Ex Ante vs by 0co00L2

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									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 + 212r11/2r21/2 + 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 + trt] + 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 + trt] + 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
Cy       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

								
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