Consider a Cobb-Douglas production function of the following form:

f(K,L) =K(1/3)L(1/3),

where there are factors of production capital and labor with amountsKandL, respectively. The marginal rate of transformation associated with this production function is -(L/K). Let the associated factor prices berandw, respectively; the total cost isC. Suppose the firm has chosen to produce an output levely.

• Obtain the factor demandsK* andL* as functions of output and factor prices.
• GivenK*, sketch the factor demand curve forKas a function of its factor price. Do the same for the other input,L.
• ObtainC*the total cost associated with a cost-minimizing firm producingyand facing factor pricesrandwas a function of output and factor prices.