Kent sells lemonade in a competitive market on a busy street corner. His production function is

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

where output q is gallons of lemonade, K is the pounds of lemons he uses and L is the number of labour-hours spent squeezing them. The corresponding marginal products are

MPL = 1L(?2/3)K(1/3)3

MPK = 1L(1/3)K(?2/3)3

Every pound of lemons cost r and the wage rate of lemon squeezers is w. (35 points)

a. Prove that this production process have decreasing returns to scale.

b. On a graph with hours lemon-squeezing (L) on the horizontal axis and pounds of lemons (K) on the vertical axis, illustrate an isoquant that represents a particular production level q?. What is the equation of the isoquant?

c. What is the equation for a slope of an isoquant? Is it constant? What does the slope indicate? Explain.

d. What are the conditions that identify the cost minimizing bundle for any output? Illustrate on a clearly labelled diagram.

e. Set up the cost minimization problem and solve for the conditional input demands as functions of the exogenous variables.

f. Derive the cost function (as a function of only the exogenous variables).

g. Continue with total cost function derived in previous part and derive the average cost. Is it upward or downward sloping? Explain.