# 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

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)

optimal level of output, y*

Profit Maximization

output prices

Total Revenue R(y)

Total Cost, c(y)

**Last week**

**Last time**

**Today**

LONG RUN

SHORT RUN

LONG RUN

SHORT RUN

SHORT RUN

# Profit Maximization

Optimize by taking derivative and setting equal to zero:

Profit is total revenue minus total costs:

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

LONG RUN

SHORT RUN

# Total Profit Analysis

# Average Profit Analysis

(multiply by y/y)

(simplify)

(by definition of AR and AC)

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!

# Marginal Profit Analysis

Increasing production **raises** profits

Increasing production **lowers** profits

# Profit as a

Function of Inputs

Total Revenue

Optimize by taking derivatives with respect to each choice variable

and setting equal to zero:

Total Cost

Marginal **Revenue** Product of each input

Marginal Cost of each input

# Conditional vs. Profit-Maximizing Input Demands

Conditional Demands in the Long Run

Profit-Maximizing Demands in the Long Run

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

Profit-Maximizing Demands for Variable Input in the Short Run

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 \(y^*\)**.

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?

#### Econ 50 | Spring 22 | Profit Maximization and Supply

By Chris Makler

# Econ 50 | Spring 22 | Profit Maximization and Supply

Profit Maximization and Supply

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