VIEWS: 6 PAGES: 16 POSTED ON: 6/9/2012 Public Domain
Ex Ante vs. Ex post cost functions, Pope and Just, J. of Econometrics, 1996 Builds from Pope’s (1982) and Taylor’s (1989) critique that dual methods may not be robust when considered under uncertainty. Without uncertainty, we use the information of the agents’ objective along with r, x, and y to clearly uncover the technology constraint. What happens when we add a random element in production Central Assumption: y = f(x, ) - where is random disturbance in the production process - most likely a weather shock - all inputs are chosen before the revelation of the shock 1. Traditional or Ex post cost functions What data is used in estimating a traditional, ex-post cost function? r, x, y = y + which yields c(r,y,) = minx {r’x | x v(y, )} - where r is a price vector -where v(y, ) are all points satisfying y f(x,) we define the technology, via the input requirement set, as if it we had a restricted input requirement set in which we knew the mystery input of What is the objective that a cost-minimizing producer really solves? minx r’x s.t. E[f(x, )] y minx {r’x | x v( y )} = c~(r, y ) Ex-ante Cost function where the producer uses the information r, f(x, ) and E[] to solve the problem. What is truly needed, then, is to have at our disposal y during the estimation process so that estimation could take place as normal. - generally speaking, we don’t know y so this simple fix doesn’t work What Happens if when we ignore this problem and use realized output instead of y ? Pope and Just work this through for an ex-ante translog cost function: ln c~(r, y ) = 0 + r ln r + g ln y + ½ rr (ln r)2 + gg ½ (ln y )2 + gr (ln r) (ln y ) substituting y = y - yields: ln c~(r, y ) = 0 + r ln r + g [ln y - ln ] + ½ rr (ln r)2 + gg [½ (ln y)2 + ln y ln + ½ (ln )2] + gr [ln r ln y + ln r ln ] simplifying yields Pope and Justs’ equation (8a): ln c~(r, y ) = 0 + r ln r + g ln y + ½ rr (ln r)2 + ½ gg (ln y)2 + gr ln r ln y + 0 where 0 = (- g - gr ln r - gg ln y + gg ln ) ln Using Shephards Lemma yields cost shares: S ln c~/ ln r = r rr ln r + gr ln y = r rr ln r + gr ln y - gr ln (8b) = r rr ln r + gr ln y + 1 where 1 = - gr ln Typically one would then use GLS to estimate: (8a) ln c~(r, y ) = 0 + r ln(r) + g ln (y) + ½ rr (ln r)2 + ½ gg (ln y)2 + grln(r) ln(y) + 0 + u0 (8b) ln c~/ ln r = r rr ln r + gr ln y + 1 + u1 where u0 and u1 are econometric (measurement) errors. What’s wrong with this approach? appears in 0, 1 and y, - hence regressors are correlated to errors o leads to bias o the bias does not disappear if we add more observations leads to inconsistent estimates Wouldn’t some type of instrumental variable approach circumvent these problems? Need something correlated to y but not to 0 or 1 Available instruments include r and x, both of which are correlated to 0 Using x as an instrument in estimating only the share equations (8b) would yield consistent estimates of those parameters - still inefficient because ignoring structural information - can’t recover all parameters of cost function because they don’t all show up in the share equation Pope and Just’s Solution Use the observed x’s, y’s and r’s along with our assumption concerning the parametric form of the cost function to calculate y . - Then we have y and can use this to estimate the parameters of the cost function. - the value of y is a byproduct of the cost function estimation procedure. Requires us to ask, what was the agent trying to produce (i.e., what was her y )? Suppose we assumed the agent’s technology was encapsulated by the following ex ante cost function (Generalized Leontief): c~(rt, y t; B) = ( y ) (rt1/2)T rt1/2 where () is some function of y and, for the two input case: rt = [r11/2 r21/2]T is a vector of the square roots of prices = [11 12, 12 22] is a matrix of parameters For example: = ln( y )[11r1 + 212r11/2r21/2 + 22r2] Assuming efficient production and nonstochastic input prices, we can solve for the output the producer expected ( y ) if we know the x’s, r’s and parameters because: c~(rt, y t; B) = rTx Or, more formally, Just and Pope embed this in a distance function argument that would also accommodate unobserved prices: ~ y max y | min 1 c ( r , y ; B ) r T x 1 y r This simply adds one more search. - If prices aren’t given to you, search over all prices until you find the set of prices that puts the observed inputs on the isoquant for that given level of expected output o i.e., choose prices to make sure the distance function related to your cost function equals 1 - Keep ratcheting up expected output until you can’t find a set of prices that will take you to the isoquant If the prices are observed, however, you don’t need to do this extra step Simplest example: Leontief technology y = min[1x1, 2x2] + - If I know the i’s and you told me what the xi’s were, I could tell you y quite easily. - If I knew behavior was cost-minimization, I would only need to know one of the xi’s Consider our two-price, generalized Leontief example. It’s a bit more complicated, but 1 y = 1 ( ) where is the largest characteristic root of the matrix: 11 12 x x1 / 2 x1 / 2 1 1 2 12 22 x1 / 2 x1 / 2 1 2 x2 If (y) = y and 12 = 0, then this simplifies to a Leontief function and we choose the largest diagonal element of the above equation. To estimate, just plug y into the Generalized Leontief functional form: c~ = (-1(1/ ))(rt1/2)T rt1/2 = (1/ )(rt1/2)T rt1/2 and then minimize the sum of squared errors: min { S2 = t [ct – (1/ )(rt1/2)T rt1/2]2 } where is a nonlinear calculation done for each data point prior to the minimization step. One can apply nonlinear least squares to recover consistent parameter estimates for - but we don’t recover because it drops out when we plug in y . Just and Pope advise adding another primal estimation equation: 1 yt = 1 ( ) t t which turns the estimation into a nonlinear system with cross- equation restrictions – more difficult but feasible. Monte Carlo Exercise: They test four estimators Method 1: estimate ct = (1/ )(rt1/2)T rt1/2 + ut 1 (ex-ante) y t = 1 ( ) t t Method 2: estimate: ct = yt (rt1/2)T rt1/2 + ut (ex-post) Method 3: estimate: ct = y t (rt1/2)T rt1/2 + ut (infeasible single) Method 4: estimate ct = y t (rt1/2)T rt1/2 + ut (infeasible system) y t = y t t Monte Carlo Setup: Each synthetic data set has a sample size of 45 Assumed production - requires two inputs - technology has constant returns to scale (no (y), just y), o hence, only 3 parameters 500 repetitions per method drew random numbers from a uniform distribution to stand in for exogenous prices, and expected outputs drew serially independent normal random numbers to stand in for cost function disturbance (ut) and production disturbance (t) with no covariance between ut and t tried several different variance levels for the randomly generated ut and t Monte Carlo Results: Ex ante vs. Ex post – - both give biased estimates but the ex ante estimates are consistent - bias and mean squared error of ex ante estimates are always smaller, no matter what the underlying variances were - particularly, as the variance of t increases, yt becomes a worse proxy for y t and drives the bias of the traditional ex post method higher Application: Ex ante vs. Ex post Pope and Just’s apply the two methods to US agriculture using an aggregate, annual time series. They turn to McElroy’s AGEM system (Additive General Error Model from her 1987 JPE article): This model postulates: ct = (1/ )[(rt1/2)T rt1/2 + trt] + rt ut ˆ xt = (1/ )[(rt1/2)T ˆ ˆ rt -1/2 + t] + ut ˆ - where rt -1/2 is a diagonal matrix with elements ri,t-1/2 - the model features an adjustment of technical change Note the ut is now a disturbance or error in input demand choice that the agent pays for; - i.e., it reappears in the cost function multiplied by r. For the ex-ante method, they estimate the system of demand equations For the ex-post method, they estimate the cost function with observed output in place of expected output Results of Application Estimates vary greatly between two models and signs of the same parameters occasionally differ The ex ante approach yields a better fit to the data The ex ante approach yields input demand that are less elastic - - Pope and Just claim this is more intuitive and more consistent with recent empirical findings based on the profit function (Antle 1984) and with old primal estimate (Grilliches) than with recent ex post cost function estimates (Binswanger 1974) Giancarlo Moschini’s critique: J. of Econometrics 100(2001:February):357-380 Recall that Pope and Just estimate one of the two following systems: ct = (1/ )(rt1/2)T rt1/2 + ut 1 y t = 1 ( ) t t or ct = (1/ )[(rt1/2)T rt1/2 + trt] + rt ut ˆ xt = (1/ )[(rt1/2)T ˆ ˆ rt -1/2 + t] + ut Recall that is the maximal root of the matrix: 11 12 x x1 / 2 x1 / 2 1 1 2 12 22 x1 / 2 x1 / 2 1 2 x2 hence it can really be written as: (x) The same goes for . ˆ Hence, unless the x’s are exogenous, the expected output y t will not be exogenous and - both sets of estimating equations will be subject to simultaneity difficulties. - x appears on both sides of the equation What is Moschini’s solution? Broaden the scope and assume the more realistic assumption that producers are profit maximizers Use observed output price as an instrument for unobserved expected output - Can use assumption of functional form for ex ante cost to solve the p = MC relationship for y t - Plug this expression for y t instead of the Pope-Just expression that relies upon x’s His expressions are: y * = s(p,w; ) expected output where is a vector of parameters x = h( s(p,w; ), w; ) profit-max demands His Example: CES ex ante Cost function 1 /(1 ) 1 n n Cy i wi wi ei i 1 i 1 where wiei terms correspond to the AGEM from McElroy and used by Pope and Just: producers pay for their demand errors. Derived demands are: /(1 ) wi 1 n xi i y k wk ei k 1 Can solve for y t as a function of x’s just as Pope and Just did: / (1 ) n 1 / ( x e )( 1) / y g ( x e; ) i i i i 1 But can also invoke f.o.c. with respect to output (p=MC) and solve for expected output as a function of p: 1 /( 1) 1 /(1 )( 1) p n w1 y s( p, w; ) i i i 1 So he plugs this back into the expression for the xi’s: /( 1) [1 /(1 )][ /(1 )] p wi 1 n xi i k wk ei k 1 Monte Carlo Experiment four inputs 25 observations for each synthetic data set (this is varied) generate random disturbances for input demands and realized output chooses 2000 random generated data sets estimate 5 models - the infeasible model where input demands are estimated using expected output - the ex post estimation where input demands are estimated using observed instead of expected output - Pope and Just’s ex ante cost estimation using the expression where expected output is a function of the x’s o involves input demands and output equations - An extension of Pope and Just (Pope and Just, 1998 AJAE) using instrumental variables in an attempt to remove some simultaneity bias - Moschini’s approach that exploits p=MC to isolate an expression for expected output as a function of input and output price. Monte Carlo Results Moschini’s approach (of course) resulted in less bias and better fit than any other approach for all parameters Surprisingly, the simple ex post estimator did next best in terms of the ’s (cost share weights) and (elasticity of substitution), out performing the ex ante cost approaches In terms of the elasticity of scale parameter (), the ex ante approaches did second best