Production & Costs Continued…Agenda:

I. Consumer and Producer Theory:

similarities and differences

II. Isoquants & The Marginal Rate of Technical Substitution

III. Diminishing vs. Decreasing Returns

IV. Isocosts

V. Putting it together: Optimal

Production & Examples

x

y

U x p

U y p

x

y

Y p

X p

Indifference curves

Utility curves

Budget constraint

Different notationSame meaning!

Consumer Utility Maximization

The Production Mountain

Quantity per unit of labor holding capital constant

Quantity per unit of capital holding labor constant

Isoquants:Combinations of capital and labor that produce a given

quantity

Long Term we can vary both capital and labor

( , ; )

( , ; )

Q f K L time

A f p l time

The Marginal Rate of Technical SubstitutionIf we change the amount of capital we use, how much do we need to change the amount of labor to make the same quantity?

0K LQ MP K MP L

K LMP K MP L

L

K

MPK

L MP

Airplane Game IsoquantsIsoquants

paper

labor

1

X

Y

1

1

Isoquants vs. Indifference Curves

Isoquants Indifference Curves

Convexity from diminishing marginal rate of technical substitutionMore is betterQuantity is a cardinal measureCan only change both capital and labor in the long run.

Convexity from preference assumptionMore is betterUtility is an ordinal measureIndividuals make trade-offs both at one time and over time

Diminishing vs. Decreasing returns

All Isoquants are convex and slope down: diminishing MRTSQuantity increases at a decreasing rate as all inputs increase: decreasing returns

+60+90

+60

+60+60

+40

+20

Isocost LinesWhat is an equation to represent the total cost of production?

C=PKK + PLLCan we re-arrange this to fit the equation for a line in (L,K) space?

L

K K

PCK L

P P

What is the optimal input combination GIVEN cost or quantity?

“No matter what the structure of industry may be… (for profit or not for profit) … the objective of most producers is to produce any given level and quality of output at the lowest possible cost. Equivalently, the producer wants to produce as much output as possible from a given expenditure on inputs.” (Frank p. 233)

Maximize Q given C Minimize C given Q*

*

L L

K K

MP PK w

L MP P r

r = cost of capital

w = cost of labor

* *K LMP MP

r w

Marginal products per dollar

Duality

Example:

If the MRTS between capital and labor is 1/2, the interest rate is 5% (use 5) and the wage rate is $10 per hour, is the firm maximizing production?

1 10

2 5

K w

L r

How should the firm adjust its mix of capital and labor?

21/2

The firm is spending more than it has to!Use LESS labor, MORE capital

The firm could be making more for the same cost!

Example:

If the marginal product of labor is 5 and the marginal product of capital is 2, the price of labor is $20 and the cost of capital is 4%, is the firm optimizing production?

5 20

2 4

5 2

20 4

L

K

L K

MP w

MP r

or

MP MP

w r

To increase the

marginal product of labor, reduce

labor.

To decrease the marginal product of capital, increase capital.

You can NOT control interest rates or reduce wages in perfect capital or labor markets.

(that said… change term structure, reduce benefits, training, perks…)