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

\min wL + rK \text{ s.t. } f(L,K) = q
\min{4L + K}
\text{s.t. }5\sqrt{LK} = 100

Conditional Demand for Inputs

Cost Minimization Subject to a Utility Constraint

Cost Minimization Subject to an Output Constraint

\min p_1x_1 + p_2x_2 \text{ s.t. } u(x_1,x_2) = U
\min wL + rK \text{ s.t. } f(L,K) = q
\text{solutions}: x_1^*(p_1,p_2,U), x_2^*(p_1,p_2,U)
\text{solutions}: L^*(w,r,q), K^*(w,r,q)

Hicksian Demand

Conditional Demand

Example

f(L,K) = L^\frac{1}{4}K^\frac{1}{4}

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

f(L,K) = \min\{2L, K\}

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

c(q) = wL^*(w,r,q) + rK^*(w,r,q)

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

\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,q)
K^*(r|w,q)

Profit-Maximizing Demands in the Long Run

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

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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