Lecture 3Production and Cost Function EstimationBronwyn H. HallEconomics 220C, UC BerkeleySpring 2005Spring 2005 Economics 220C 2Outline• Production, Cost, and Profit functions–uses• Data and estimation issues– Panel data– specification– Exit and selection• parametric• Semi-parametric (Olley-Pakes)Spring 2005 Economics 220C 3Why study them?• One piece of supply/demand framework– Needed for any equilibrium computation– Form influences model (e.g. learning by doing, networks)• Used to evaluate efficiency effects of policy– Regulation - increasing returns, cost complementarities– Mergers – cost reduction, synergies• Productivity analysis– Impact of deregulation– Impact of public infrastructure– Impact of non-market production externalitiesSpring 2005 Economics 220C 4Functions• Production– Output = f(inputs, technical efficiency)•Cost – Dual to production, assuming cost minimization given output– Cost = f(output, prices, technical efficiency)•Profit– Profit = Revenue - cost function = f(output, prices, technical efficiency)– Similar to cost function, unless a demand model used to construct revenue functionSpring 2005 Economics 220C 5ProductionStart with Cobb-Douglas for firms (or plants) indexed by iProperties of estimator of (α,β) depend on the relationship between inputs and disturbance.Why is this very simple form useful?– First order log-log approx., constant elasticity• Identification of higher orders sometimes difficult– Corresponds to growth accounting framework– Easy to add additional inputsor where iiiiii i iiiQALKqalkaαβαβµε==+ +=+Spring 2005 Economics 220C 6Drawbacks to Cobb-Douglas• Elasticity of substitution always one• All tech change is neutral• Multiproduct firms – merger and antitrust analysis– Cost synergies of interest– Explore subadditivity of cost functionSpring 2005 Economics 220C 7Some alternativesflexible3--Generalized Leontieff (dual)flexible35 (8 with t)translogσ=1/(1+ρ) for all inputs23CES1for all inputs12Cobb-DouglasElasticity of substitution# params if CRS, symmetry imposed# params (2 inputs)Functional formSpring 2005 Economics 220C 8Variances in productivity• Empirical facts1.Large variance in productivity aiacross firms2.Productivities highly correlated over time (within firm)• Suggests that input choices might depend on the disturbance • Sources of dependence– True technology or management differences– Measurement error (inputs or outputs)– External factors (weather, strikes, breakdowns, etc.)• How do input choices react to these shocks?Spring 2005 Economics 220C 9Panel production function• Now assume we have several periods of data for each firm– Add time dummies– Consider two types of transitory error (transmitted and not transmitted) where uit t it it itit i it itqlkueδαβαη=+ + +=++Spring 2005 Economics 220C 10Production function errorWhat’s in uit = αi+εit= αi+ηit+eit?– αi= “permanent” differences in firm productivity (perhaps due to market power or varying product mix), known to firm when it chooses both variable and fixed inputs.– ηit = transitory differences in firm productivity (due to demand or supply shocks), known to firm when it chooses variable inputs, but not fixed (capital) inputs.– eit= transitory measurement error (the econometrician’s problem, but not the firm’s).Spring 2005 Economics 220C 11Measurement problems• Production: – Output usually sales (turnover or revenue) divided by a price index• Most plants and firms have multiple output types• Same price for different firms with different product mix• For individual firms, reinterpret result as revenus productivity– Labor input usually hours or person-years• No quality adjustment, although some exceptions– Capital aggregates investment of different types at different times using simple depreciation models.• Errors in quantity measurement usually mean errors in corresponding price (dual forms)Spring 2005 Economics 220C 12Endogeneity• If inputs respond to shocks (ηitor αi), OLS estimates will be biased– more serious for inputs that adjust quickly like labor and materials• Some solutions– Use panels and try to remove αi(more later)– Find instruments• Lagged values of inputs problematic given serial correlation• Prices if you can find variance across firms unrelated to disturbanceSpring 2005 Economics 220C 13ExampleSelected large U.S. manufacturing firms, 10 years of data from 1986 to 1995.– y = log sales – output measure– l = log employment – labor measure– k = log gross P&E – capital measureyit= λt+ αkit+ βlit+ uitSubtracting labor from both sides of the eq provides an easy test for scale economies:yit -lit= λt+ α(kit–lit)+ (α+β-1)lit+ uituit= αi+εitSpring 2005 Economics 220C 14Log salesSelected U.S. Manufacturing Firms 1986-1995Log employment-50 50510AmCyAmCyAmCyAmCyAmCyAmCyAmCyAmCyB&JB&JB&JB&JB&JB&JB&JB&JB&JB&JChevChevChevChevChevChevChevChevChevChevCokeCokeCokeCokeCokeCokeCokeCokeCokeCokeH-DH-DH-DH-DH-DH-DH-DH-DH-DH-DPlazaPlazaPlazaPlazaPlazaPlazaP&GP&GP&GP&GP&GP&GP&GP&GP&GP&GStrikerStrikerStrikerStrikerStrikerSpring 2005 Economics 220C 15Log salesSelected U.S. Manufacturing Firms 1986-1995Log capital-50 5 100510AmCyAmCyAmCyAmCyAmCyAmCyAmCyAmCyB&JB&JB&JB&JB&JB&JB&JB&JB&JB&JChevChevChevChevChevChevChevChevChevChevCokeCokeCokeCokeCokeCokeCokeCokeCokeCokeH-DH-DH-DH-DH-DH-DH-DH-DH-DH-DPlazaPlazaPlazaPlazaPlazaPlazaP&GP&GP&GP&GP&GP&GP&GP&GP&GP&GStrikerStrikerStrikerStrikerStrikerSpring 2005 Economics 220C 16Log output-labor ratioSelected U.S. Manufacturing Firms 1986-1995Log capital-labor ratio24 6 84567AmCyAmCyAmCyAmCyAmCyAmCyAmCyAmCyB&JB&JB&JB&JB&JB&JB&JB&JB&JB&JChevChevChevChevChevChevChevChevChevChevCokeCokeCokeCokeCokeCokeCokeCokeCokeCokeH-DH-DH-DH-DH-DH-DH-DH-DH-DH-DPlazaPlazaPlazaPlazaPlazaPlazaP&GP&GP&GP&GP&GP&GP&GP&GP&GP&GStrikerStrikerStrikerStrikerStrikerSpring 2005 Economics 220C 17Panel data estimatorsyit-yi,t-1= (Xit –Xi,t-1)β + uit-ui,t-1or ∆yit= ∆xitβ + ∆uitFirst differences (FE)yit=a + Xitβ + uit=a + Xitβ + αi+εitVar(uit) = σα2+σε2Variance components (RE)yit-yi= (Xit-Xi)β + (uit-ui)Within (FE)yi= a + Xiβ + ui where i subscript denotes firm meansBetweenyit= a + Xitβ + uitTotalSpring 2005 Economics 220C 18OLS Production function estimates.205.319.916.598.549R22.123 (<1.00)0.104 (.000)0.828 (.000)--0.182 (.000)Durbin-Watson.132.411.150.293.329s.e.0.176