1. A firm produces an output y using labor and capital, whose quantities are denoted respectively by xL and xK. Suppose this firm’s production function is given by f(L,K)=50√(KL)+K+L. Answer the following questions.
a. Derive the marginal product of labor and the marginal product of cap- ital.
b. Is the marginal product of labor diminishing everywhere? Does it ever take negative values?
c. Does the production process of this firm exhibit increasing, constant, or decreasing returns to scale?
2. Let B denote the number of bicycles produced from xF units of bicycle frames and xT units of tires. Suppose that every bicycle needs exactly two tires and one frame. Draw an isoquant that represents this production process and write down a possible production function for bicycles.
3. Suppose a firm’s production function is given by f(x1, x2) = 6√x1 + 8√x2. Let w1=$1 be the price of input 1 and let w2 =$4 be the price of input 2. The price of the firm’s output, which the firm takes as given, is p = $8. Calculate the profit-maximizing quantity of output.
4. Suppose a firm’s production function is given by f(x1,x2) = √x1x2. Let w1 = $10 and w2 = $15. If the firm’s goal is to maximize profits, in what proportions should it use the inputs 1 and 2?