Lecture # 12aLecture # 12a

Costs and Cost MinimizationCosts and Cost Minimization

Lecturer: Martin ParedesLecturer: Martin Paredes

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1. Long-Run Cost Minimization (cont.) The constrained minimization problem Comparative statics Input Demands

2. Short Run Cost Minimization

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Example: Linear Production Function Suppose : Q(L,K) = 10L + 2K Suppose:

Q0 = 200

w = € 5r = € 2

Which is the cost-minimising choice for the firm?

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Example (cont.): Tangency condition

MRTSL,K = MPL = 10 = 5MPK 2

w = 5 r 2

So the tangency condition is not satisfied

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Example: Cost Minimisation: Corner Solution

L

K

Isoquant Q = Q0

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Example: Cost Minimisation: Corner Solution

L

K

Isoquant

Isocost line

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Example: Cost Minimisation: Corner Solution

L

K

Direction of decreasein total cost

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Example: Cost Minimisation: Corner Solution

L

K

Cost-minimising choice

•A

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A change in the relative price of inputs changes the slope of the isocost line.

Assuming a diminishing marginal rate of substitution, if there is an increase in the price of an input: The cost-minimising quantity of that

input will decrease The cost-minimising quantity of any

other input may increase or remain constant

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If only two inputs are used, capital and labour, and with a diminishing MRTSL,K :

1. An increase in the wage rate must:a. Decrease the cost-minimising quantity of

laborb. Increase the cost-minimising quantity of

capital.2. An increase in the price of capital must:

a. Decrease the cost-minimising quantity of capital

b. Increase the cost-minimising quantity of labor.

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Example: Change in the wage rate

L

K

Q0

0

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Example: Change in the wage rate

L

K

Q0•

0-w0/r

A

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Example: Change in the wage rate

L

K

Q0•

•

0-w0/r

-w1/r

A

B

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A change in output moves the isoquant constraint outwards.

Definition: An expansion path is the line that connects the cost-minimising input combinations as output varies, holding input prices constant

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Example: Expansion Path with Normal Inputs

L

K

TC0/w

TC0/r

Q0•

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Example: Expansion Path with Normal Inputs

L

K

TC0/w TC1/w

TC1/r

TC0/r

Q0

•

• Q1

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Example: Expansion Path with Normal Inputs

L

K

TC0/w TC1/w TC2/w

TC2/r

TC1/r

TC0/r

Q0

••

• Q1

Q2

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Example: Expansion Path with Normal Inputs

L

K

TC0/w TC1/w TC2/w

TC2/r

TC1/r

TC0/r

Q0

••

•

Expansion path

Q1

Q2

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As output increases, the quantity of input used may increase or decrease

Definitions: If the cost-minimising quantities of labour

and capital rise as output rises, labour and capital are normal inputs

If the cost-minimising quantity of an input decreases as the firm produces more output, the input is an inferior input

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Example: Labour as an Inferior Input

L

K

TC0/w

TC0/r

Q0

•

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Example: Labour as an Inferior Input

L

K

TC0/r

Q1•

•

Q0

TC0/w TC1/w

TC1/r

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Example: Labour as an Inferior Input

L

K

TC0/r

Q1•

•

Q0

TC0/w TC1/w

TC1/rExpansion path

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Definition: The input demand functions show the cost-minimising quantity of every input for various levels of output and input prices.

L = L*(Q,w,r)

K = K*(Q,w,r)

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Definition: The input demand curve shows the cost-minimising quantity of that input for various levels of its own price.

L = L*(Q0,w,r0)

K = K*(Q0,w0,r)

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0 L

K

L

Q = Q0

Example: Labor Demand

w

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0 L

K

L

•

• Q = Q0

w1/r

Example: Labor Demand

w

w1

L1

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0 L

K

L

••

• • Q = Q0

w1/rw2/r

Example: Labor Demand

w

w2

w1

L2 L1

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0 L

K

L

••

•

• • • Q = Q0

w1/rw2/r

w3/r

Example: Labor Demand

w

w3

w2

w1

L3 L2 L1

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0 L

K

L

w

••

•L*(Q0,w,r0)

• • • Q = Q0

L3 L2 L1

Example: Labor Demand

w1/rw2/r

w3/r

w3

w2

w1

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Example: Suppose : Q(L,K) = 50L0.5K0.5

Tangency condition MRTSL,K = MPL = K = w

MPK L r

=> K = w . L r

=> This is the equation for expansion path

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Example (cont.): Isoquant Constraint:

50L0.5K0.5 = Q0

=> 50L0.5(wL/r)0.5 = Q0

=> L*(Q,w,r) = Q . r 0.5 50 w

K*(Q,w,r) = Q . w 0.5 50 r

( )( )

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Example (cont.): So, for a Cobb-Douglas production function:

1. Labor and capital are both normal inputs2. Each input is a decreasing function of its

own price.3. Each input is an increasing function of

the price of the other input

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Definition: The firm’s short run cost minimization problem is to choose quantities of the variable inputs so as to minimize total costs… given that the firm wants to produce an

output level Q0

under the constraint that the quantities of some factors are fixed (i.e. cannot be changed).

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Cost minimisation problem in the short run:

Min TC = rK0 + wL + mM subject to: Q0=F(L,M,K0)

L,Mwhere: M stands for raw materials

m is the price of raw materials Notes:

L,M are the variable inputs.wL+mM is the total variable

cost.K0 is the fixed inputrK0 is the total fixed cost

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Solution based on: Tangency Condition: MPL = MPM

w m

Isoquant constraint: Q0=F(L,M,K0)

The demand functions are the solutions to the short run cost minimization problem:

Ls = L(Q,K0,w,m)

Ms = M(Q,K0,w,m)

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Hence, the short-run input demands depends on plant size (K0).

Suppose K0 is the long-run cost minimizing level of capital for output level Q0…

… then, when the firm produces Q0, the short-run input demands must yield the long run cost minimizing levels of both variable inputs.

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1. Opportunity costs are the relevant notion of costs for economic analysis of cost.

2. The input demand functions show how the cost minimizing quantities of inputs vary with the quantity of the output and the input prices.

3. Duality allows us to back out the production function from the input demands.

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4. The short run cost minimization problem can be solved to obtain the short run input demands.

5. The short run input demands also yield the long run optimal quantities demanded when the fixed factors are at their long run optimal levels.