Sep 9, 2010 ... Sensitivity to units makes interpretation impossible. Need unit free measure of
substitution. Following RGD Allen (1938), John Hicks (1932).
1 Miscellaneous Notes on Production Economics Compiled by Peter F. Orazem September 9, 2010 I. Implications of convex isoquants Two input case , along an isoquant 0 along an isoquant Slope of the isoquant = , , MRTS is the marginal rate of technical substitution A convex shape requires that inputs have diminishing marginal productivity, the necessary assumption for downward sloping derived demand for factor inputs. This requires that as we move down an isoquant toward more x1 and less x2, f1 falls and f2 increases, and so
0.
x2
Slope =
x1
Note that
, ,
; and by Young’s theorem,
2 If term in brackets < 0, then
?
0 (I.1)
0 implying convex isoquants and diminishing marginal rate
of technical substitution. Diminishing MRTS is consistent with stage 2 of production in 1 input case.
2
II. Elasticities of substitution is a measure of substitutability, but it is sensitive to units of the inputs. Example: Cobb-Douglas ;
K measured in horsepower or dollars changes slope, as does L measured in hours, weeks, … Sensitivity to units makes interpretation impossible Need unit free measure of substitution. Following RGD Allen (1938), John Hicks (1932) ,
%∆
,
%∆
, %∆
%∆
2
,
·
· ·
?
· 0
·
(II.1)
This is symmetric and unit free If 2 indifferences are convex.
0, then
,
0 which implies that x1 and x2 are substitutes if
With two inputs, diminishing returns, and either cost minimization or profit maximization, the two inputs have to be substitutes. With more than two inputs, complements are possible but at least one pair of inputs has to be substitutes as shown below.
3 III. Generalization of the elasticity of substitution The general form of the elasticity of substitution ·
|
| | |
(III.1)
Where |Fij| is the i,j cofactor of F, the Hessian matrix of the production function Estimating the elasticity of substitution using the primal is complicated because it requires both first and second order derivatives of the unknown production function to estimate the elements of the matrix F. 0 f1 f1 f11 F= f2 f12 . . fn f1n
f2 …. fn f12 …. f1n f22 …. f2n f2n
….
fnn
Special case of constant returns to scale (CRS) The general form of the elasticity of substitution is unwieldy, and so we often impose restrictions that simplify the form. Output elasticities sum to 1 1 which means that (III.2) With constant returns, each factor is paid its marginal contribution to production. To see this, multiply both sides by output price p
Total revenue pq is equal to each factor’s marginal revenue product (equal to input price at optimum) times the amount of the input. Using (III.2), we can define factor shares without knowing input or output prices by / /
(III.3)
And so with constant returns to scale, factor shares equal the ratio of the output elasticity to the scale elasticity.
4 The imposition of constant returns to scale simplifies the estimation of the elements of F somewhat, although it is difficult to demonstrate that without using a particular specification which we will demonstrate with the translog production function below. From the general form above, we can write the CRS form of the elasticity of substitution as ·
|
| | |
Which does not simplify much. However, in the two input form, we can show some substantial simplifications. ·
(III.4)
This can be further simplified by imposing the restrictions implied by implies 0 0 0
which
0
Inserting these into the denominator of the second term in (III.4), we have 2 Placing this into the CRS form of the elasticity of substitution, we have ·
(III.5)
IV. Translog Production Function To estimate the elements of matrix F, we need first and second partials of the production function. The translog is a second-order approximation to an unknown production function q = f(x1, x2, …, xn) ln
∑ ∑
∑
(IV.1)
First derivatives of (IV.1) will yield output elasticities ∑ So
(IV.2)
5 Second derivatives of (IV.1) are of the form
Rearranging:
And so ·
·
Similarly ·
1
If you wish, you can populate the Hessian matrix of the production function, F, using the estimated first and second partial derivatives of f(•) and compute the elasticities of substitution The CRS form of the translog production function imposes the restriction that the output elasticities must add up to 1. Inspection of (IV.2) reveals that the restrictions implied by CRS are ∑ 1; ∑ 0 . We can also impose symmetry so that . Often the restriction improve the precision of the estimates by reducing collinearity among the first- and second-order terms and reducing the number of parameters to be estimated.
6 V. Generalized Cost Minimization Assume the firm wants to produce a level of output the firm is a price taker in input markets. The firms objective is to choose inputs x1 through xn so as to minimize the cost of producing . ,
,…,
First order conditions: 0 . . . 0 ,
,…,
0
For inputs for which equality holds, long-run optimum for any two inputs xi and xj
Firm will set marginal cost of all inputs equal so that , . Assume all first order conditions hold with equality. Totally differentiating, we get 0 . . . 0 Rearranging and applying Young’s theorem dλ
-
-f1 -λf11 -λf12 …. -λf1n
dx1
-dw1
-f2 -λf12 -λf22 …. –λf2n . . . -fn -λf1n -λfn2 …. –λfnn
dx2
0
-f1
-f2
H
….
–fn
dxn
=
-dw2
-dwn
0
7 The second order condition for cost minimum is that |H| < 0 Using the implicit function rule, |H|≠0 implies that there are well defined reduced form equations ruling input demand of the form , , ,…, , , ,…, . . . , , ,…, If there are multiple constraints, the condition for a minimum is sgn|H| =(-1)m where m is the number of constraints. In two input case, H is 0 -f1 -f2 -f1 -λf11 -λf12 -f2 -λf12 -λf22
2
