Input and Output Decisions of a Competitive Firm

Christopher Makler

Stanford University Department of Economics

Econ 50 | Lecture 18

pollev.com/chrismakler

What determines how much someone is paid?

 

(In real life, not this course.)

Take everything we're about to do with a gigantic grain of salt.

Theory of the Firm

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

Exogenous Variables

Endogenous Variables

technology, \(f(L,K)\)

level of output, \(q\)

input prices \(w, r\)

Cost Minimization

Profit Maximization

cost function, \(c(w,r,q)\)

revenue function \(r(q)\)

\text{Optimal Output }q^*

Special Case: Competitive Firm

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

Exogenous Variables

Endogenous Variables

technology, \(f(L,K)\)

level of output, \(q\)

input prices \(w, r\)

Cost Minimization

Profit Maximization

cost function, \(c(w,r,q)\)

market price \(p\)

\text{Supply }q^*(w,r,p)
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)
L^*(w\ |\ p)

Assumptions

We will be analyzing a
competitive (price-taking) firm 
in the short run

  • output price \(p\)
  • wage rate \(w\)

Competitive (Price-Taking) Firms

Demand and Inverse Demand

Demand curve:

quantity as a function of price

Inverse demand curve:
price as a function of quantity

QUANTITY

PRICE

Special case: perfect substitutes

Demand and Inverse Demand

Demand curve:

quantity as a function of price

Inverse demand curve:
price as a function of quantity

QUANTITY

PRICE

Special case: perfect substitutes

For a small firm, it probably looks like this...

c(q) = 64 + {1 \over 4}q^2
r(q) = pq = 12q
\text{Let's assume }p = 12:
\text{Profit function is:}
\text{Total cost}
\pi(q) = 12q - [64 + {1 \over 4}q^2]
\text{Take derivative with respect to } q \text{ and set equal to zero:}
\pi'(q) = \ \ \ \ \ \ - \ \ \ \ \ \ = 0
12
{1 \over 2}q

Price

MC

\(q\)

$/unit

P = MR

12

MC = {1 \over 2}q

24

\Rightarrow q^* = 24

Output Supply as a Function of \(p\) with Fixed \(w\)

When price is fixed at 12

For a general price

1. Costs and Revenues

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(q) = 12q - (64 + \tfrac{1}{4}q^2)
\pi^\prime(q) = 12 - {1 \over 2}q = 0
r(q) = pq
\pi(q) = pq - (64 + \tfrac{1}{4}q^2)
\pi^\prime(q) = p - {1 \over 2}q = 0
q^* = 24
q^*(p) = 2p

Profit-Maximizing Output Choice when \(w = 8\), \(r = 2\), and \(\overline K = 32\)

c(q) = wL(q) + r\overline K = {1 \over 4}q^2 + 64
r(q) = pq = 12q
c(q) = wL(q) + r\overline K = {1 \over 4}q^2 + 64

NUMBER

FUNCTION

f(L,K) = \sqrt{LK}, \overline K = 32
64 + {1 \over 4}q^2
{1 \over 2}q
pq
\pi(q) = \ \ \ \ \ - [\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]
\pi^\prime(q) = \ \ \ - \ \ \ \ \ \ \ \ = 0
p

TR

TC

MR

MC

Take derivative and set = 0:

Solve for \(q^*\):

q^*(p) = 2p

SUPPLY FUNCTION

When \(p = 4\), this function says that the firm should produce \(q = 8\).
If it does this...

When \(p = 12 , w = 8, \overline K = 32\) 

For a general \(p, w\), with \(r= 2, \overline K = 32\)

1. Costs and Revenues

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(q) = 12q - (\tfrac{1}{4}q^2 + 64)
\pi^\prime(q) = 12 - {1 \over 2}q = 0
r(q) = pq
\pi(q) = pq - (\tfrac{w}{32}q^2 + 2 \times 32)
\pi^\prime(q) = p - {2w \over 32}q = 0
q^* = 24
q^*(p\ | w) = {16 p \over w}

Now's let's also let wage be a variable

c(q) = {1 \over 4}q^2 + 64
r(q) = pq = 12q
c(q) = w \times {q^2 \over 32} + 2 \times 32

NUMBER

FUNCTION

When \(p = 12 , w = 8, \overline K = 32\) 

For a general \(p, w,r\) and \(\overline K\)

1. Costs and Revenues

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(q) = 12q - (\tfrac{1}{4}q^2 + 64)
\pi^\prime(q) = 12 - {1 \over 2}q = 0
r(q) = pq
\pi(q) = pq - (\tfrac{w}{\overline K}q^2 + r\overline K)
\pi^\prime(q) = p - {2w \over \overline K}q = 0
q^* = 24
q^*(p\ | w, \overline K) = {\overline K p \over 2w}

Now's let's also let wage and capital be a variable

c(q) = {1 \over 4}q^2 + 64
r(q) = pq = 12q
c(q) = w \times {q^2 \over \overline K} + r\overline K

NUMBER

FUNCTION

Profit-Maximizing Demand for Labor

Three Approaches

  • Plug \(q^*(p,w)\) into conditional labor demand \(L^c(q)\)
  • Write profit as a function of labor, solve for optimal quantity of labor to hire
  • Choose point along production function where the last worker's marginal product of labor is equal to their real wage.
f(L,K) = \sqrt{LK}, \overline K = 32
= 8 \left({p \over w}\right)^2

PROFIT-MAXIMIZING
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

The profit-maximizing labor demand is

CONDITIONAL LABOR DEMAND FUNCTION

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

TR

TC

MRPL

MC

Take derivative and set = 0:

Solve for \(L^*\):

L^*(w) = {1152 \over w^2}

PROFIT-MAXIMIZING
LABOR DEMAND FUNCTION

Profit as a function of output \(q\)

Profit as a function of labor \(L\)

c(q) = {wq^2 \over \overline K} + r \overline K

1. Costs for general \(w\) and revenue for general \(p\)

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(q) = pq - [{wq^2 \over 32} + 32r]
\pi^\prime(q) = p - {wq \over 16} = 0
r(L) = p\sqrt{32L}
\pi(L) = p\sqrt{32L} - [wL + 32r]
\pi^\prime(L) = p \times \sqrt{8 \over L} - w = 0
q^*(p\ |\ w) = {16 p \over w}
L^*(w\ |\ p) = {8p^2 \over w^2}
\text{Profit }\pi = pq - [wL + rK]
r(q) = p \times q
c(L) = wL + r \overline K

PROFIT-MAXIMIZING OUTPUT SUPPLY

PROFIT-MAXIMIZING LABOR DEMAND

Profit as a function of output \(q\)

Profit as a function of labor \(L\)

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

1. Costs for general \(w\) and revenue for general \(p\)

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)
\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]
r(q) = p \times q
c(L) = wL + r \overline K

[cost of labor required for \(q\) units of output]

[revenue of output produced by \(L\) hours of labor]

MARGINAL COST (MC)

MARGINAL REVENUE PRODUCT OF LABOR (MRPL)

"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

Profit as a function of output \(q\)

Profit as a function of labor \(L\)

p = w \times {1 \over MP_L}
w = p \times MP_L
MP_L = \tfrac{2}{3}

Is it true?

Are people paid the value of their marginal product?

What determines the value of people's work?

What should?

What determines how much people are paid?

Review: Theory of the Firm

  • Derive LR conditional demands for labor and capital: set \(MRTS = w/r\), plug into \(q = f(L,K)\)
  • Derive SR conditional demand for labor by solving \(q = F(L\ |\ \overline K)\) for \(L(q\ |\ \overline K)\)
  • From conditional input demands, find costs by plugging into \(c(q) = wL(q) + rK(q)\)
  • Find profit-maximizing quantity
    • In general: set MR = MC
    • Special case of constant elasticity: use IEPR
    • Special case of competitive firm: use P = MC
  • Find profit-maximizing labor demand in one of two ways:
    • Plug \(q^*(p,w)\) into \(L^c(q)\)
    • Set \(w = p \times MP_L\) (or equivalently, \(MP_L = w/p\))
  • Note: you may assume all profit-maximization problems on the exam are in the short run.

Given a Cobb-Douglas production function, you should be able to: