# Output Supply and Labor Demand

Christopher Makler

Stanford University Department of Economics

Econ 50

## Optimization

What is an agent's optimal behavior for a fixed set of circumstances?

x_1^*
x_2^*
q^*
L^*

Utility-maximizing bundle for a consumer

Profit-maximizing quantity for a firm

Profit-maximizing input choice for a firm

## Comparative Statics

How does an agent's optimal behavior change when circumstances change?

x_1^*
x_2^*
q^*
L^*

Utility-maximizing bundle for a consumer

Profit-maximizing quantity for a firm

Profit-maximizing input choice for a firm

x_1^*(p_1\ |\ p_2,m)
x_2^*(p_2\ |\ p_2,m)
q^*(p\ |\ w,r)
L^*(w\ |\ p,r)

## Assumptions

We will be analyzing a
competitive (price-taking) firm

• output price $$p$$
• wage rate $$w$$
• rental rate $$r$$

# Today's Agenda

## Edge Cases

f(L,K) = \sqrt{LK}, \overline K = 32
64 + {wq^2 \over 32}
{wq \over 16}
pq
\pi(q) = \ \ \ \ \ - [\ \ \ \ \ \ \ \ \ \ \ \ \ \ ]
\pi^\prime(q) = \ \ \ - \ \ \ \ \ \ \ \ = 0
p

TR

TC

MR

MC

Take derivative and set = 0:

Solve for $$q^*$$:

q^*(p\ |\ w) = {16p \over w}

SUPPLY FUNCTION

### Profit as a function of labor

c(q) = wL(q) + r \overline K

1. Total costs = cost of required inputs

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(q) = pq - [wL(q) + r\overline K]
\pi^\prime(q) = p - w \times {dL \over dq} = 0
r(L) = p \times f(L)

1. Total revenue = value of output produced

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(L) = p\times f(L) - [wL + r\overline K]
\pi^\prime(L) = p \times {dq \over dL} - w = 0
p = w \times {dL \over dq}
p \times {dq \over dL} = w

### Profit as a function of labor

r(q) = p \times f(L)

1. Total revenue = value of output produced

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(q) = p\times f(L) - [wL + r\overline K]
\pi^\prime(q) = p \times {dq \over dL} - w = 0
p \times {dq \over dL} = w

"Keep producing output as long as the marginal revenue from the last unit produced is at least as great as the marginal cost of producing it."

p = MC

### Profit as a function of labor

"Keep producing output as long as the marginal revenue from the last unit produced is at least as great as the marginal cost of producing it."

"Keep hiring workers as long as the marginal revenue from the output of the last worker is at least as great as the cost of hiring them."

p = MC
w = MRP_L
f(L,K) = \sqrt{LK}, \overline K = 32
wL + 64
w
p\sqrt{32L}
\pi(L) = \ \ \ \ \ \ \ \ \ \ \ \ \ - [\ \ \ \ \ \ \ \ \ \ \ \ \ \ ]
\pi^\prime(q) = \ \ \ \ \ \ \ \ \ \ - \ \ \ \ = 0
p\sqrt{8 \over L}

TR

TC

MRPL

MC

Take derivative and set = 0:

Solve for $$L^*$$:

L^*(w\ |\ p) = 8 \left({p \over w}\right)^2

LABOR DEMAND FUNCTION

L^*(w\ |\ p) = 8 \left({p \over w}\right)^2

LABOR DEMAND FUNCTION

f(L,K) = \sqrt{LK}, \overline K = 32
L^*(w\ |\ p) = 8 \left({p \over w}\right)^2

LABOR DEMAND FUNCTION

q^*(p\ |\ w) = {16p \over w}

SUPPLY FUNCTION

# Edge Cases

Edge Case 1:

Multiple quantities where P = MC

Edge Case 2:

Corner solution at $$q = 0$$

"The supply curve is the portion of the MC curve above minimum average variable cost"

# Summary

A competitive firm takes input prices $$w$$ and $$r$$, and the output price $$p$$, as given.
We can therefore characterize its optimal choices of inputs and outputs
as functions of those prices: the supply of output $$q^*(p\ |\ w)$$,
and the demand for inputs (e.g. $$L^*(w\ |\ p)$$).

We can find the optimal input-output combination either by finding the optimal quantity of output and determining the inputs required to produce it, or to find the profit-maximizing inputs and determine the resulting output. These two methods are equivalent.

Profit is increasing when marginal revenue is greater than marginal cost, and vice versa.
In most cases, the profit-maximizing choice occurs where
$$MR = MC$$.
If $$p$$ is below the minimum value of AVC, the profit-maximizing choice is $$q = 0$$.
In which MR or MC is discontinuous, logic must be applied. (There is an old exam question on the homework that explores this...and this kind of thing often shows up on exams...)

By Chris Makler

• 375