# The Firm in the Long Run

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## Today's Agenda

Part 1: Cost Minimization

Part 2: Profit Maximization

Cost minimization

Conditional demands

Expansion paths

Total costs

Profits as a function of inputs

Conditional vs. profit max demand

Effect of price changes

# Cost Minimization

### Cost Minimization Subject to an Output Constraint

#
**Conditional** Demand for Inputs

### Cost Minimization Subject to a Utility Constraint

### Cost Minimization Subject to an Output Constraint

Hicksian Demand

Conditional Demand

**Example**

Find the **tangency condition** that sets MRTS = w/r

Plug that value of K back into the **isoquant constraint** q = f(L,K)

Solve for K as a function of L.

Solve for \(L(q)\)

Plug \(L(q)\) back into the relationship between K and L to find \(K(q)\).

Lagrange Method

Find \(L^*\) and \(K^*\)

for \(w = 10\), \(r = 6\), and \(q = 60\)

Find \(L^*(w,r,q)\) and \(K^*(w,r,q)\)

# Long-Run Expansion Path

# Long-Run Total Cost of \(q\) Units

Conditional demand for labor

Conditional demand for capital

"The total cost of producing q units in the long run

is the cost of the cost-minimizing combination of inputs

that can produce q units of output."

# 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, \(q\)

Capital and Labor required to produce

the **profit-maximizing** amount of output, \(q^*(w,r,p)\)

# Effects of Price Changes

### 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?

#### The Firm in the Long Run

By Chris Makler

# The Firm in the Long Run

LR conditional and profit-maximizing demands for inputs

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