Identifying demand shocks from production data Carlos Daniel Santos Nova School of Business and Economics March 6, 2018
Abstract Standard productivity estimates contain a mixture of cost e¢ ciency and demand conditions. 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 that contains information on prices and demand conditions. Finally, we generate Monte Carlo 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 on 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). 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). Attention is focused on 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 in which materials are chosen after labor. 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 labor, together with the output equation is, however, insu¢ cient to identify the demand shocks. We need to introduce auxiliary assumptions similar to those 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 Monte Carlo exercises to evaluate the model’s performance and sensitivity. 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
2
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 relationship 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 tackles 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) and another is less ‡exible (say labor), the asymmetry of responses provides information about the unexpected "news". The intuition is simple. Intermediate inputs adjust as a response to both demand and supply shocks. In the fully ‡exible case, intermediate inputs and labor should be equated on the margin and the ratio of materials to labor equals the elasticity of technical substitution. However, in the presence of adjustment costs, there is a wedge between the two inputs. This wedge is a function of the demand shock, a larger demand shock generates a larger departure of the input ratio from the elasticity of technical substitution.
3
This allows identi…cation of the demand shock. 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.1 Under optimality, the ratio of the input share (ratio of intermediate inputs to revenues) to the output elasticity of the intermediate input (e.g., 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).2 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 nonparametrically identify the production function. From the optimality conditions, the optimal input share is a function of the output elasticity. We can then nonparametrically estimate it in a …rst step regression. While I do not attempt to estimate the input elasticities, 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 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. 2 As shown by Santos et al. (2016), the introduction of persistent demand shocks and the use of price data solves the unidenti…cation problem.
4
of the input share regression. One important distinction from earlier work is that while Gandhi et al. (2016) and De Loecker (2011) use a transformation of the …rm’s …rstorder 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), who use 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.
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 labor (L). Assumption 2 (A.2) Information: (i) Firms observe their productivity before any decision is made. (ii) Demand conditions are observed after labor is chosen (adjustment costs to labor prevent its instantaneous adjustment to match demand conditions). For the remainder of the article I will consider three inputs: capital, labor, and materials. Assumptions (A.1) and (A.2) generate an exclusion restriction since the demand 5
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 materials independently from labor. Labor is thus allowed to be correlated with the predictable part of demand. Firms with greater 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 to labor. For example, in some countries, labor regulations will imply that labor 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 labor. 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 labor hiring/…ring decisions where labor becomes immediately available. 2. Demand shock is observed: Conditional on the capital existing in the beginning of the period and the labor 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 labor subject to adjustment costs. 6
Demand Inverse demand for the product of …rm i in period t is pit = p(qit ; Zit ; "dit ) 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 labor hiring decision. This information set contains the demand level from last period ("di;t 1 ), together with all other state variables, such as 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.
7
By the end of the period the …rm chooses the level of materials that 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 gives3 E"d where ql =
@p(qit ; Zit ; "dit )qit @qit j @qit @Lit @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 labor and quadratic adjustment costs to labor. 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 labor. 3
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
8
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 [ 1
=
m:l |{z}
Supply (ETS)
where
m:l
E"d | it
+ 1] p(qit ; Zit ; "dit )qm d p;q + 1] p(qit ; Zit ; "it )ql j
p;q
d p;q (qit ; Zi;t ; "it ) + 1 d p;q (qit ; Zi;t ; "it ) + 1
{z
(3) it
p(qit ; Zi;t ; "dit )ql p(qit ; Zi;t ; "dit )ql j
it
Demand
}
is the elasticity of technical substitution (ETS).4 The ratio in expression (3)
depends on the ETS and the ratio of marginal return to labor to expected marginal return to labor (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 toward 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. Alternatively pM M it Mit h it i h = = L pit Lit E"d qqml j it Lit E"d
Mit Mit m:l Lit j
it
i
= Lit
Mit E"d [ m:l Mit j
it ]
(4)
The last step is possible because Lit is known at the time Mit is being chosen. When 4
The ETS,
ml
=
ql M qm L
where
ql qm
is the marginal rate of technical substitution.
9
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. 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 ! were
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 labor 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, nonparametric
case with pit = p(qit ; Zi;t 1 ; "dit ). For convenience let us abstract from age as this adds nothing substantial to the analysis and we would need to keep track of one extra state 10
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 labor adjustments, an extreme form of adjustment costs. In practice, …rms adjust their workforces 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 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 (the quotes emphasize that in practice there is no "transmission" since both temporary employment and materials are simultaneously determined). 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). 11
When LTit = 0 (binding constraint) employment is quasi-…xed and we are back in the original case. Note that the ratio of temporary employment 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
it ]
(8)
;
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
12
(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 the up to 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 labor 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 use either 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 .
13
Furthermore, we can have di¤erences in the quality of inputs or their usage (e.g. number of workers being a bad proxy for labor 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. At 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
14
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 on 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 report demand conditions directly. 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 for both 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 15
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 in order 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 (Panels A and C) and changes in market shares (Panels B and C). Changes in demand conditions are more important than 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 at both the industry and the more disaggregated …rm level.
16
.1
.8 .3
-.3
.4
-.2
.5
-.1 all_ind
.6
Input ratio (detrended) .6 .5 .7
0
.7
.8
Figure 1: Input ratio vs. self reported demand conditions: whole manufacturing sector.
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
Notes: Left: Scatter of (year-average) self-reported demand shocks against the input ratio. Right: Time-series of (year-average) changes to market conditions and estimated demand shocks (via input ratio). 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 Notes: Standard deviation, serial correlation (orders 1 to 5), and cross sectional dependence for the estimated demand shock.
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 independent, 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 independence.
17
18
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
0.176* 0.106**
0.158***
0.262***
Other Transp.
1,339
0.058* 0.031
0.049**
0.094***
Publish.
0.028 0.074*
0.083***
0.114***
Furniture
1,808
0.067** 0.020
0.032*
0.084***
Chemic.
0.070* 0.066**
0.090***
0.155***
Misc.
1,342
0.056** 0.013
0.030*
0.070***
Plastics
Table 2: Correlation of estimated demand shocks with market conditions.
Observations 1,808 858 2,417 1,959 405 1,776 1,218 555 1,261 572 Notes: Results for linear regression of the estimated demand shocks, by industry on self-reported shocks (Panel A) , Mkt. Share (Panel B), and both (Panel C). 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
d i;t
Dep. Var:
Coef. t-stat p-value -0.026 -1.465 0.143 Constant 0.003 3.627 0.000 Serial correlation tests AR(1) -10.676 0.000 AR(2) 1.592 0.112 Firms 2,808 Observation 22,259 Notes: Dynamic panel data results for di;t . GMM-type instruments for the di¤erenced equation: di;t 2 , di;t 3 , di;t 4 . d i;t 1
Table 3: Test for serial correlation: Dynamic panel data (GMM) estimates.
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 labor. We can thus (i) compare the proxy for demand and supply shocks with prices and physical output and, (ii) evaluate the sensitivity 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.1) in the Appendix. TFP is the residual from the production function equation, including the …xed e¤ects. We start with the e¤ect on prices and output. 3.2.1
E¤ects on prices and output
The …rst 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 correlation 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
19
to demand (Table 4).5 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. 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*** 0.263*** 0.114* 0.0981*** 0.262*** 0.327*** 0.0746 0.227*** 0.663*** 0.348*** 0.282*** 0.623*** 0.341*** 0.455*** 0.537*** 0.353*** 0.610*** 0.683*** 0.130 0.325***
1.409** 0.555** 0.290 0.373** 0.448** 0.785*** 0.0427 0.817*** 0.845*** 0.732** 0.495*** 0.421 0.336*** 0.452** 1.014* 0.538** 0.692** 0.734** 0.309* 0.515
ln(Prices) Demand TFP shock 0.0126 -0.00314 0.0253* -0.00281 -0.0141 0.00701 0.0912* 0.0214* 0.0583*** 0.0257 0.0193* 0.140*** 0.0276*** 0.0365*** -0.00956 0.0121 0.0183 0.0453*** -0.0179 0.00336
-0.204*** -0.136*** -0.0967** -0.0518*** -0.0339* -0.0260 -0.217*** -0.0825 -0.117** -0.0392 -0.107** -0.170*** -0.0838*** -0.0313** -0.108 -0.0869*** -0.0401** -0.0392* -0.0702*** -0.0680*
Notes: Linear regression of the log of de‡ated revenues (output) and log of the price index on the growth of TFP and demand. S.e.’s clustered at the …rm level. * p<0.1; ** p<0.05; *** p<0.001
Table 4: Correlation of estimated demand and TFP with prices and quantities, by industry.
3.2.2
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 be correlated with only materials. Table (5) reports the results from estimating a log-linear 5
A zero correlation of prices and the shocks to demand implies a ‡at inverse supply schedule (e.g. constant marginal costs).
20
approximation to the optimal input choices: d ^ it + "m ln pM it it Mit = m0 + m1 ^ it + m2 ln Kit + m3 !
^ it + "lit : ln pLit Lit = l0 + l1 ^dit + l2 ln Kit + l3 ! 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 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 estimating the demand function directly. This is possible because our data contain information on …rm level 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
+g1 [ln (qi;t 1 ) + ln (pi;t 1 )
21
d it ).
(12)
ln(Zi;t ) a
ln(agei;t 1 )
z
ln(Zi;t 1 )] +
d it
22
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.
1101
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
514
1.045*** 0.767*** -0.179 4.293***
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.
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.
Table 5: Input demand equations (with estimated demand shocks).
Notes: OLS estimates for the regression of the wage bill (top panel) and materials (bottom panel) on estimated demand shocks, 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
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
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. That is, 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 P P a s . In the …rst year for which the stock is Zia = as=0 Yis (1 )a s = Yia as=0 1
company enters the sample the demand stock at that period is a function of the revenues
in that year multiplied by a term that depends on the age of the …rm and growth rate, a 1 a 1 (1 ) . We use a growth rate of = 1:063 since 6.3% is the average Zia = Yia 1 1 (1 ) 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 in which the set of instruments is current and lagged TFP, age, and capital stock plus the lagged values of 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 3.2.2. 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 in price. The shocks to demand are obtained as the residual ( cdit ) 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
23
via input ratio. They are also positively correlated with the self-reported demand shock, which signals that the three measures have strong collinearity. This is con…rmed with a principal component analysis of the three shocks, which reveals that one factor can explain 45% of the total variation. Finally, we can test the same set of predictions that was conducted in Table (5): shocks to productivity are expected to have an e¤ect on both inputs while shocks to demand should be correlated with only materials. The di¤erence is that 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 labor 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
Materials to wage bill Whole sample (i) 0.0453*
Number of workers LT > 0 Whole sample (ii) (iii) 0.0548* 0.0197
LT > 0 (iv) 0.0398
Whole sample (v) 0.926**
0.612***
0.619***
0.124*** 0.250*** 0.452***
30,034 4,079
23,267 3,675
24,734 3,460
Hours LT > 0 Whole sample (vi) (vii)
LT > 0 (viii)
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: Estimates of Equation (6). Linear regression of ratio of materials to wages on shocks to demand and ratio of temporary employment, for varying subsamples. Firm and year speci…c (…xed) e¤ects included. S.e.’s clustered at the …rm level. Columns (iii), (iv), (vii) and (viii) add the control for the estimated demand shock. *** p<0.001, ** p<0.01, * p<0.05
Table 6: Estimates for Equation (6). Our …nal robustness exercise consists of evaluating how temporary labor responds to demand shocks. We have shown in Section (2.3.2) how temporary employment can be incorporated in the model. The ratio of temporary to permanent employment plays the same role as the input ratio - Equation (7). The input ratio is simultaneously deter24
mined with the ratio of temporary to permanent employment - Equation (6). Temporary employment and materials should be equated at the margin. We estimate both Equations (6) and (7) using two measures of ‡exible labor: the fraction of overtime hours to total hours and the fraction of temporary employees to total employees.6 Table (6) 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 from Equation (6). From Equations (6) and (7), the ratio of temporary to permanent employment "transmits" the demand shocks to the input ratio. The use of quotes emphasizes that there is no (causal) transmission since both inputs are simultaneously determined and are both a function of the demand shock. In columns (iii), (iv), (vii), and (viii) we add as covariates the shock to demand (both the self-reported and estimated as the residual of the demand function). The coe¢ cient on the temporary to permanent employment becomes substantially smaller and in most cases insigni…cant, con…rming the simultaneity. We further investigate this mediating role by estimating equation (7) for how the news about demand correlates with the ratio of temporary employment to permanent employment in Table (7). Both measures of temporary employment are positively correlated with the demand shock. Overall, the evidence is consistent with the optimal equations for temporary employment, (6) and (7). Temporary employment is "equivalent" to materials in the sense that temporary employment and materials (both ‡exible inputs) are simultaneously determined and should be equated at the margin. The logic applied to the input ratio applies to the temporary to permanent employment. 6
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 labor market in Spain. Temporary work is used as a way to avoid entering long-term contracts that have relatively large …ring costs.
25
ln(LT =LP )
Dep. Var.: workers (ii)
(i) Shock to demand input ratio via demand function Self-reported Constant
(iii)
(iv)
hours (v)
(vi)
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***
Observations 19,992 19,105 19,092 8,952 8,391 8,385 Firms 3,192 3,082 3,082 1,542 1,474 1,474 Notes: Linear regression of ratio of temporary to total employment (number of workers and hours) on shocks to demand. Firm and year speci…c (…xed) e¤ects included. S.e.’s clustered at the …rm level. *** p<0.001, ** p<0.01, * p<0.05
Table 7: Regression estimates for adjustable employment on demand shocks.
4
Monte Carlo simulations
In this 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 Technology and CES demand functions as speci…ed above qit = ea0 ! it Mit Lit Kit
p(qit ; Zit ; "dit ) = qit Zi;tz 1 "dit
and
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
1
+
d it
and
ln ! it = g! ln ! it
1
+
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
26
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. In Figure (2) we report the …t (scatter plot and correlation) between the true simulated demand shock (
d it )
and the estimate. We present four 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 (3)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 that 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 (3) shows that the correlation between the true and estimated demand shock is reduced while still remaining large at 93%. Case (iii) introduces quadratic adjustment costs to labor of the form
( Lit )2 . Li;t 1
Every time the company wants to adjust
its labor 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 that remain above 99.4%. Finally, case (iv) considers cubic demand ln (pit ) =
1
ln(qit ) +
2
ln(qit )2 +
1
3
ln(qit )3
z
ln(Zi;t )
ln("dit ) :
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
27
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
1
ln(qit ) + 3
3
ln(qit )2 ) ) is -13 at ln q = 10; -4 at
ln q = 20, and -1.7 at ln q = 30.
N=500, T=15 1
Corr = 99.94
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
2
True demand shock
Estimated demand shockEstimated demand shock (via input ratio) (via input ratio)
Estimated demand shockEstimated demand shock (via input ratio) (via input ratio)
Figure 2: Simulated vs. estimated shocks with varying sample sizes. 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
Notes: Scatterplot and correlation between simulated and estimated demand shocks using four di¤erent sample sizes (N and T).
(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
2
Estimated demand shock Estimated demand shock (via input ratio) (via input ratio)
Estimated demand shock Estimated demand shock (via input ratio) (via input ratio)
Figure 3: Simulated vs. estimated shocks with varying parametrizations. 1 Corr = 93.01 0
−1 −2
0.1
0
1
2
Corr = 99.04 0
−0.1 −2
True demand shock
−1
True demand shock
(iv) Cubic demand
−1
0
1
2
True demand shock
Notes: Scatterplot and correlation between simulated and estimated demand shocks using 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) 28
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.7
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 compares with other measures of demand shocks, including self reported and estimated residuals 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.
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, and EARIE2013 for useful comments and suggestions.
References [1] Ackerberg, D., Benkard, L., Berry, S., and Pakes, A. (2003) "Econometric Tools for Analyzing Market Outcomes", Handbook of Econometrics, Vol 6. [2] Ackerberg, D., Caves, K. and Frazer, G. (2006) "Structural Identi…cation of Production Functions", manuscript. [3] Bond, S. and Måns Söderbom, (2005) "Adjustment costs and the identi…cation of Cobb Douglas production functions", IFS Working Papers, W05/04. 7
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.
29
[4] Blundell, R. and Matzkin, R., (2013) "Control Functions in Nonseparable Simultaneous Equations Models", UCL working paper. [5] Chen, X., Chernozhukov, V., Lee, S., and Newey, W. (2014) "Local Identi…cation of Nonparametric and Semiparametric Models", Econometrica, 82(2), 785-809. [6] Chernozhukov, V., Imbens, G., and Newey, W. (2007) "Instrumental variable estimation of nonseparable models", Journal of Econometrics, 139 (1), 4-14. [7] Cobb, C. and P. H. Douglas (1928) "A Theory of Production", The American Economic Review, 18 (Supplement), 139-165. [8] Collard-Wexler, A. and De Loecker, J. (2012) "Reallocation and Technology: Evidence from the U.S. Steel Industry", manuscript. [9] De Loecker, J. (2011) "Recovering markups from production data", International Journal of Industrial Organization, 29, 350-355. [10] De Loecker, L. and Warzynski, F. (2012) "Markups and Firm-Level Export Status", The American Economic Review, 102(6), 2437-2471. [11] Doraszelski, U. and Jaumandreu, J. (2013) "R&D and productivity: Estimating endogenous productivity", Review of Economic Studies. [12] Eaton, J., Kortum, S., and Kramarz, F. (2011) "An Anatomy of International Trade: Evidence from French Firms", Econometrica, 79(5), 1453-1498. [13] Foster, L., Haltiwanger, J., and C. Syverson (2008) "Reallocation, Firm Turnover, and E¢ ciency: Selection on Productivity or Pro…tability?", American Economic Review, 98 (1), 394-425. [14] Foster, L., Haltiwanger, J., and Syverson, C. (2016) "The Slow Growth of New Plants: Learning about Demand?", Economica, 83(329), 91-129. [15] Gale,D. and Nikaido, H. (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 Symposium, 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.
30
[21] Klette, T. (1999) "Market Power, Scale Economies and Productivity: Estimates from a Panel of Establishment Data", The Journal of Industrial Economics, 47(4), 451476. [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 Andrews, W. H. (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.
31
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 nonparametric 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 us 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 that 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
1
+ pit qit
< 1 is the discount factor and we now have an extra term that includes the
continuation value. (1
) Zi;t
) Zi;t
1
Notice that if we let yit = p(qit ; Zi;t 1 ; "dit )qit , and since Zit =
+ 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 gives8 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 work. 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 8
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-
dynamic 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 (labor and materials). Consider the previous model but let us assume that prices are given, as in Gourio and Rudanko (2014). Let us 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 that 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 that 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: the customer base and the potential buyers. The customer base n decays at rate
n.
This means that in every period a fraction
n
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 3
follow a …rst-order Markov process, 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
h
n ) (1
nit + s1it bit @V (:) @n0
bit sit
) (1
E [Vn (ni;t+1 ; ! i;t+1 ; bi;t+1 )jbit ; nit ; ! it ] bit sit
)=
(1
)
i . The …rst term, p (1
bit sit
)
(A.3)
+ 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 labor 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
since for the Cobb Douglas production function pM it Mit = pLit L
pLit @M @L
=0 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
1=
bit
e!it Lit
E"it
nit + s1it
bit
1=
e!it Lit
nit + s1it bit 1 e
= E"it
1=
"it
nit + s1it bit 1 e
"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, the number of potential buyers, and 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 nonparametric 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 0
B sit = @(1
2
6 ) bit 4p + (1
e n ) vn (bit ; nit ; ! it )
1
pM it e!it Lit
1=
nit +
bit sit
! (1
)
31 1+1 7C 5A
which illustrates both the endogeneity of sit with respect to "it (via bit ) and the validity 6
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 ex-
tra cost that increases with the number of temporary workers and their regular cost, P
pLit
LT it . LP it
The cost of a temporary worker is less than the cost of a regular worker for
small amounts but increases with the amount of temporary workers. The total cost is 2 (LTit ) LT T LP T LP . The "…scal" advantage is related with the thus pit Lit = (1 ) pit Lit + pit LP 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. By the end of the period the …rm chooses the level of materials that maximizes the following objective function
max p(qit ; Zit ; "dit )qit
LT it ;Mit
pM it Mit
T
pLit LTit
s.t. qit = q(Kit ; Li;t ; Mit ; ! i;t ) Lit = LTit + LPit T
pLit
P
= pLit
1
+
LTit LPit
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
[
p;q
+ 1] p(qit ; Zit ; "dit )qm = pM it
7
(A.4)
[ where
p;q
P
d L p;q + 1] p(qit ; Zit ; "it )ql = pit
< 0 is the elasticity of demand, qm =
1
@q(:) @M
+2 and ql =
LTit LPit
(A.5)
@q(:) . @L
We can rewrite the
reduced form optimal solutions as
Mit = m(! it ; LPit ; Kit ; Zit ; "dit )
= lT (! it ; LPit ; Kit ; Zit ; "dit ):
LTit
Putting the two equations together T
pM 1 1 pLit it Mit = = LP LP pit 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 labor 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
pM it Mit
T
pLit LTit j
it
s.t. qit = q(Kit ; Lit ; Mit ; ! it )
i
P
pLit LPit
Lit = LTit + LPit which gives
E"d
@p(qit ; Zit ; "dit )qit @qit j @qit @LPit
it
= E"d [
p;q
+ 1] p(qit ; Zit ; "dit )ql j
P
it
= pLit :
Combining the FOCs for LTit and LPit , we obtain the following restriction
8
(A.7)
P
E"d pitL
1
+2
LTit LPit
P
j
= pLit :
it
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 labor, the only di¤erence is the cost that 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
it ]
(A.8)
:
The two ratios of materials to labor and temporary employment to labor 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,
A.3
LT it . LP it
Higher order process (extension)
The input ratio is
9
E"d [LT it j
it
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 an n-dimensional vector of information known to the …rm
at period t. Again, write ~dit =
1
ln
1 d ( +1) it
and normalize E
d it
~dit = 0,
pM it Mit = 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 function and the production function. 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 equations9 . 9
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 two unobservables (! it , "dit ) and the vector of information known to …rm i when it chooses labor,
it .
correlated with all the observables due to serial correlation (note
it
This is potentially includes ! i;t 1 , "di;t
1
and the other state variables) and three equations.10 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 labor (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 approach 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 follows a separable …rst-order Markov process, "dit = d
g("di;t 1 )e it . The news term is, by assumption, serially independent. 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 …rm
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
10
Notice that if we allow adjustment costs to labor, lagged employment would enter the input demand function for labor, which 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 indepen-
dent 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
As long as there
and (Li;t 1 ; Ki;t 1 ; Zi;t 1 ) become valid
instruments simply by inverting the equation ! i;t
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 equation m ~ (:) directly to identify the distribution of the demand shocks
d it
(again up to
standard normalization and location restrictions. A.4.1
Invertibility
We can express the previous problem of obtaining a solution for git = g(
it )
and ! it as
an invertibility problem. Inverting the labor 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 obtain the distribution of
d it
by Mit = m(L ~ it ; Kit ; Zit ;
d it )
directly. 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,11 @[
@
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. labor (slope) to be di¤erent from same di¤erential e¤ect for the supply shock. This is because labor/intermediate input choices "control" for one of the shocks (gi;t or ! it ). In the nonparametric 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 11 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 that 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. That is, inverting the labor 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
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 1188 Notes: Production function estimates. Regression reults with …rm speci…c (…xed) e¤ects. Standard errors clustered at …rm level. * p<0.05; ** p<0.01; *** p<0.001
Capital Labor Materials Services Constant
Sector
N
Capital Labor 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.
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
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 1201 562 Notes: OLS estimates for the regression of the wage bill (top panel) and materials (bottom panel) on self-reported demand shocks, 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: GMM estimates of the demand model. Instrument set includes 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
0.015 0.260**
0.264**
Non-metal Minerals
683
0.008 0.438***
0.439***
Meat
-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
0.118 0.344***
0.352***
Other Transp.
1305
0.03 0.415***
0.422***
Publish.
0.097*** 0.093***
0.100***
Furniture
1734
0.007 0.400***
0.335**
Chemic.
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 537 1221 563 Notes: Results for linear regression of the demand shock (via input ratio) on the demand shock (via demand function residual), 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.
^d
^d
Sector
Observations
Self-reported mkt.
^d
^d
Sector
A.5.6
Input regression estimates (estimated demand residuals)
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.
0.206* 0.686*** -0.582 5.150***
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
Timber bill) 0.073 0.501*** 0.815** 6.305***
ln(materials)
Foot. ln(wage 0.056 0.431*** 0.468** 7.665***
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.
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
Table A.5: Input demand equations (with estimated demand residuals).
N 1613 694 2191 1773 359 1647 1049 516 1140 553 Notes: OLS estimates for the regression of the wage bill (top panel) and materials (bottom panel) on estimated demand shocks (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.
A.6
Monte Carlo results
Figure A.2: Simulated vs. estimated shocks with varying parametrizations (
Corr = 99.9
0.5 0 −0.5 −1 −2
0
2
1 0.5
−0.5 −1 −2
Estimated demand shock (via input ratio)
Estimated demand shock (via input ratio)
Corr = 99.9
0.5 0 −0.5
0
0
2
True demand shock
(iii) Labour quadratic adj. costs
−1 −2
Corr = 84.6
0
True demand shock
1
= 0:75).
(ii) Translog production function Estimated demand shock (via input ratio)
Estimated demand shock (via input ratio)
(i) Baseline case 1
z
2
True demand shock
(iv) Cubic demand 0.1 0.05
Corr = 99.7
0 −0.05 −0.1 −2
0
2
True demand shock
Notes: Scatterplot and correlation between simulated and estimated demand shocks using four di¤erent parametrizations ( z = 0:75).
27
Figure A.3: Simulated vs. estimated shocks with varying parametrizations ( = 0:66). (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
2
True demand shock
(iv) Cubic demand 0.1 Corr = 99.7 0.05 0 −0.05 −0.1 −2
0
2
True demand shock
Notes: Scatterplot and correlation between simulated and estimated demand shocks using four di¤erent parametrizations ( = 0:66).
28
Figure A.4: Simulated vs. estimated shocks with varying parametrizations ( = 0:35 and = 0:7). (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 0.5 0 −0.5 −1 −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 −1.5 −2
0
0
2
True demand shock
(iii) Labour quadratic adj. costs 1
Corr = 93.6
2
True demand shock
(iv) Cubic demand 0.1 0.05
Corr = 99.1
0 −0.05 −0.1 −2
0
2
True demand shock
Notes: Scatterplot and correlation between simulated and estimated demand shocks using four di¤erent parametrizations ( = 0:35 and = 0:7).
29
