Output Supply
and Labor Demand

Christopher Makler

Stanford University Department of Economics

Econ 50 | Lecture 19

The Competitive Firm

L^c(w,r,q)
K^c(w,r,q)

Exogenous Variables

Endogenous Variables

technology, f()

level of output, q

conditional
input demands

Cost Minimization

Isoquant

Isocost

lines

factor prices (w, r)

q^*(w,r,p)

profit-maximizing output supply

Profit Maximization

output price, p

Total Revenue

Total Cost

profit-maximizing input demands

total cost

TC(w,r,q)
L^*(w,r,p)
K^*(w,r,p)

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

Supply of Output

Demand for Labor

Short-Run Responses to Price
and Wage Changes

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 quantity

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
\text{Profit }\pi = pq - [wL + rK]

Profit as a function of quantity

Profit as a function of labor

1. Total revenue = value of output produced

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

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

Profit as a function of quantity

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

the conditional labor demand
for the profit-maximizing supply:

L^*(w\ |\ p) = L^c(q^*(p\ |\ w))
L^c(q) = {1 \over 32} q^2
= {1 \over 32}\left(16p \over w\right)^2
= 8\left(p \over w\right)^2

The profit-maximizing labor demand is

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"

Long Run Supply

TC^{LR}(w,r,q) = 2\left(\sqrt{wr}\right)q^2

LONG RUN

SHORT RUN

f(L,K) = L^\frac{1}{4} K^\frac{1}{4}
\pi(q) = pq - 2\left(\sqrt{wr}\right)q^2
\pi(q) = pq - \left[w\frac{q^4}{\overline{K}} + r\overline{K}\right]
\pi'(q) = p - 4\left(\sqrt{wr}\right)q = 0
\pi'(q) = p - 4w\frac{q^3}{\overline{K}} = 0
q^*(w,r,p) = \frac{p}{4\sqrt{wr}}
TC(w,r,q) = w\frac{q^4}{\overline{K}} + r\overline{K}
q^*(w,r,p) = \left(\frac{\overline K p}{4w}\right)^\frac{1}{3}
\text{Sanity check: how do these depend on }w, r, p,\overline K?
\text{Step 2. Take the derivative, set it equal to zero, and solve for the optimal choice, }y^*:
\text{Step 1. Write down the expression of profit as a function of the choice variable, }y:

Supply Elasticities

LONG RUN

L^c(w,r,q) = w^{-{1 \over 2}}r^{1 \over 2}q^2
K^c(w,r,q) = w^{1 \over 2}r^{-{1 \over 2}}q^2
q^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

SHORT RUN

L^c(q|\overline K) = {1 \over \overline K}q^4
q^*(w,r,p) = \left({\overline K \over 4}\right)^{1 \over 3}w^{-{1 \over 3}} p^\frac{1}{3}

LONG RUN

L^c(w,r,q) = w^{-{1 \over 2}}r^{1 \over 2}q^2

SHORT RUN

L^c(q|\overline K) = {1 \over \overline K}q^4

What is the output elasticity of conditional labor demand in the short run and long run?

Intuitively, why this difference?

\epsilon_{L^c,q}=
\epsilon_{L^c,q}=
2
4

In the long run, the firm uses
both labor and capital to increase output;
in the short run, it only increases labor.

Supply Elasticities

LONG RUN

SHORT RUN

L^c(w,r,q) = w^{-{1 \over 2}}r^{1 \over 2}q^2
K^c(w,r,q) = w^{1 \over 2}r^{-{1 \over 2}}q^2
q^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p
L^c(q|\overline K) = {1 \over \overline K}q^4
q^*(w,r,p) = \left({\overline K \over 4}\right)^{1 \over 3}w^{-{1 \over 3}} p^\frac{1}{3}

LONG RUN

q^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

SHORT RUN

q^*(w,r,p) = \left({\overline K \over 4}\right)^{1 \over 3}w^{-{1 \over 3}} p^\frac{1}{3}

What is the price elasticity of supply
 in the long run and short run?

Intuitively, why this difference?

\epsilon_{q,p}=
\epsilon_{q_s,p}=
1
{1 \over 3}

In the long run, the firm can adjust its capital to keep its costs down; so its marginal cost rises less steeply, and its supply curve is flatter.

Supply Elasticities

LONG RUN

Work with a partner:
How would the firm respond to a
6% increase in the wage rate in the long run?

L^c(w,r,q) = w^{-{1 \over 2}}r^{1 \over 2}q^2
K^c(w,r,q) = w^{1 \over 2}r^{-{1 \over 2}}q^2
q^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

pollev.com/chrismakler

L^c(w,r,q) = w^{-{1 \over 2}}r^{1 \over 2}q^2
K^c(w,r,q) = w^{1 \over 2}r^{-{1 \over 2}}q^2
q^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

How would the firm respond to a
6% increase in the wage rate in the long run?

A 6% increase in w decreases q by 3%.

and decreases by 6% (due to the -3% change in q),
for a total decrease of 9%.

K increases by 3% (due to the +6% change in w)

\epsilon_{L^c,w}=-{1 \over 2}
\epsilon_{L^c,q}=+2
\epsilon_{q,w}=-{1 \over 2}

L decreases by 3% (due to the +6% change in w)

\epsilon_{K^c,w}=+{1 \over 2}
\epsilon_{K^c,q}=+2

and decreases by 6% (due to the -3% change in q),
for a total decrease of 3%.

L^c(w,r,q) = w^{-{1 \over 2}}r^{1 \over 2}q^2
q^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

How would the firm respond to a
6% increase in the wage rate in the long run?

A 6% increase in w decreases q by 3%.

and decreases by 6% (due to the -3% change in q),
for a total decrease of 9%.

L decreases by 3% (due to the +6% change in w)

L^*(w,r,p) = w^{-{1 \over 2}}r^{1 \over 2}[y^*(w,r,p)]^2
= w^{-{1 \over 2}}r^{1 \over 2}\left[{1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p\right]^2
= {1 \over 16}w^{-{3 \over 2}}r^{-{1 \over 2}}p^2

Note: we can calculate the LR profit-maximizing demand for labor:

\epsilon_{L^c,w}=-{3 \over 2}

What did we just show?

  • If there is a direct causality \(X \rightarrow Y\), elasticity measures how Y responds to X.
  • If there is a chain of causality \(X \rightarrow Y \rightarrow Z\), the elasticity composes just like a function does (like the chain rule for elasticity)

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...)

Econ 50 | Spring 2023 | Lecture 19

By Chris Makler

Econ 50 | Spring 2023 | Lecture 19

Output Supply and Labor Demand for a Competitive Firm

  • 371