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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