Lecture 2: Measuring Firm Heterogeneity

October 23, 2017

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The Dual Problem: Cost Function A natural alternative to production function estimation approach is cost function. It is much less used • producer-level factor prices are rarely observed • lack of instrumental strategy for output • the output also needs to be cleanly measured (i.e. homogenous) In a few regulated industries, such as electricity, public transportation, etc., it might still be feasible to implement. • U.S electric generating plants: a transition from cost-of-service regulation towards market-oriented environment • Water, bus, electricity, etc. are the typical candidates of this approach

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General framework • Minimize variable cost, subject to an output constraint ¯ it , V¯it ) minVit C(W s.t.QP it

≤ F (Ki , Vit , it )

where V¯it are variable inputs, and Ki is allowed to be quasi-fixed ¯ it are factor prices. (capacity etc.). W • Earlier literature (i.e Christensen and Greene etc.), avoids specifying F (.), instead use a Translog Cost Function as an approximation to arbitrary continuously differentiable function. X 1 2 ¯ it , QP lnC(W αj lnWitj it ) = α0 + αQ lnQit + γQQ (lnQit ) + 2 j +

X 1 XX lnWitk lnWitj + γQj lnQit lnWitj + it 2 j j k

• The source of endogeneity is obvious here, especially for Qit . Earlier regulations are used to argue that cov(Qit , it ) = 0 • Shapard’s Lemma to derive input demand - identification not solid. 3

Fabrizio et al • Parametric assumption of F (.). minLit ,Mit Wit Lit + Sit Mit s.t.QP it

L

M

≤ Q0 (Ki )(Lit )γ (Mit )γ exp(P it )

• Measurement equation assumes actual output QA = QP exp(A ) • Variable input demand A ln(Lit ) = ln(λγ L ) + ln(QA it ) − it − ln(Wit )

• Further enrich the shock to include regulatory regime, year, and plant permanent heterogeneity L L L A L ln(Lirt ) = ln(QA irt ) − ln(Wirt ) + αi + δt + φr − irt + irt

L irt is interpreted as unobserved productivity shock (or measurement error), although how this enters input demand is not clearly specified. 4

Identification

• Facing similar identification challenge: endogeneity and selection • Search for an IV for ln(QA irt ): state-level electricity demand as an instrument. Unlikely to be correlated with how efficiently each plant is run. Run robustness with dynamic panel style IVs (i.e. lagged output). • Argue away the “selection” problem – exit is rare for electricity generating plants – and keep the unbalanced panel. • The impact of restructuring is identified off the different timing in different locations + differential impact on plants of alternative ownerships.

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Index Numbers: Input Optimization Several key ideas in Index Numbers: approximation and optimization • an aggregator function f is "flexible" if it can provide a second order approximation to an arbitrary twice differentiable linearly homogenous function r

) 0 r 0 r • define a quantity index Q(p0 , pr ; x0 , xr ). if ff (x (x0 ) = Q(p , p ; x , x ) for any period r = 1, 2, ..R then we say Q is "exact" for f . QN • for instance, i=1 (x1i /x0i )si is exact for Cobb-Douglas aggregator function, where si = p0i x0i /p0 x0 • a quantity index Q is "superlative" if it is exact for an f which is flexible.

• Focus on a homogeneous translog function, PN PN PNwhich is "flexible": lnf (x) = α0 + j=1 αi lnxn + 1/2 i=1 j=1 γij lnxi lnxj , where PN PN n=1 αn = 1,γij = γji and i γij = 1 for j = 1, 2, ..., N . (Christensen,Jorgenson, and Lau (1971)).

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Main result from Diewert (1976) • assume that xr > 0N is a solution to the aggregator maximization problem maxx {f (x) : pr x = pr xr , x ≥ 0N }, and f is the homogeneous translog function. • using first order conditions for the maximization problem and Quadratic Approximation Lemma, PN it can be shown that: ln[f (x1 )/f (x0 )] = n=1 12 [s1n + s0n ]ln[x1n /x0n ], where srn = prn xrn /pr xr for period r = 0, 1. QN 1 0 • It is immediate that Q0 (p0 , p1 ; x0 , x1 ) = n=1 [x1n /x0n ]1/2[sn +sn ] is "superlative" quantity index. • The above maximization problem can be interpreted naturally as producer’s problem, where srn is the nth share of cost in period r. • Technological progress can be measured by 1 + τ = [y 1 /y 0 ]/

N Y

1

0

[x1n /x0n ]1/2[sn +sn ]

n=1

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The Index number has several advantages • minimal assumption of functional form of production function • easy-to-implement and often a useful first-cut Why this literature disappears from “standard” IO applications? • disconnect from econometric literature - no distinction between data vs. model (i.e. no structural error) • all input factors are optimized in a static fashion • firms are typically price takers The translog functional form though has re-gained some popularity in a few recent applications. • A recent paper by Gandhi Navarro Rivers (2017) has utilized some of the ideas and combine it with modern econometrics.

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Gandhi Navarro Rivers • Note that ACF starts with value-added production function. Implicitly assumed Leontiff structure. • Additional identification challenges if trying to estimate a Gross Production Function, say with material inputs • Not a trivial issue: input-output linkages, firm-to-firm trade

relationships, etc. • ACF doesn’t work in this case • Recall that mit = m(kit , lit , ωit ) • If mit is directly an input factor in gross production function, which

variable can serve as instrument to identify its coefficient? • A particularly insightful equation here is

mit = m(kit , lit , g(m−1 (kit−1 , lit−1 , mit−1 )) + ξit )

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Gandhi Navarro Rivers

• Despite the apparent abundance of potential IVs, after condition on current and lagged inputs, the only variation left in mit is the transitory productivity shock ξit . • ξit is orthogonal to all potential IVs (lagged inputs) by construction. • In other words, once the lagged material inputs were used to “proxy” for unobserved productivity shocks, we run out of a “relevant” IV to identify the material production function coefficient. • The only viable remedy under ACF framework would be input prices, but we know it is usually quite problematic

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Gandhi Navarro Rivers

• Remedy surprisingly simple: Use the FOC of material inputs • With any gross production function f (kit , lit , mit ), the profit maximization implies sit ≡ ln(

pm ∂f (kit , lit , mit ) t Mit ) = lnC + ln( ) − it pt Yit ∂mit

• The above share equation serves as an additional cross-equation restriction on the production function. • In fact, the material input share non-parametrically identifies the input elasticity of material, subject to a constant.

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