Profit Maximization

Christopher Makler

Stanford University Department of Economics

Econ 50 : Lecture 14

pollev.com/chrismakler

When does the production function

f(L) = 10L^a

exhibit diminishing marginal product of labor?

Diminishing marginal product:
as you use more of a unit, holding other inputs constant,
each additional unit produces less additional output.

\frac{\partial MP_L}{\partial L} < 0
\text{Check: when does }f(L,K) = L^aK^b \text{ have diminishing }MP_L?
MP_L = \frac{\partial f(L,K)}{\partial L} = aL^{(a-1)}K^b
\frac{\partial MP_L}{\partial L} = a(a-1)L^{(a - 2)}K^b

This has the same sign as \((a-1)\); so it's negative when \(a < 1\).

Today's Agenda

  • Wrap up discussion of long-run vs. short-run costs from last time
  • Revenue as a function of quantity
  • Profit as a function of quantity
  • Punting elasticity discussion to next time...

Friday

Monday

Wednesday

Cost Minimization for Consumers

Cost Minimization for Firms

Short-Run and Long-Run Costs; Cost Curves

Profit Maximization for a Firm

A Competitive Firm's
Supply Curve

Market Supply and Demand

Midterm II

Market Equilibrium in the Long Run

Market Equilibrium in the Short Run

Unit II: Firms and Markets

Decomposing a Price Change into Income & Substitution Effects

Week 4

Week 5

Week 6

Week 7

Efficiency of Markets: Consumer & Producer Surplus

Midterm I

Theory of the Firm

Labor

Firm

🏭

Capital

⏳

⛏

Customers

🤓

p
w
r
q
L
K

Firms buy inputs
and produce some good,
which they sell to a customer.

PRICE

QUANTITY

Labor

w
L

Capital

r
K

Output

p
q

Firm

🏭

Costs

wL + rK

Revenue

pq

Theory of the Firm

Firms buy inputs
and produce some good,
which they sell to a customer.

PRICE

QUANTITY

Labor

w
L

Capital

r
K

Output

p
q

Costs

wL + rK

Revenue

pq

Profit

\pi = pq - (wL + rK)

The difference between their revenue and their cost is what we call profits, denoted by the Greek letter \(\pi\).

Theory of the Firm

Costs

c(q) = wL(q) + rK(q)

Revenue

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

Profit

\pi(q) = r(q) - c(q)

Theory of the Firm

Our approach will be to write costs, revenues, and profits all as functions of the ouput \(q\).

Long-Run vs. Short-Run

Highly Precise and Technical Definitions

 Long Run

 Short Run

Something is fixed

That thing isn't fixed

In the context we're talking about today:

Capital is fixed at some amount \(\overline K\),
labor is variable

All inputs are variable

c^{LR}(q) = wL^c(w,r,q) + rK^c(w,r,q)

Conditional demand for labor

Conditional demand for capital

TC(q\ |\ \overline{K}) = wL(q\ |\ \overline{K}) + r\overline{K}

Variable cost

Fixed cost

Cost of the lowest-cost way of producing \(q\) units if you can vary both labor and capital

Cost of producing \(q\) units if your capital is fixed, so you can only scale production by adding labor

Costs of Production

 Long Run (can vary both labor and capital)

 Short Run with Capital Fixed at \(\overline K \)

  • Increasing marginal product:
    MPL is increasing in L
  • Constant marginal product:
    MPL is constant in L
  • Diminishing marginal product:
    MPL is decreasing in L

Scaling Production

  • Increasing returns to scale (IRS):
    doubling all inputs more than doubles output.
  • Constant returns to scale (CRS):
    doubling all inputs exactly doubles output.
  • Decreasing returns to scale (DRS):
    doubling all inputs less than doubles output.

 Long Run (can vary both labor and capital)

 Short Run with Capital Fixed at \(\overline K \)

f(L,K) = \sqrt{LK}
L^c(q) = \sqrt{\frac{r}{w}}q
K^c(q) = \sqrt{\frac{w}{r}}q
TC^{LR}(q) = wL^c(q) + rK^c(q)
=2\sqrt{wr}q
L^c(q | \overline K) = {q^2 \over \overline K}
TC(q) = wL^c(q | \overline K) + r \overline K
={wq^2 \over \overline K} + r \overline K

CONSTANT RETURNS TO SCALE => CONSTANT MC (LINEAR FUNCTION)

DIMINISHING MPL =>
INCREASING MC (CONVEX FUNCTION)

 Long Run (can vary both labor and capital)

 Short Run with Capital Fixed at \(\overline K \)

f(L,K) = \sqrt{LK}

 Long Run (can vary both labor and capital)

TC^{LR}(q)=2\sqrt{wr}q

 Short Run with Capital Fixed at \(\overline K \)

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

Let's fix \(w= 8\), \(r = 2\), and \(\overline K =32\)

TC^{LR}(q)=8q
TC(q)=64 + {1 \over 4}q^2

Relationship between
Short-Run and Long-Run Costs

TC(q | \overline{K}) = \text{Cost if capital is fixed at }\overline{K}
TC(q,K^c(q)) = \text{Cost if capital is fixed at optimal }K \text{ for producing }q

What conclusions can we draw from this?

TC(q,\overline{K}) \ge TC^{LR}(q)
= TC^{LR}(q)
TC^{LR}(q) \text{ is the lower envelope of the family of }TC(q)\text{ curves}

Total, Fixed and Variable Costs

c(q\ |\ \overline{K}) =

Fixed Costs \((F)\): All economic costs
that don't vary with output.

Variable Costs \((VC(q))\): All economic costs
that vary with output

explicit costs (\(r \overline K\)) plus
implicit costs like opportunity costs

r\overline{K}
wL(q,\overline{K})
+
TC(q) =
+
F
VC(q)

e.g. cost of labor required to produce
\(q\) units of output given \(\overline K\) units of capital

TC(q) = {wq^2 \over \overline K} + r\overline K
VC(q) = {wq^2 \over \overline K}
F = r\overline K

pollev.com/chrismakler

Generally speaking, if capital is fixed in the short run, then higher levels of capital are associated with _______ fixed costs and _______ variable costs for any particular target output.

\text{Total Cost}: TC(q) = F + VC(q)
\text{Average Cost}: ATC(q) = \frac{TC(q)}{q} = \frac{F}{q} + \frac{VC(q)}{q}

Fixed Costs

Variable Costs

Average Fixed Costs (AFC)

Average Variable Costs (AVC)

Average Costs

\text{Total Cost}: TC(q) = F + VC(q)
\text{Average Cost}: ATC(q) = \frac{F}{q} + \frac{VC(q)}{q}

Average Costs

TC(q) = 64 + {1 \over 4}q^2
ATC(q) = {64 \over q} + {1 \over 4}q
\text{Total Cost}: TC(q) = F + VC(q)
\text{Marginal Cost}: MC(q) = TC'(q) = 0 + VC'(q)

Fixed Costs

Variable Costs

Marginal Cost

(marginal cost is the marginal variable cost)

\text{Total Cost}: TC(q) = F + VC(q)
\text{Marginal Cost}: MC(q) = TC'(q)

Marginal Cost

TC(q) = 64 + {1 \over 4}q^2
MC(q) = {1 \over 2}q

Relationship between Marginal Cost and Marginal Product of Labor

TC(q) = wL^c(q | \overline K) + r \overline K
{dTC(q) \over dq} = w \times {dL^c(q) \over dq}
= w \times {1 \over MP_L}

Revenue

Profit

The profit from \(q\) units of output

\pi(q) = r(q) - c(q)

PROFIT

REVENUE

COST

is the revenue from selling them

minus the cost of producing them.

Revenue

We will assume that the firm sells all units of the good for the same price, \(p\). (No "price discrimination")

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

The revenue from \(q\) units of output

REVENUE

PRICE

QUANTITY

is the price at which each unit it sold

times the quantity (# of units sold).

The price the firm can charge may depend on the number of units it wants to sell: inverse demand \(p(q)\)

  • Usually downward-sloping: to sell more output, they need to drop their price
  • Special case: a price taker faces a horizontal inverse demand curve;
    can sell as much output as they like at some constant price \(p(q) = p\)

Demand and Inverse Demand

Demand curve:

quantity as a function of price

Inverse demand curve:
price as a function of quantity

QUANTITY

PRICE

\text{revenue} = r(q) = p(q) \times q
\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p(q)
\displaystyle{\text{average revenue} = \frac{r(q)}{q} = p(q)}

If the firm wants to sell \(q\) units, it sells all units at the same price \(p(q)\)

Since all units are sold for \(p\), the average revenue per unit is just \(p\).

By the product rule...
let's delve into this...

Total, Average, and Marginal Revenue

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p
dr = dp \times q + dq \times p
p
p(q)
q

The total revenue is the price times quantity (area of the rectangle)

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p
dr = dp \times q + dq \times p
p
p(q)
q
dp
dq

The total revenue is the price times quantity (area of the rectangle)

If the firm wants to sell \(dq\) more units, it needs to drop its price by \(dp\)

Revenue loss from lower price on existing sales of \(q\): \(dp \times q\)

Revenue gain from additional sales at \(p\): \(dq \times p\)

Demand

Inverse Demand

q(p) = 20 - p
p(q) = 20-q

Revenue

r(q) =
MR(q)=
AR(q)=
20q - q^2
20 - 2q
20 - q

pollev.com/chrismakler

Suppose instead that the firm faced the demand function
(not inverse demand!)

\(q(p) = 20 - 2p\).

What would their marginal revenue function \(MR(q)\) be?

Correctness matters on this one...

Demand

Inverse Demand

q(p) = 20 - p
q(p) = 20 - 2p
p(q) = 20-q
p(q) = 10 - \tfrac{1}{2}q

Revenue

r(q) =
MR(q)=
AR(q)=
r(q) =
MR(q)=
AR(q)=
20q - q^2
20 - 2q
20 - q
10q - \tfrac{1}{2}q^2
10 - q
10 - \tfrac{1}{2}q

Profit

\pi(q) = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -

Optimize by taking derivative and setting equal to zero:

\pi'(q) = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \ \ \ \ \ \ \ \ \ = 0
\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ =

Profit is total revenue minus total costs:

"Marginal revenue equals marginal cost"

(64 + {1 \over 4}q^2)
(20q - q^2)
r'(q)
c'(q)
{1 \over 2}q
20 - 2q
r'(q)
c'(q)
{1 \over 2}q
20 - 2q
40 - 4q = q
40 = 5q
q^* = 8

CHECK YOUR UNDERSTANDING

p(q)=20-2q
c(q)=10+5q+{1 \over 2}q^2

Find the profit-maximizing quantity.

Average Profit Analysis

\pi(q) = r(q) - c(q)

Multiply right-hand side by \(q/q\): 

\pi(q) = \left[{r(q) \over q} - {c(q) \over q}\right] \times q
= (AR - AC) \times q

Profit is total revenue minus total costs:

"Profit per unit times number of units"

AVERAGE PROFIT

Next Time

  • How does demand elasticity affect a firm's ability to mark up its price above marginal cost?
  • Extreme example: a firm facing perfectly elastic demand (i.e., a "price taker" or "competitive firm")
  • Derive supply curve