This document contains 30 questions related to producer theory and the theory of the firm. The questions cover various topics including production functions, cost minimization, profit maximization, elasticities, and comparative statics. Students are asked to analyze production functions, solve optimization problems, examine how input demands and costs change in response to price changes, and prove various relationships between theoretical concepts.

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2. Producer Theory Practice Set 2
V.Bardis
2. Using each of the production functions above, solve the profix maximization problem of
the firm assuming the firm is a price taker in all markets, that is, find the firm's market input
demands, supply and maximum profit function. If a solution does not exist for some (or all)
values of the parameters, explain why.
1. For each of the production functions given below where q denotes output and x=(x1,x2) is the input
vector
(a) calculate the elasticity of scale and determine whether the function exhibits decreasing, constant or
increasing returns to scale;
(b) find the elasticity of substitution.

3. 3. Consider the functions given below where p and wi denote the output price and price of input i
respectively. Determine whether each function is a (maximum value of) profit function. If it is, fin d the
firm's supply and market input demands and, if possible, `recover' the firm's cost function, conditi onal
input demands and its production function.
4. Assuming the Hessian matrix of the firm's production function is negative definite, examine how a n
increase in the price of one of its inputs will affect the demand for each input and the output supp lied.
5. For each of the production functions in question 1, solve the cost minimization problem of the firm,
that is, find the conditional input demands and the (minimum value of) cost function.
6. Assuming the firm's production function is strictly quasi-concave, carry out the comparative statics
with respect to the price on input 1 in the context of cost mininimization, that is, determine how an
increase in the price will affect the firm's conditional input demands and its cost of production.
a)
b)
c)
d)
7. For each of the functions given below where wi is the price of the input i and q denotes output, answer
the following:
a) Determine whether the function is a firm's (minimum value of) cost function and, if it is,
b) find and compare the firm's marginal cost and average cost and determine the type of returns-to-scale;
c) find the firm's conditional input demands;
d) find the firm's production function and calculate the elasticity of scale to confirm your result in part (b).
1)
2)
3)

4. 8. State and prove the equation that relates the slopes of the Market and Conditional Input demands of
a price-taking firm, that is, decompose the total effect of an input price change into an "output effect"
and a "substitution effect". Restate the equation in its elasticity form. Where applicable, confirm that
the equation holds for each of the functions given in question 1.
9. Prove that the price (point) elasticities of the firm's (i) market demand for an input and (ii) conditional
input demand for an input must sum to zero.
11. Show that if the production function is H.O.D(k), then the cost function is H.O.D(1/k) in output (q).
13. In the context of the cost minimizing firm, define the term "substitution effect". Can the substitution
effect be positive? Prove it.
10. Can the (mininum value of) cost decrease as output increases (i.e., can marginal cost be negativ e)?
Prove it.
15. Other things equal, can marginal cost
(i) be greater in long-run than in the short run? Prove it.
(ii) be steepter in the long-run than in the short run? Prove it.
12. Show that if the production function of a (multi-input) firm exhibits decreasing returns, its marginal cost
must exceed its average cost. (To simplify, assume two inputs and that the production function is strictly
quasi-concave with |BH|>0.)
14. Define inferior input and analyze the output effect of an increase in the price of such an input.
Illustrate with the help of well-defined diagrams.
16. Suppose a firm can produce output by using plant 1 or plant 2 or both. In each of the four cases
below, find the firm's minimum value of cost as a function of total output q given the cost functions of
the individual plants. In each case, identify "short-run cost" versus "long-run cost" and confirm whether
the claim in question 15 holds.
a)
b)
c)
d)
17. If a (two-input) production function is homothetic and differentiable, must the elasticity of
substitution be constant? If "yes", prove it. If "no", provide a fully worked out counterexample. What if
the function is homogeneous?

5. 18. Assuming that the cost function (of a multi-input firm) is twice-differentiable, show that average cost
must equal marginal cost when average cost is minimum. What condition is required for average cost to
have a minimum point? What is the sufficient condition for the minimum point of AC to be unique?
19. Consider two firms which use the same two inputs and face the same input prices. The production
function of firm 1 is function (1) (Cobb-Douglas) in question 1 such that a=1, b=1 and A=1.
The production function of firm 2 is function (4) (C.E.S) in question 1 such that gamma=1/2, a=b=A=1
and r = -1. Each firm is a cost minimizer.
a) Assuming input prices are initially equal, determine which firm will experience the greater percent
increase in the total cost of one unit of output if the price of input 1 increases by 5%, other things
equal.
21. True or False: If the three-input production function q=f(x1,x2,x3) is twice-continuously
differentiable and its Hessian matrix is negative definite (and so f is strictly concave), then " f13 >0 "
implies that inputs 1 and 3 are gross complements.
20. True or False: If the two-input production function q=f(x1,x2) is twice-continuously differentiable and
its Hessian matrix is negative definite (and so f is strictly concave), then " f12 >0 " implies that inputs 1
and 2 are gross complements.
22. For the production function given below
a) Find the elasticity of scale and use it to predict the relation between average cost and marginal
cost.
b) Show that the function is quasi-concave.
c) Find the condtional input demands and the cost function.
Next, assuming the price of input 1 is 1 and the price of input 2 is 2,
d) plot the "output expansion path" and
e) plot the AC and MC curves
so that the plots reflect your results in part (c).
23. Find the elasticity of substitution of the production function given in question 22 (assuming both
inputs' values are positive, otherwise the elasticity is not defined) and compare it to the elasticity of
substitution of the Cobb-Douglas function. Which function has greater "ease" of substitution?
Suppose, firm 1 and firm 2 produce the same good and face the same input prices but firm 1's
production function is the function above, whereas firm 2's production function is the Cobb-Douglas
function q = x1 x2 , where q denotes output. If the price of input 1 increases, which firm do you expect to
experience the greater percent cost increase, assuming both produce the same amount of the good
and use positive amounts of both inputs ? One would think that the easier it is to substiture i.e., to
move away from the input that becomes more expensive, the smaller will be the (percent) cost increase
resulting from the rise in the input's price.
Calculate the percent cost increase for each firm assuming q = 9, w1 = 1 and w2 = 16 and then the price
of input 1 changes to 2, i.e., w1' = 2. Is the intuition stated above confirmed? If not, why do you think
that is?

6. 26. True or False: The product of the derivative of the conditional demand of input i with respect to
output and the derivative of the supply function with respect to the price of the good supplied is
equal to the derivative of the market demand for input i with respect to the price of output.
Firm 1 Firm 2
24. Find the market demand for input L (labour) of each of the two firms below assuming the two firms
operate in the same market, both are price takers in all market and each firm is a profit maximizer.
Determine which demand is more elastic with respect to the price of input L, other things equal.
25. For the two firms in question 24, find the output supply of each of each firm and determine which
supply is more elastic with respect to the price of output, other things equal.
27. Show that the "gross" substitution matrix of a firm with two inputs is negative definite and
discuss the meaning of this result. (You may assume that the function is twice-continuously
differentiable and strictly concave with |H|>0 and |BH|>0.)
28. Show that the substitution matrix of a firm with two inputs is negative definite and discuss the
meaning of this result. (You may assume that the function is twice-continuously differentiable and
strictly concave with |H|>0 and |BH|>0.)
29. True or False: In the context of the standard theory of the firm, if an input is inferior (normal), then
demand for the input must decrease (increase) when the price of output increases.
30. Derive the aggregate production function in an industry with N price-taking firms such that each
firm uses two inputs, L and K, and has the same production function, function (1) in question 1 with
a=b=1/4 and A=1.

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