Consider a bus company that has the following production function: T(L,K) = L, where T is the number of passengers served by the company (i.e., output), L is labor, and K is capital. Assume that the unit price for L is 1 and that for K is 4. The company is required to serve T passengers.
a. Write down the cost-minimization problem for the bus company.
b. Derive the slope of the isoquant curve, i.e., derive MRTSL,K and the slope of iso-cost curve.
c. Derive the optimality condition (tangency condition), which shows the relationship between L and K at the cost-minimizing point.
d. Substitute the optimality condition into the production function, and express cost-minimizing L a function of T. Is L * increasing with T? In the same way, express cost-minimizing K as a function of T. Substitute these functions (L * and K * ) into the cost expression C =wL + rK Express the minimized cost as a function of T, i.e., derive the cost function. Illustrates the relationship between Cand T by drawing a curve. Determine whether the firmâs cost function exhibits IRTS or DRTS and briefly explain.
Consider a bus company that has the following production function: T(L,K) = L, where T is the number of passengers served by the company (i.e., output), L is labor, and K is capital. Assume that the unit price for L is 1 and that for K is 4. The company is required to serve T passengers.
a. Write down the cost-minimization problem for the bus company.
b. Derive the slope of the isoquant curve, i.e., derive MRTSL,K and the slope of iso-cost curve.
c. Derive the optimality condition (tangency condition), which shows the relationship between L and K at the cost-minimizing point.
d. Substitute the optimality condition into the production function, and express cost-minimizing L a function of T. Is L * increasing with T? In the same way, express cost-minimizing K as a function of T. Substitute these functions (L * and K * ) into the cost expression C =wL + rK Express the minimized cost as a function of T, i.e., derive the cost function. Illustrates the relationship between Cand T by drawing a curve. Determine whether the firmâs cost function exhibits IRTS or DRTS and briefly explain.
