Problem:
Consider a firm that uses two inputs. The quantity used of input 1 is denoted by x_1 and the quantity used of input 2 is denoted by x_2. The firm produces and sells one good using the production function f(x_1,x_2)=4x_1^0.5+3x_2^0.5. The final good is sold at price P=$10. The prices of inputs 1 and 2 are w_1=$2 and w_2=$3, respectively. The markets for the final good and both input goods are treated as competitive markets by the firm, that is, it takes prices as given.
a) Show whether the production function has increasing, decreasing, or constant returns to scale.
b) Draw the isoquant for an output level of 12. Clearly label the axes and the curve and show any two input bundles on the curve by indicating their coordinates.
Now consider the long run, where the quantity of input 2 can be varied.
c) According to your answer in part a), does the firm have a profit maximising plan in the long run? If no, explain why. If yes, is the plan unique?
d) Write down the firm's profit function and the firm's long run profit maximisation problem. Find the firm's optimal use of input 1, input 2, the associated optimal quantity of the output, and the firm's profit level.