Profit Maximization and Comparative Statics

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Today's Agenda

Part 1: Profit Maximization

Part 2: Output Supply & Input Demands

Solving for the optimal quantity

Total profit analysis

Average profit analysis

Marginal profit analysis

Profit-maximizing demands for inputs

Output supply as a function of p

Labor demand as a function of w

Movements along vs. shifts of curves

(optimization)

(comparative statics)

Producer Theory

L^*(w,r,y)
K^*(w,r,y)

Exogenous Variables

Endogenous Variables

technology, f()

level of output, y

factors used

total cost, c(y)

Cost Minimization

Production Set

Choice

Rule

factor prices (w, r)

y^*(w,r,p)

optimal level of output, y*

Profit Maximization

output prices

Total Revenue R(y)

Total Cost, c(y)

Last week

Last time

Today

\text{Ongoing Mathematical Example: }f(L,K) = L^\frac{1}{4} K^\frac{1}{4}
c(w,r,y) = 2\sqrt{wr}y^2
c_s(w,r,y) = w\frac{y^4}{\overline{K}} + r\overline{K}
\text{Last time we found the LR and SR total cost functions: }
\text{Now let's fix }w=r=10\text{ and }\overline K = 25:
c(y) = 20y^2
c_s(y) = 250+0.4y^4

LONG RUN

SHORT RUN

\text{Ongoing Mathematical Example: }f(L,K) = L^\frac{1}{4} K^\frac{1}{4}
c(y) = 20y^2
c_s(y) = 250+0.4y^4

LONG RUN

SHORT RUN

F
c_v(y)
\text{Ongoing Mathematical Example: }f(L,K) = L^\frac{1}{4} K^\frac{1}{4}
c_s(y) = 250+0.4y^4

SHORT RUN

SMC(y) = c'_s(y) = 1.6y^3
SAC(y) = \frac{c_s(y)}{y} = \frac{250}{y} + 0.4y^3

Profit Maximization

\pi(y) = r(y) - c(y)
\pi(y) = p \times y - c(y)

Optimize by taking derivative and setting equal to zero:

\pi'(y) = p - c'(y) = 0
\Rightarrow p = c'(y)

Profit is total revenue minus total costs:

For a price taker (or competitive firm), revenue equals price times quantity:

c(w,r,y) = 2\left(\sqrt{wr}\right)y^2

LONG RUN

SHORT RUN

\text{Ongoing Mathematical Example: }f(L,K) = L^\frac{1}{4} K^\frac{1}{4}
\pi(y) = py - 2\left(\sqrt{wr}\right)y^2
\pi(y) = py - \left[w\frac{y^4}{\overline{K}} + r\overline{K}\right]
\pi'(y) = p - 4\left(\sqrt{wr}\right)y = 0
\pi'(y) = p - 4w\frac{y^3}{\overline{K}} = 0
y^*(w,r,p) = \frac{p}{4\sqrt{wr}}
c_s(w,r,y) = w\frac{y^4}{\overline{K}} + r\overline{K}
y_s^*(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:

Total Profit Analysis

c_s(y) = 250+0.4y^4
r(y) = py = 200y
\text{Let's assume }p = 200:
\text{Profit function is:}
\text{Ongoing mathematical example: short run}
\pi(y) = 200y - 250 - 0.4y^4
\text{Profit is maximized at } y = 5.

Average Profit Analysis

\pi(y) = r(y) - c(y)
= \left[\frac{r(y) - c(y)}{y}\right]y
= \left[\frac{r(y)}{y} - \frac{c(y)}{y}\right]y
= \left[AR - AC\right]y

(multiply by y/y)

(simplify)

(by definition of AR and AC)

= \left[p - AC\right]y
\text{If firm is a price taker, so }r(y) = py:

Profit may be seen as an area in a unit costs graph.

Note: We can also see total revenues and total costs as areas in a unit cost graph!

-
=
-
=
\pi(y)
r(y)
c(y)
-
=
AR(y) \times y
AC(y) \times y
(AR - AC) \times y

Marginal Profit Analysis

\pi(y) = r(y) - c(y)
\pi'(y) = r'(y) - c'(y)
\pi'(y) > 0

Increasing production raises profits

Increasing production lowers profits

r'(y) > c'(y)
\pi'(y) < 0
r'(y) < c'(y)

Profit as a
Function of Inputs

\pi(L,K) = r(L,K) - (wL + rK)
\pi(L,K) = p \times f(L,K) - (wL + rK)

Total Revenue

Optimize by taking derivatives with respect to each choice variable
and setting equal to zero:

Total Cost

\frac{\partial \pi(L,K)}{\partial L} = p \times MP_L - w = 0
\frac{\partial \pi(L,K)}{\partial K} = p \times MP_K - r = 0
\Rightarrow pMP_L = w
\Rightarrow pMP_K = r

Marginal Revenue Product of each input

Marginal Cost of each input

Conditional vs. Profit-Maximizing Input Demands

Conditional Demands in the Long Run

L^*(w|r,y)
K^*(r|w,y)

Profit-Maximizing Demands in the Long Run

L^*(w|r,p)
K^*(r|w,p)
=L^*(w|r,y(w,r,p))
= K^*(r|w,y(w,r,p))

Capital and Labor required to produce
a fixed amount of output, y

Capital and Labor required to produce
the profit-maximizing amount of output, y*(w,r,p)

Conditional Demand for Variable Input in the Short Run

L(y,\overline K)

Profit-Maximizing Demands for Variable Input in the Short Run

L^*(w,r,p,\overline K)
=L^*(y^*(w,r,p,\overline K),\overline K)

Labor required to produce
a fixed amount of output, y,

given a fixed amount of capital

Labor required to produce
the profit-maximizing amount of output, y*(w,r,p,K),

given a fixed amount of capital

Summary: Profit Maximization

Last Tuesday we established
the relationship between inputs and outputs
via the production function.

Last Thursday we used that production function to
to solve the firm's cost-minimization problem for a specific output \(y\) 
and used this to derive its conditional input demands and cost function.

Today we embedded that cost function
into a profit-maximization problem 
to determine the optimal output \(y^*\).

y = f(L,K)
c(w,r,y) = wL(w,r,y) + rK(w,r,y)
p = \frac{\partial c(w,r,y)}{\partial y} \Rightarrow y^*(w,r,p)

The optimal quantity of output also implies profit maximizing levels of inputs.

What happens in the short run run when:

  • The price increases
  • The wage rate increases
  • The rental rate of capital increases

What about the long run?