For a number of questions in this problem set, it will be helpful to know that the derivative of the function y = Ax, where A is some constant number, is (∂y/∂x)=A. The derivative of the function y= Ax², where A is constant number, is (∂y/∂x)=2Ax

1. Suppose that a firm has the production function f(x₁,x₂) = (√x₁)+x_2²

a. Describe the direction of the marginal product of factor 1 (does it increase/decrease/stay constant as factor 1 increases)? How about factor 2? Note that MP₁= 1/(√x₁) and MP₂=2x₂

b. Does this production function satisfy the definition of increasing returns to scale, constant returns to scale or decreasing returns to scale? Support your answer, using examples of combinations of inputs.

2. Universal Widget produces high quality widgets at its plant in Gulch, Nevada, for sale throughout the world. The cost function for total widget production (q) is given by Total Cost = TC(q) = 0.25q²

Widgets are demanded only in Australia (where the demand curve is given by q = 100 - 2P) and Lapland (where the demand curve is given by q = 100 - 4P). If the Universal Widget can control the quantities supplied to each market, how many should it sell in each location in order to maximize total profits? What price will be charged in each location?

3. A firm producing baseball bats has a production function given by q= 2√KL. In the short run, the firm's amount of capital equipment is fixed at K = 100. The rental rate for k is v = $1 and the wage rate for L is w = $4.

a. Calculate the firm's shortrun total cost function. Calculate the shortrun average cost function.

b. The firm's shortZrun marginal cost function is given by SMC = q/50. What are the STC, SAC, and SMC for the firm if it produces 25 baseball bats? Fifty? 200?

c. (optional) Graph the average cost and marginal cost curves for the firm. Where does the SMC curve intersect the SAC curve?

4. Suppose that the oil industry in Utopia is perfectly competitive and that all firms draw oil from a single (and practically inexhaustible) pool.

Each competitor believes that he or she can sell all the oil he or she can produce at a stable world price of $10 per barrel and that the cost of operating a well for one year is $1,000.

Total output per year (Q) of the oil field is a function of the number of wells (N) operating in the field. In particular,

Q = 500N - N²

and the amount of oil produced by each well (q) is given by q = Q/N = 500 - N. You can think of N as the number of firms.

a. Describe the equilibrium output and the equilibrium number of wells in this perfectly competitive case. Is there a divergence between private and social marginal cost in the industry?

Hint: This is a hard problem. The trick in this problem is to consider the industry from two different standpoints: first, as an individual oil firm and second, as the industry as a whole. Since total industry output, and individual firm output, is dependent in part on the number of firms, each firm will effect the other firms' profits.

For part a, you are asked to think about the competitive equilibrium. In this case, there is no overseer (government) Z each firm does what is best for itself. In a competitive industry, oil firms will keep entering the industry until their profit dries up Z or they just break even. The trick is to find out what N must be in order for each individual firm's profits to be zero... If the "i"th firm is dealing with a profit function of πi = P ∗ qi - 1000, what will "N" be when this profit is zero? At that N, no new firm will enter. Note that each firm only considers their own situation, without considering what their entry will do to other firms' profits...

b. Suppose the government nationalizes the oil field. How many oil wells should it operate? What will total output be? What will the output per well be?

Hint: Now, instead of considering the situation from the individual firm's point of view, consider the industry as a whole. The industry as a whole would like to maximize INDUSTRY profits, not firm profits. Think about what industry profits would look like:

πindustry = P ∗ (500N - N²) - 1000N.

What N maximizes this equation? At this alternative N, how does industry output compare to industry output in part (a)?

c. As an alternative to nationalization, the Utopian government is considering an annual license fee per well to discourage overdrilling. How large should this license fee be to prompt the industry to drill the optimal number of wells?

Hint: To discourage overdrilling, we need a fee "F" such that each individual firm will feel the pinch, and only the right number of firms will enter (ie the number of firms that we found in part b). So imagine each firm's new profit equation:

πi = P ∗ q_i - 1000 - F .

At the optimal N (from part b), set this equation equal to zero and solve for F.

Remember, you can contact me for any derivatives you may need, as you will need to "maximize" certain equations.