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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
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 \(K^*(w,r,q)\)
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."
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 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, \(q^*(w,r,p)\)