! i;t+1 ; "i;t+1 )]
s.t. bit = h(bit 1 )e"it ni;t+1 = (1
bit
pM it
it+1
5
1 C A
By the envelope theorem, the …rst order condition becomes
pM it
E"it
@Mit @Lit
pLit
since for the Cobb Douglas production function pM it Mit = pLit L
=0 @M @L
M , L
=
the solution becomes
Mit E"it (Mit )
which is exactly the same solution we had before in Equation (4). The di¤erence is that now,
Mit E"it (Mit )
is a more complicated expression. In particular,
Mit = E"it (Mit )
nit + s1it
bit
1=
e!it Lit
E"it
nit + s1it
bit
e!it Lit
nit + s1it bit
= E"it
1=
1=
nit + s1it bit nit + s1it bit 1 e
= E"it
1=
"it
nit + s1it bit 1 e
1=
"it
1=
and we see that, as before, the input ratio is a function of the demand shock (and not a function of the productivity shock) but is now also a function of the demand state variables, namely the customer base and the number of potential buyers as well as the (endogenous) number of sellers. To address the endogeneity of sit (since sit is correlated with "it ), we can use Lit (or Kit in the model extended with capital) as a valid instrument as demonstrated in the …rst order condition in Equation (A.3). The previous equation can be estimated by non-linear or non-parametric IV depending on whether we can obtain a solution to E"it
nit + s1it bit 1 e
"it
1=
.
Call vne (bit ; nit ; ! it ) = E [Vn (ni;t+1 ; ! i;t+1 ; bi;t+1 )jbit ; nit ; ! it ]. Note that the …rst order condition for sit is
6
0
B sit = @(1
2
6 ) bit 4p + (1
e n ) vn (bit ; nit ; ! it )
1
pM it e!it Lit
nit +
1=
bit sit
! (1
)
31 1+1 7C 5A
which illustrates both the endogeneity of sit with respect to "it (via bit ) and the validity
of Lit as an instrument.
A.2
Non-Fixed Labor/temporary work (extension)
In this section we extend the previous results to the case with two types of employment: permanent and temporary. The cost of hiring an extra temporary worker equals the cost of hiring a regular worker with a "…scal" advantage ((1
P
) pLit ) plus an extra cost P
which increases with the number of temporary workers and their regular cost, pLit
LT it . LP it
The cost of a temporary worker is smaller than the cost of a regular worker for small amounts but increases with the amount of temporary workers. The total cost is thus 2 T P P (LTit ) . The "…scal" advantage is related with the pLit LTit = (1 ) pitL LTit + pitL P L it
money a company can save when it hires a temporary workers on social security, taxes and other expenses (e.g. medical insurance, etc). In our case, this rationalizes the use of temporary workers in small amounts. Companies …rst maximize for permanent employment and later optimize for temporary employment and materials. As such By the end of the period the …rm chooses the level of materials which maximizes the following objective function
max p(qit ; Zit ; "dit )qit
LT it ;Mit
s.t. qit = q(Kit ; Li;t ; Mit ; ! i;t ) Lit = LTit + LPit T
pLit
P
= pLit
1
+
7
LTit LPit
pM it Mit
T
pLit LTit
where pM it is the price of materials. Assume an interior solution for temporary employment, LTit > 0. When this constraint binds we are in the original solution where temporary employment is not possible. From the …rst order conditions
[
[ where
p;q
p;q
+ 1] p(qit ; Zit ; "dit )qm = pM it
P
p;q
+ 1] p(qit ; Zit ; "dit )ql = pLit
1
< 0 is the elasticity of demand, qm =
(A.4)
+2
@q(:) @M
LTit LPit
and ql =
(A.5) @q(:) . @L
We can rewrite
the reduced form optimal solutions as
Mit = m(! it ; LPit ; Kit ; Zit ; "dit )
LTit
= lT (! it ; LPit ; Kit ; Zit ; "dit )
Putting the two equations together T
pM 1 pLit 1 it Mit = = P P L pitL Lit m:l pit m:l
1
+2
LTit LPit
(A.6)
The input ratio is now also a function of the wedge created by the labour market friction, together with the ETS. In the beginning of the period the …rm chooses the level of employment that maximizes the expected returns
h max E"d p(qit ; Zit ; "dit )qit LP i;t
s.t. qit = q(Kit ; Lit ; Mit ; ! it ) Lit = LTit + LPit which gives
8
pM it Mit
T
pLit LTit j
it
i
P
pLit LPit
E"d
@p(qit ; Zit ; "dit )qit @qit j @qit @LPit
= E"d [
it
p;q
+ 1] p(qit ; Zit ; "dit )ql j
P
it
= pLit
(A.7)
Combining the FOCs for LTit and LPit , we obtain the following restriction
P
E"d pitL
1
+2
LTit LPit
P
j
it
= pLit
This is because the bene…t the company derives from one unit of temporary labor is identical to the bene…t it derives from one unit of permanent labour, the only di¤erence is the cost which it wants to equate. Replacing the cost of temporary workers we obtain LTit j LPit
E"d or we can write this as E"d LTit j
it
=
2
it
=
2
LPit so the company would like to keep the ratio
of temporary to permanent work as a constant LTit LTit = 2 E"d [LTit j LPit
(A.8)
it ]
The two ratios of materials to labour and temporary employment to labour are a function of
LT it E"d [LT it j
it
] pM 1 it Mit = LP pit Lit m:l
1
+
LTit E"d [LTit j
it ]
P
pLit LTit LTit = P 2 E"d [LTit j pLit LPit
it ]
Given that Lit = LTit +LPit , even the parametric case does not deliver an elegant analytic solution. However, we can see that the case with temporary employment delivers similar results to the case without it. The materials to wages ratio is a function the the ETS and the temporary employment to the expected temporary employment. The ratio
LT it
] is a function of the demand shock. We can also use the ratio of temporary to permanent employment,
LT it . LP it
9
E"d [LT it j
it
A.3
Higher order process (extension)
The input ratio is pM it Mit = pLi;t Li;t
E
h
"dit "dit
1 ( +1)
1
1
1 ( +1)
j
it
i
The input ratio is a sole function of the ETS and the demand component. For simpli…cation, let the demand shock be a separable s order process, "dit = g( where
d i;t
d d i;t 1 ; ::: i;t s ;
d it )
= g(
d d d i;t 1 ; ::: i;t s ) it
(A.9)
= ("dit ; zit ) and z is a n-dimensional vector of information known to the …rm at
period t. Again, write ~dit =
d it
ln
1
1 ( +1)
and normalize E
d it
~dit = 1,
pM it Mit = ln ( = ) + ln ~dit pLi;t Li;t
As we can see from above, the restriction to a …rst order Markovian process is immaterial.
A.4
Nonparametric Identi…cation
In this section I discuss how the assumptions and observed data allow us to nonparametrically identify the distribution of demand shocks. This is instructive, to explain that identi…cation of demand shocks does not depend on the parametrization presented in Section 2.4. Instead, identi…cation is obtained from meaningful variation together with the economic restrictions. There is a total of four equations: the two …rst order conditions, the demand and production functions. This is the case when prices and quantities are observed. However, since prices (and quantities) are not observed we need to combine the production and demand functions into a revenue function. This delivers three equations 8
: 8
In this section, to reduce the number of variables in the notation, input prices (PitM ; PitL ) are included in Zit .
10
yit = p(qit ; Zit ; "dit )q(Kit ; Lit ; Mit ; ! it ) Mit = m(! it ; Lit ; Kit ; Zit ; "dit ) Lit = l(! it ; Kit ; Zit ;
it )
Unfortunately, this is not su¢ cient to identify the demand shock nor is this su¢ cient to identify productivity. The reason is that we have 2 unobservables (! it , "dit ) and the vector of information know to …rm i when it chooses labor,
it .
correlated with all the observables due to serial correlation (note
This is are potentially it
includes ! i;t 1 , "di;t
1
and the other state variables) and 3 equations.9 If we attempt to estimate any of the three equations individually, we would face endogeneity of potentially all the explanatory variables since (yit ; Mit ; Lit ; Kit ; Zit ) are correlated with (! it ; "dit and
it ).
Note that if we
allow both inputs to be fully ‡exible, the equation for labour (l(:)) becomes a function of "di;t instead of
it
and both components, ! it and "di;t , enter all three equations. This is
the reason why demand and supply shocks cannot be separately identi…ed. Furthermore, when prices and quantities are observed, the two equations allow us to separately identify the demand from the supply conditions p(qit ; Zit ; "dit ) and q(Kit ; Lit ; Mit ; ! it ) . This is the approached followed in Santos, Costa and Brito (2016). In order to make progress, let the demand shock be separable in "dit = g( where
d it
it )e
d it
= git e
d it ,
d it
is the ’news’to demand or the unexpected changes to demand conditions. One
simple case is when the demand follow a separable …rst order Markov process, "dit = d
g("di;t 1 )e it . The news term is, by assumption, serially uncorrelated. Out of all the information contained in the information set, g(
it )
is now a su¢ cient statistic for "dit .
This single index restriction is important because it links "dit with
it .
When the …rms
chooses Lit , it must forecast "dit . However, given the separability assumption on the demand shock, git = g(
it )
is now a su¢ cient statistic. In the general case, there is now
9
Notice that if we allow adjustment costs to labour, lagged employment would enter the input demand function for labour that would become, l(! it ; Kit ; Zit ; Li;t 1 ; it ).
11
a system with four equations
yit = y(Kit ; Lit ; Mit ; ! it ; Zit ; "dit ) Mit = m(! it ; Lit ; Kit ; Zit ; "dit ) Lit = l(! it ; Kit ; Zit ; git ) "dit = g(
it )e
d it
= git e
d it
where y(Kit ; Lit ; Mit ; ! it ; Zit ; "dit ) = p(q(Kit ; Lit ; Mit ; ! it ); Zit ; "dit )q(Kit ; Lit ; Mit ; ! it ). In Section 2.4 we used the ratio of the amount spent on the two inputs. This was because taking the ratio eliminates ! it and git from the ratio equation. The ratio is then a function of the state variables and the demand shock
d it .
The demand shock is independent from
all the state variables, which guarantees identi…cation of both the equation of interest and the shock distribution. In the general case, taking the ratio does not eliminate ! it and git . Instead, we can characterize the reduced form solutions obtained from the …rst order conditions. First we can eliminate "dit by replacing with g(
it )e
d it
which leaves three
equations d
yit = y(Kit ; Lit ; Mit ; ! it ; Zit ; git e it ) d
Mit = m(! it ; Lit ; Kit ; Zit ; git e it ) Lit = l(! it ; Kit ; Zit ; git ) From the last two equations we can solve for ! it and git as functions of the remaining variables (Mit ; Lit ; Kit ; Zit ). We study the technical conditions for invertibility in the next subsection. Plugging these solutions into the …rst equation
yit = y(Kit ; Lit ; Mit ; Zit ; ! it ; g(
it )e
d it
) d
= y(Kit ; Lit ; Mit ; Zit ; ! (Mit ; Lit ; Kit ; Zit ) ; g (Mit ; Lit ; Kit ; Zit ) e it ) From equation (A.10) the distribution of "news" to demand term,
d it
(A.10)
is identi…ed up
to standard normalization and location restrictions on the nonparametric functions (see 12
Matzkin, 2007) as long as all the variables (Mit ; Lit ; Kit ; Zit ) are independent from
d it .
That is coherent with the model for the state variables (Lit ; Kit ; Zit ). However, from the d it .
the model’s assumptions, we know that Mit is not independent from is serial correlation in ! it , lagged values of Mi;t
1
instruments simply by inverting the equation ! i;t
As long as there
and (Li;t 1 ; Ki;t 1 ; Zi;t 1 ) become valid 1
=m ~
1
(:). Equation (A.10) can be
estimated by nonparametric instrumental variables when the demand shock is separable (Newey and Powell, 2003). When the demand shock is nonseparable, we need to consider further restrictions to implement the nonparametric IV estimator (see for example, Chen et al., 2014 and Blundell and Matzkin, 2013). Note that in some cases, the equation for intermediate inputs can be written as Mit = m(L ~ it ; Kit ; Zit ;
d it ).
This is the parametric case presented in Section 2.4. In this case we
do not require invertibility conditions that guarantee that we can write ! it and g(
it )
as functions of the remaining variables (Mit ; Lit ; Kit ; Zit ). In such case, we can use the d it
(again up to
it )
and ! it as an
equation m ~ (:) directly to identify the distribution of the demand shocks standard normalization and location restrictions. A.4.1
Invertibility
We can express the previous problem of obtaining a solution for git = g(
invertibility problem. Inverting the labour demand function (git = l 1 (! it ; Kit ; Zit ; Lit ), see Levinshon and Petrin (2003) for invertibility conditions), and substituting it in the materials input function, we obtain
d
Mit = m(! it ; Lit ; Kit ; Zit ; l 1 (! it ; Kit ; Zit ; Lit )e it ) = m(! ~ it ; Lit ; Kit ; Zit ;
(A.11)
d it )
The next step is inverting the m ~ function with respect to ! it to obtain
! it = m ~ 1 (Mit ; Lit ; Kit ; Zit ;
d it )
Replacing ! it back in git = l 1 (! it ; Kit ; Zit ; Lit ) we obtain the expression for git . We 13
can thus solve for git and ! it as functions of (Mit ; Lit ; Kit ; Zit ;
d it )
and replace git and ! it
in the revenue Equation (A.10). We will discuss below the case where m ~ is not a function of !. In this situation we can directly obtain the distribution of
d it
by Mit = m(L ~ it ; Kit ; Zit ;
d it ).
This is the parametric
case presented in Section 2.4. A.4.2
Technical conditions for invertibility
Monotonicity of the …rst order conditions is not su¢ cient to obtain invertibility. Take the system formed by the two …rst order conditions in equations (1) and (2) where "dit = g(
it )e
d it
. If the determinants of all principal submatrices of the Jacobian 2 6 4
d d M p;q +1]p(qit ;Zit ;"it )qm pit @"it @git @"dit E"d [[ p;q +1]p(qit ;Zit ;"dit )ql ] pL it @git
@[
@[
d M p;q +1]p(qit ;Zit ;"it )qm pit @! it
E"d [[
d p;q +1]p(qit ;Zit ;"it )ql
]
pL it
@! it
3 7 5
are non vanishing, it follows by Theorem 7 in Gale and Nikaido (1965) that the system is invertible. We thus obtain a solution to this system in (git ; ! it ) as functions of the remaining variables Mit ; Kit ; Lit ; Zit ;
d it
. A necessary condition to obtain a solution
is,10 @[
@
R
[
p;q +1]p(qit ;Zit ;g( @"dit p;q +1]p(qit ;Zit ;g(
d it )e it )qm
it )e
d it )ql
@"dit @git df (
@[ d) it
6=
@git
p;q +1]p(qit ;Zit ;g(
d it )e it )qm
@! it @
R
[
p;q +1]p(qit ;Zit ;g(
d it )e it )ql
df (
d) it
@! i;t
This condition states that we require the di¤erential e¤ect of the demand shock on material vs labour (slope) to be di¤erent from same di¤erential e¤ect for the supply shock. This is because labour/intermediate input choices "control" for one of the shocks (gi;t or ! it ). In the non-parametric case, productivity shocks have an e¤ect on the input ratio and the input ratio varies with both demand and supply shocks (e.g. non Hicks neutral productivity). If the di¤erential e¤ects are the same for supply and demand 10 Notice that when the determinat is zero and invertibility condition holds with equality (and we cannot solve the system explicitly), the productivity term drops from the equation for Mit , i.e. Mit = m(L ~ it ; Kit ; Zit ; dit ) and the equation is clearly no longer invertible in ! it . However, in this case the problem is actually simpli…ed and guarantees the demand shock is identi…ed since we just need to nonparametrically regress Mit on (Lit ; Kit ; Zit ). This is the collinearity case of ACF (2006). In this case only the two FOCs and the separability condition (A.3) are required.
14
shocks, changes to the ratio allow us to distinguish one shock from the other. When the condition is not veri…ed, it implies that both shocks have a symmetric e¤ect on the two …rst order conditions such that varying one of them is in fact equivalent to varying the other. The system is not invertible. However, this is equivalent to the parametric case presented in Section 2.4. It is particularly useful because we su¢ ce with the two …rst order conditions. I.e., inverting the labour demand equation e¤ectively "controls" for the two shocks ! and "d and
d
mit = m(! it ; Lit ; Kit ; Zit ; l 1 (! it ; Kit ; Zit ; Lit )e it ) = m(L ~ it ; Kit ; Zit ;
d it )
and m ~ is not a function of !. We can nonparametrically regress Mit against Lit ; Kit ; Zit to identify the distribution of
d it .
15
A.5 A.5.1
Results Year-averaged estimated demand shocks and self-reported changes in market conditions, by industry.
2005
Corr=54% 1990
1995
2000
2005
2000
-.2-.1 0 .1 .2 .3
-.6-.4-.2 0 .2
2005
2000
1995
2005
-.2-.10 .1.2.3
-.4-.2 0 .2 .4
2005
1995
2000
1995
2000
2005
1995
2005
2000
-.3-.2-.10 .1 .2
2005
1995
1995
2000
2005
Corr=87% 1990
1995
2000
2005
Plastic and rubber products
2000
2005
Corr=72% 1990
1995
2000
2005
Vehicles and accessories
Corr=69% 1990
2005
Nonmetal mineral products
Corr=50% 1990
2000
Corr=78% 1990
Timber
Corr=51% 1990
2000
Paper
2005
1995
Industrial and agricultural equipment
Corr=38% 1990
Textiles and clothing
Corr=62% 1995
2000
1990
Miscellaneous
Corr=79% 1990
Printing and publishing
1990
1995
1995
Corr=82%
2005
Corr=77% 1990
Other transportation materials
Corr=26% 1995
2005
Corr=28% 1990
Office mach etc
1990
2000
Meat related products -.3-.2-.1 0 .1 .2
-.2 0 .2 .4
Leather fur and footwear
1995
2000
Furniture
Corr=18% 1990
1995
-.3-.2-.10 .1 .2
2000
Corr=66% 1990
-.4-.2 0 .2 .4
1995
2005
-.2-.10 .1.2.3
Corr=88% 1990
2000
Food and tobacco -.1 0 .1 .2 .3
-.4 -.2 0 .2
Fabricated metal products
1995
-.4 -.2 0 .2
Corr=70% 1990
Electric materials and accessories
2000
2005
-.4 -.2 0 .2
2005
-.4-.2 0 .2 .4
2000
-.4-.2 0 .2 .4
1995
-.2-.10 .1 .2 .3
Corr=89% 1990
Chemicals -.2-.1 0 .1 .2
Beverage -.4-.2 0 .2 .4
-.4 -.2 0 .2
Basic metal products
Corr=73% 1990
1995
2000
2005
-----(solid) - Reported changes in Market conditions - - -(dashed) - Estimated Demand shocks
Figure A.1: Year-averaged estimated demand shocks and self-reported changes in market conditions, by industry.
16
A.5.2
Production function estimates
17
18
0.034* 0.439*** 0.354*** 0.128*** 6.208***
Non-metal Minerals
642
0.024 0.167*** 0.520*** 0.096*** 5.583***
Meat
0.025 0.265*** 0.541*** 0.144*** 4.157***
Basic Metal
2240
0.035*** 0.202*** 0.621*** 0.072*** 3.991***
Food
0.035** 0.419*** 0.442*** 0.075*** 5.635***
Fabricat. Metal
461
0.192*** 0.282*** 0.172*** 0.186*** 6.694***
Drinks
720
0.064*** 0.202*** 0.487*** 0.080*** 5.261*** 610
0.126*** 0.341*** 0.303*** 0.078*** 6.474***
0.001 0.314*** 0.524*** 0.068*** 5.473***
-0.006 0.275*** 0.511*** -0.003 7.037***
-0.023* 0.437*** 0.412*** 0.074*** 6.881***
Dep. Variable: Physical Output Equip. O¢ c./data Electric. Precision
2375
0.043*** 0.483*** 0.247*** 0.082*** 7.753***
Dep. Variable: Physical Output Text. Foot. Timber
1188
0.092*** 0.385*** 0.456*** 0.019*** 5.481***
Vehicles
730
0.096*** 0.547*** 0.297*** 0.099*** 6.166***
Paper
538
0.02 0.261*** 0.599*** 0.112*** 3.874***
Other Transp.
1235
0.049*** 0.302*** 0.389*** 0.028*** 7.258***
Publish.
1186
0.026* 0.364*** 0.382*** 0.068*** 6.683***
Furniture
1736
0.050*** 0.203*** 0.479*** 0.156*** 4.949***
Chemic.
Table A.1: Production function estimates, …xed e¤ects results by industry.
N 1731 776 2402 1903 396 1729 Notes: Production function estimates with …xed e¤ects regression. Standard errors clustered at …rm level. * p<0.05; ** p<0.01; *** p<0.001
Capital Labour Materials Services Constant
Sector
N
Capital Labour Materials Services Constant
Sector
543
0.065** 0.425*** 0.278*** 0.090*** 7.241***
Misc.
1306
0.011 0.415*** 0.367*** 0.120*** 6.447***
Plastics
A.5.3
Input regression estimates (self reported shocks to demand)
19
20
0.150*** 0.453*** 0.025 8.100***
0.073** 0.427*** 0.218* 8.105***
Non-metal Minerals
662
0.021 0.264** 0.093 12.120***
-0.092 0.138 0.134 12.479***
Meat
0.096* 0.211 0.252 13.447***
0.010 0.184* 0.162* 12.580***
Basic Metal
2402
0.130*** 0.270*** 0.080 11.181***
0.034 0.262*** 0.266** 10.503***
Food
0.212*** 0.245*** 0.120 11.202***
0.017 0.226*** 0.193** 11.037***
Fabricat. Metal
477
0.056 0.468** 0.003 8.716***
0.061 0.369*** 0.333** 9.306***
Drinks
0.205*** 0.237*** -0.094 10.992***
0.023 0.236*** 0.260*** 10.909***
Equip.
2600
0.244*** 0.288*** 0.029 9.706***
0.042 0.288*** 0.188* 9.770***
Text.
0.120 0.325** -0.971* 9.789***
bill) -0.007 0.242*** 0.268** 11.493***
Electric.
687
-0.039 0.183* 0.416 12.042***
0.227* 0.510*** 0.047 7.920***
0.197*** 0.434* -0.029 9.354***
ln(materials)
O¢ c./data Precision ln(wage 0.075 0.446*** 0.290* 8.616***
809
bill) -0.064 0.191*** 0.412*** 11.058***
Timber
ln(materials)
ln(wage -0.064 0.236*** 0.201* 10.055***
Foot.
1201
0.131** 0.229*** 0.307 13.116***
0.033 0.178* 0.295* 13.014***
Vehicles
756
0.193** 0.368*** 0.051 9.990***
0.038 0.266*** 0.065 10.754***
Paper
562
0.399* 0.087 0.058 14.775***
0.099* 0.214* 0.188 12.066***
Other Transp.
1350
0.130** 0.246*** 0.266** 10.720***
0.001 0.228*** 0.396*** 11.003***
Publish.
Table A.2: Input demand equations (with self reported demand shocks).
N 1832 778 2530 1968 407 1786 Notes: Fixed e¤ects estimates of the wage bill (top panel) and materials (bottom panel) on the self-reported demand shock, TFP and capital stock, by industry. Year dummies included. S.e.’s clustered at the …rm level. All variables in logs. * p<0.05; ** p<0.01; *** p<0.001
Self reported demand ln(Capital) TFP Cons.
Dep. Var:
Dep. Var: Self reported demand ln(Capital) TFP Cons.
Industry
N
Self reported demand ln(Capital) TFP Cons.
Dep. Var:
Dep. Var: Self reported demand ln(Capital) TFP Cons.
Industry
1269
0.203*** 0.431*** 0.099 8.139***
0.036 0.346** 0.128 8.958***
Furniture
1793
0.067 0.423*** 0.159 9.854***
-0.011 0.371*** 0.162 9.659***
Chemic.
598
0.342** 0.038 -0.220 13.523***
-0.001 0.162* 0.242 11.639***
Misc.
1366
0.046 0.274*** 0.008 11.283***
0.021 0.291*** 0.190 10.260***
Plastics
A.5.4
Demand estimates: GMM results
21
22
-3.240*** -1.868*** 1.794*** 0.990***
Non-metal Minerals
-2.235*** -1.048*** 1.117*** 0.956***
Basic Metal
1,994
-1.143*** -0.505*** 1.085*** 0.854***
-3.529** 0.358 -0.285 1.004***
557
Food
Meat
-1.720*** -1.111*** 1.116*** 0.941***
Fabricat. Metal
389
-1.896*** -0.314*** 1.068*** 0.867***
Drinks
-1.037*** -1.077*** 1.146*** 0.950***
Equip.
2,150
-4.212*** -2.450*** 2.420*** 0.996***
Text.
545
O¢ c./data Electric. Precision Quantity -0.432** -0.973*** -1.320*** -1.082*** 1.088*** 1.055*** 0.943*** 0.953***
666
Timber Quantity -14.69*** -5.903*** 0.0487 -2.158*** 0.321 1.731*** 1.046*** 0.969***
Foot.
-13.87*** -0.642*** 0.842*** 1.032***
Vehicles
626
-1.818*** -0.610*** 0.991*** 0.852***
Paper
-0.499 -0.350* 0.998*** 0.740***
Other Transp.
1,109
-2.648*** -1.698*** 1.656*** 1.008***
Publish.
-14.64*** -1.048*** 1.268*** 1.037***
Furniture
1,497
-1.015** -1.467*** 1.522*** 0.991***
Chemic.
Table A.3: Demand estimates, IV results per industry.
N 1,481 649 2,055 1,624 328 1,503 985 471 1,054 Notes: Results of GMM estimates of the demand model using as instrument current and lagged TFP, age and capital stock and lagged output, prices and demand stock. Year dummies included. S.e.’s clustered at the …rm level. * p<0.05; ** p<0.01; *** p<0.001
g1
z
a
Dep. Var:
Industry
N
g1
z
a
Industry Dep. Var:
506
-1.240*** -0.547*** 1.145*** 0.865***
Misc.
1,114
-2.439*** -1.192*** 1.075*** 0.940***
Plastics
A.5.5
Correlation of the demand shock estimated via demand function with the self reported demand change and the demand shock estimated via input ratio
23
24
0.008 0.438***
ln(^ d ) Self-reported mkt.
0.015 0.260**
ln(^ d ) Self-reported mkt. -0.01 0.386***
0.384***
Basic Metal
2473
0.034 0.443***
0.456***
Fabricat. Metal
511
754
O¢ c./data Precision Panel A 0.525*** 0.507*** Panel B 0.024 0.013 0.517*** 0.504***
Equip.
2472
Dep. Variable: Estimated Demand Shocks Food Drinks Text. Foot. Panel A 0.512*** 0.256 0.307*** 0.123*** Panel B -0.007 0.064 0.063* 0.06 0.514*** 0.241 0.290*** 0.116***
0.039 0.528***
0.539***
Electric.
663
-0.028 0.446***
0.440***
0.044* 0.096***
0.100***
Vehicles
705
0.037 0.256*
0.260*
(via input ratio) Timber Paper
537
0.118 0.344***
0.352***
Other Transp.
1305
0.03 0.415***
0.422***
Publish.
1221
0.097*** 0.093***
0.100***
Furniture
1734
0.007 0.400***
0.335**
Chemic.
563
0.049 0.620***
0.644***
Misc.
1248
-0.065 0.635**
0.618**
Plastics
Table A.4: Correlation of the demand shock estimated via demand function with the self reported demand change and the demand shock estimated via input ratio.
Observations 1709 770 2326 1866 386 1712 1135 Notes: Results for linear regression of the estimated demand shocks, by industry. Each panel reports separate results. Standard errors clustered at the …rm level. * p<0.1; ** p<0.05; *** p<0.01
0.264**
Non-metal Minerals
ln(^ d )
Sector
683
0.439***
ln(^ d )
Observations
Meat
Sector
A.5.6
Input regression estimates (estimated shocks to demand)
25
26
0.401*** 0.731*** 0.540* 4.199***
-0.001 0.716*** -0.083 3.806***
Non-metal Minerals
607
0.601*** 0.748*** 0.263 3.940***
0.130 0.711*** -0.448 3.660***
Basic Metal
2177
0.774*** 0.903*** 0.205 1.846***
0.195* 0.693*** 0.380** 4.002***
0.096 0.620*** 1.070*** 5.172***
0.553*** 0.688*** 0.517 5.783*
Food
Meat
0.679*** 0.731*** 0.034 4.218***
0.100 0.628*** 0.140 5.079***
Fabricat. Metal
433
0.612 0.600*** 1.735*** 6.548***
0.211 0.812*** 0.640* 1.959
Drinks
0.863*** 0.830*** -0.430 3.193***
0.120 0.651*** 0.439** 5.219***
Equip.
2364
0.680*** 0.690*** 1.576*** 4.575***
0.226** 0.574*** 0.480*** 5.833***
Text.
bill) 0.077 0.566*** 0.745*** 6.645***
Electric.
590
0.279** 0.645*** 0.969 5.417***
0.155 0.692*** 0.536 4.889
0.503*** 0.654*** 0.942* 6.062***
ln(materials)
O¢ c./data Precision ln(wage 0.026 0.565*** 0.663** 6.805***
725
0.206* 0.686*** -0.582 5.150***
ln(materials)
Foot. Timber Total wages 0.056 0.073 0.431*** 0.501*** 0.468** 0.815** 7.665*** 6.305***
0.178* 0.882*** 0.814 1.915
0.054 0.742*** 0.399 3.585***
Vehicles
674
0.207 0.765*** 0.260 3.989***
-0.061 0.689*** -0.148 4.117***
Paper
0.631*** 0.758*** -0.516 4.419***
0.106 0.696*** 0.298 4.431***
Other Transp.
1209
0.719*** 0.626*** 0.936*** 5.170***
0.070 0.528*** 1.040*** 6.506***
Publish.
0.200** 0.438*** 1.627*** 8.219***
0.148*** 0.430*** 1.368*** 7.806***
Furniture
1627
0.569*** 0.743*** 0.212 4.688***
0.294*** 0.703*** 0.262 4.307***
Chemic.
Table A.5: Input demand equations (with estimated demand residuals).
N 1613 694 2191 1773 359 1647 1049 516 1140 Notes: OLS estimates of the wage bill (top panel) and materials (bottom panel) on the estimated demand shocks (obtained via demand function), TFP and capital stock, by industry. Year dummies included. S.e.’s clustered at the …rm level. All variables in logs. * p<0.05; ** p<0.01; *** p<0.001
Demand residual Capital TFP Cons.
Dep. Var:
Dep. Var: Demand residual Capital TFP Cons.
Industry
N
Demand residual Capital TFP Cons.
Dep. Var:
Industry Dep. Var: Demand residual Capital TFP Cons.
553
0.759* 0.571*** 1.291** 6.474***
0.150 0.610*** 0.415 5.284***
Misc.
1193
0.708*** 0.773*** 0.672 3.614***
0.008 0.657*** 0.546*** 4.684***
Plastics
A.6
Montecarlo results (ii) Translog production function
Corr = 99.9
Estimated demand shock (via input ratio)
Estimated demand shock (via input ratio)
(i) Baseline case 1 0.5 0 −0.5 −1 −2
0
1 0.5 0 −0.5 −1 −2
2
True demand shock Estimated demand shock (via input ratio)
Estimated demand shock (via input ratio)
Corr = 99.9
0.5 0 −0.5 −1 −2
0
2
True demand shock
(iii) Labour quadratic adj. costs 1
Corr = 84.6
0
2
(iv) Cubic demand 0.1 0.05
Corr = 99.7
0 −0.05 −0.1 −2
True demand shock
0
2
True demand shock
Figure A.2: Scatterplot and correlation between simulated and estimated demand shocks using four di¤erent parametrizations (
z
= 0:75).
27
(ii) Translog production function
Corr = 99.9
Estimated demand shock (via input ratio)
Estimated demand shock (via input ratio)
(i) Baseline case 1 0.5 0 −0.5 −1 −2
0
2
1 Corr = 92.8 0.5 0 −0.5 −1 −2
True demand shock Estimated demand shock (via input ratio)
Estimated demand shock (via input ratio)
(iii) Labour quadratic adj. costs 1
Corr = 99.9
0.5 0 −0.5 −1 −2
0
0
2
True demand shock
(iv) Cubic demand 0.1 Corr = 99.7 0.05 0 −0.05
2
True demand shock
−0.1 −2
0
2
True demand shock
Figure A.3: Scatterplot and correlation between simulated and estimated demand shocks using four di¤erent parametrizations ( = 0:66).
28
(ii) Translog production function
Corr = 99.9
Estimated demand shock (via input ratio)
Estimated demand shock (via input ratio)
(i) Baseline case 1 0.5 0 −0.5 −1 −2
0
2
1
Corr = 93.6
0.5 0 −0.5 −1 −2
True demand shock
Corr = 99.9
Estimated demand shock (via input ratio)
Estimated demand shock (via input ratio)
(iii) Labour quadratic adj. costs 1 0.5 0 −0.5 −1 −1.5 −2
0
0
2
True demand shock
2
(iv) Cubic demand 0.1 Corr = 99.1
0.05 0 −0.05 −0.1 −2
True demand shock
0
2
True demand shock
Figure A.4: Scatterplot and correlation between simulated and estimated demand shocks using four di¤erent parametrizations ( = 0:35 and
29
= 0:7).
