Producer Theory

Question 1

When is a firm a cost-minimiser?

Answer 1

If it produces any given output level y  0 at the smallest possible total cost.

Question 2

What is c(y)?

Answer 2

C(y) denotes the firm’s smallest possible total cost for producing y units of output.
C(y) is the firm’s total cost function.

Question 3

What is the cost-minimisation problem?

Answer 3

• Consider a firm using 2 inputs to make one output
• The production function is: Y = f(x1, x2)
• Take the output level y  0 as given
• Given the input prices w1 and w2, the cost of an input bundle (x1, x2) is: w1x1 + w2x2
  For given w1, w2 and y, the firm’s cost-minimisation problem is to solve:
  Min x1,x2  0 w1x1 +w2x2
  Subject to f(x1, x2) = y

Question 4

What are the conditional demands for inputs 1 and 2 (with regards to the cost-minimisation problem)?

Answer 4

x1(w1, w2, y) and x2(w1, w2, y)

The (smallest possible) total cost for producing y output units is therefore:
C(w1, w2, y) = w1x1(w1, w2, y) + w2x2(w1, w2, y)

Question 5

What is an iso-cost curve?

Answer 5

A curve that contains all of the input bundles that cost the same amount.
e.g. given w1 and w2, the $100 iso-cost line has the equation: w1x1 + w2x2 = 100

WHEN GRAPHING:
Vertical axis: x2
Horizontal axis: x1

Question 6

What is the slope of an iso-cost curve?

Answer 6

Slope = -w1/w2

Question 7

What is a firm’s average total production costs?

Answer 7

For positive levels y, a firm’s average total cost of producing y units is:
AC(w1, w2, y) = c(w1, w2, y)/y

Question 8

What is the long-run cost minimisation problem?

Answer 8

Min x1,x2  0 w1x1 +w2x2

Subject to f(x1, x2) = y

Question 9

What is the short-run cost minimisation problem?

Answer 9

Min x1,x2 >/ 0 w1x1 +w2x’2
Subject to f(x1, x’2) = y

The SR cost minimisation problem is the LR problem subject to the extra constraint that x2 = x’2

Question 10

Explain the relationship between cost minimisation and the lagrange.

Answer 10

Minimising cost subject to a production constraint yields the lagrangian and its FOCs:
Min L = wL + rK + λ[q – f(L, K)]
c1, c2, λ

∂L/∂L = 0 <=> w = λ(∂f/∂L)
∂L/∂K = 0 <=> r = λ(∂f/∂K)
∂L/∂λ = 0 <=> q = f(K,L)

Rearranging terms reveals:
w/r = (∂f/∂L)/(∂f/∂K) = MPL/MPK

Note: q is meant to be written as q-bar (but for neatness I left it as q)

Question 11

How do you compute the marginal productivity of capital?

Answer 11

By finding the partial derivative of the function (q or y) with respect to capital (K).

Question 12

How do you compute the marginal productivity of labour?

Answer 12

By finding the partial derivative of the function (q or y) with respect to labour (L).

Question 13

What is the point of output maximisation?

Answer 13

• The premise is that firms are always profit maximisers.
• The implicit assumption about output maximisation is that firms operate in perfectly competitive environments (the price of the output is given)
• For a given cost (C-bar) and a given output price (p), max output is equivalent to max profits.

Question 14

What happens to output maximisation in imperfectly competitive markets?

Answer 14

In imperfectly competitive markets, p is a function of output.
- For a given cost (c-bar), when the output price (p) is a function of output, max output is not longer equivalent to max profits.

Question 15

When does cost minimisation and output maximisation occur?

Answer 15

• Firms minimise costs in both perfectly and imperfectly competitive markets.
• However, only firms operating in perfectly competitive markets (price-taking firms) can maximise output.

Question 16

Explain the relationship between output maximisation and the lagrange.

Answer 16

Maximising output subject to a cost constraint yields the lagrangian and its FOCs:
Max L = f(L, K) + λ[C – wL -rK)
L, K, λ

∂L/∂L = 0 <=> λw = ∂f/∂L
∂L/∂K = 0 <=> λr = ∂f/∂K
∂L/∂λ = 0 <=> C = wL +rK

Rearranging terms reveals:
w/r = (∂f/∂L)/(∂f/∂K) = MPL/MPK

Note: c is meant to be written as c-bar (but for neatness I left it as c)

Question 17

Explain output maximisation as the dual of cost minimisation.

Answer 17

Output maximisation: max(K,L) subject to the cost constraint c-bar = rK + wL
Cost minimisation: min rK + wL subject to out being given by the production function q-bar = f(K,L)

The constraint in the max problem becomes the objective function in the min problem and the objective function in the max problem becomes the constraint in the min problem.

Finding the highest possible isoquant for a given iso-cost is the dual of finding the lowest possible iso-costs for a given isoquant.

Question 18

How do you compute the marginal productivity of capital?

Answer 18

By finding the partial derivative of the function (q or y) with respect to capital (K).

Question 19

How do you compute the marginal productivity of labour?

Answer 19

By finding the partial derivative of the function (q or y) with respect to labour (L).

Question 20

How do you find the MRTS between capital and labour?

Answer 20

By multiplying the following by -1:
The partial derivative of the function (q or y) with respect to labour (L) ÷
The partial dervative of the function (q or y) with respect to capital (K).

Question 21

What is the difference between using the lagrange for output maximisation and using the lagrange for cost minimisation.

Answer 21

Output maximisation is subject to a cost constraint:
Max L = f(L, K) + λ[C – wL -rK)
L, K, λ

Cost minimisation is subject to production constraint:
Min L = wL + rK + λ[q – f(L, K)]
c1, c2, λ