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# Aniko Oery University of California, Berkeley Section 9: Cost minimization and returns to scale Econ 100A, MICRO-ECONOMIC ANALYSIS As a first step of solving the profit-maximization problem of the firm, the firm needs to find the cheapest input combination to produce a specific output goal. Therefore, it solves the costminimization problem min L,K C(L, K) subject to f(L, K) = Q0 for a output goal Q0, where f(L, K) is the production function we introduced last time. The solution to this problem will depend on the value of Q0 and it will give us the cost function C(Q0). In this discussion section we will review how to solve the cost minimization problem. We will also briefly talk about retruns to scale. 1 The cost minimization problem Note first that the cost that the firm faces is given by C(L, K) = wL + rK where w is the price for input L and r is the price for input K. The level curves of this cost function are called iso-cost lines and describe input combinations that result in the same cost. Let us define an output goal Q0 and let us fix the isoquant that corresponds to this output level Q0, that is f(L, K) = Q0. In order to solve the cost minimization problem we only need to find the lowest iso-cost line, that touches this isoquant. As in the consumer choice theory, there are again two types of solutions: inner solutions and corner solutions. The tangency condition (TC) is given by MRT S ? MPL MPK = w r and the budget line condition is replaced by f(L, K) = Q0. Hence, the optimization problem again turns out to be equivalent to solving a system of two equations! 2 Returns to scale Returns to scale just indicate whether an increase in all inputs increases the output proportionally. If the output increases less than proportionally, that is if f(? · L, ? · K) < ? · f(L, K) for ? > 1, then we have decreasing returns to scale. If the output increases more than proportionally, that is f(? · L, ? · K) > ? · f(L, K), then we have increasing returns to scale. If output just increases proportionally to the increase in inputs, that is f(? · L, ? · K) = ? · f(L, K), then we have constant returns to scale. The measure of returns to scale is given by RT S = %?Q %? ALL inputs = ?Q Q ? ALL inputs ALL inputs . For the one input case we have RT S = MPL APL , where APL = f(L) L is the average product of labor. 1 Aniko Oery University of California, Berkeley 3 Practice Problems from the 3rd edition of the textbook ”Microeconomics” by Bersanko and Braeutigam 1. Suppose the production of airframes is characterized by a Cobb-Douglas production function: Q = LK. The marginal products for this production function are MPL = K and MPK = L. Suppose the price of labor is 10 dollars per unit and the price of capital is 1 dollar per unit. Find the cost-minimizing combination of labor and capital if the manifacturer wants to produce 121,000 airframes. 2. A manufacturing firm’s production function is Q = KL + K + L. For this production function, MPL = K + 1 and MPK = L + 1. Suppose that the price r of capital services is equal to 1, and let w denote the price of labor services. If the firm is required to produce 5 units of output, for what values of w would a cost-minimizing firm use a) only labor? b) only capital? c)both labor and capital? 3. Consider the CES production function given by Q = (K0.5 + L 0.5 ) 2 . a) Qhat is the elasticity of substitution for this production function? b) Does this production function exhibit increasing, decreasing, or constant returns to scale? c) Suppose that the production function took the form Q = (100 + K0.5 + L 0.5 ). Does this production function exhibit increasing, decreasing, or constant returns to scale? 4. A firm’s production function is initially Q = ? KL, with MPK = 0.5 ? ? L K and MPL = 0.5 ? ? K L . Over time the production function changes to Q = KL, with MPK = L and MPL = K. a) Verify that this change represents technological progress. b) Is this change labor saving, capital saving, or neutral?

Aniko Oery University of California, Berkeley

Section 9: Cost minimization and returns to scale

Econ 100A, MICRO-ECONOMIC ANALYSIS

As a first step of solving the profit-maximization problem of the firm, the firm needs to find

the cheapest input combination to produce a specific output goal. Therefore, it solves the costminimization

problem

min

L,K

C(L, K) subject to f(L, K) = Q0

for a output goal Q0, where f(L, K) is the production function we introduced last time. The

solution to this problem will depend on the value of Q0 and it will give us the cost function

C(Q0). In this discussion section we will review how to solve the cost minimization problem. We

will also briefly talk about retruns to scale.

1 The cost minimization problem

Note first that the cost that the firm faces is given by

C(L, K) = wL + rK

where w is the price for input L and r is the price for input K. The level curves of this cost

function are called iso-cost lines and describe input combinations that result in the same cost.

Let us define an output goal Q0 and let us fix the isoquant that corresponds to this output level

Q0, that is f(L, K) = Q0. In order to solve the cost minimization problem we only need to find

the lowest iso-cost line, that touches this isoquant.

As in the consumer choice theory, there are again two types of solutions: inner solutions and

corner solutions. The tangency condition (TC) is given by

MRT S ?

MPL

MPK

=

w

r

and the budget line condition is replaced by f(L, K) = Q0. Hence, the optimization problem

again turns out to be equivalent to solving a system of two equations!

2 Returns to scale

Returns to scale just indicate whether an increase in all inputs increases the output proportionally.

If the output increases less than proportionally, that is if f(? · L, ? · K) < ? · f(L, K) for ? > 1,

then we have decreasing returns to scale. If the output increases more than proportionally,

that is f(? · L, ? · K) > ? · f(L, K), then we have increasing returns to scale. If output just

increases proportionally to the increase in inputs, that is f(? · L, ? · K) = ? · f(L, K), then we

have constant returns to scale.

The measure of returns to scale is given by

RT S =

%?Q

%? ALL inputs =

?Q

Q

? ALL inputs

ALL inputs

.

For the one input case we have RT S =

MPL

APL

, where APL =

f(L)

L

is the average product of labor.

1

Aniko Oery University of California, Berkeley

3 Practice Problems from the 3rd edition of the textbook

”Microeconomics” by Bersanko and Braeutigam

1. Suppose the production of airframes is characterized by a Cobb-Douglas production function:

Q = LK. The marginal products for this production function are MPL = K and MPK = L.

Suppose the price of labor is 10 dollars per unit and the price of capital is 1 dollar per unit. Find

the cost-minimizing combination of labor and capital if the manifacturer wants to produce 121,000

airframes.

2. A manufacturing firm’s production function is Q = KL + K + L. For this production function,

MPL = K + 1 and MPK = L + 1. Suppose that the price r of capital services is equal to 1, and

let w denote the price of labor services. If the firm is required to produce 5 units of output, for

what values of w would a cost-minimizing firm use

a) only labor?

b) only capital?

c)both labor and capital?

3. Consider the CES production function given by Q = (K0.5 + L

0.5

)

2

.

a) Qhat is the elasticity of substitution for this production function?

b) Does this production function exhibit increasing, decreasing, or constant returns to scale?

c) Suppose that the production function took the form Q = (100 + K0.5 + L

0.5

). Does this production

function exhibit increasing, decreasing, or constant returns to scale?

4. A firm’s production function is initially Q =

?

KL, with MPK = 0.5

?

?

L

K

and MPL = 0.5

?

?

K

L

.

Over time the production function changes to Q = KL, with MPK = L and MPL = K.

a) Verify that this change represents technological progress.

b) Is this change labor saving, capital saving, or neutral?

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