Comparative Statics and Duality of the Cost Function

1. Comparative Statics and Duality of the Cost Function Lecture VII

2. Comparative Statics • Comparative statics with respect to changes in input prices. • The most common results of the comparative statics with respect to input prices involve intuition about derived demand functions. • From the primal approach, we expect the demand functions for each input to be downward sloping with respect to input prices.

3. Starting from the cost function: By Shephard’s lemma. • In addition, we know that if i=j then by the concavity of the cost function in input prices:

4. In addition, we know that by Young’s theorem, the Hessian matrix for the cost function is symmetric: • The Hessian matrix for the cost function is also singular.

5. Euler’s Theorem: Euler’s theorem is based on the definition of homogeneity: Differentiating both sides with respect to t and applying the chain rule yields:

6. Letting t=1 then yields • Coupling this result with the observation that if a function is homogeneous of degree r, then its derivative is homogeneous of degree r-1. We know that the input demand functions are homogeneous of degree zero in prices. Thus,

7. Multiplying this expression by xi*(w,y) yields • Given that we know that eii< 0, this result imposes restrictions on the cross-price elasticities.

8. Briefly, let us prove the homogeneity of the marginal cost function. Starting with the marginal cost, differentiate with respect to each price:

9. Comparative statics with respect to output levels. • Following from the restrictions on the cross price elasticities above, the comparative statics with respect to output levels imply that not all inputs can be inferior or regressive. • An inferior input is an input whose use declines as production increases while the use of a normal input increases as production increases.

10. Like the cross-price results above, we start with the sum of the differences of individual demand functions with respect to the level of output:

11. In order to develop the effect of output on total cost, we start with the original Lagrangian from the primal problem: Solving for the output using the first-order condition on the Lagrange multiplier and differentiating the solution then yields:

12. From the first set of first-order conditions, we see

13. Which proves the definition of l consistent with the envelope theory. Therefore, • Given this optimum l, we can then sum over the initial first-order conditions:

14. Introducing the notion of an elasticity of scale:

15. Chambers defines as the cost flexibility (the ratio between the marginal and average costs).

16. Remember the elasticity of scale in the production function: which is the ratio between the marginal physical and average physical products.

17. What is developed here is not quite the same, but is actually the elasticity of size. • It answers the question: Do I build one large plant or several small ones? • Defining y*=y/m is related to the homogeneity of the cost function in terms of scale. Taking

18. If n(y*,w) > 1 then e(y,x(w,y)) < 1 there are no efficiencies to centralization {diseconomies of scale}. • If n(y*,w) < 1 then e(y,x(w,y)) > 1 there are efficiencies to centralization {economies of scale}.

20. Just like the MPP-APP comparison, we can envision the ratio be marginal cost and average cost. It is clear that • Also, it is apparent that average cost equals marginal cost at the minimum of the average cost curve:

21. Duality between the Cost and Production Functions • In our discussion of the primal we demonstrated how the production function placed restrictions on economic behavior. • The question posed in duality is whether the optimizing behavior can be used to recover or reconstruct the properties of the production function.

22. Minkowski’s Theorem: A closed, convex set is the intersection of half-spaces that support it: • The half space H(m,k) is defined as

23. Thus, based on the definition of a cost function we have: This definition actually recovers the original production set:

24. If V*(y)=V(y), the original technology can be recovered.