Question

**Only [Harder] Question**

Problem 2. Consider a firm that has a cost function of c(y) = 5y 2 + 50, 000. In other words, this is a firm with a fixed cost of $50,000 (which might be something like the cost of rent on the firm’s building, which they have to pay whether they produce any output or not) and a variable cost of $5Y 2 , (which we’ll think of as the cost of the labor and machinery necessary to produce output).

(a) Let’s begin by graphing the firm’s cost curves.

(i) Compute the firm’s marginal cost curve and average cost curve.

(ii) Find the minimum of the average cost curve and prove that it is equal to the point where marginal cost is equal to average cost. In other words, prove that the marginal cost curve passes through the minimum of the average cost curve. [Hint: you do know how to do this, because we’ve been doing maximization and minimization all semester. Just set up a simple minimization problem, miny AC(y), compute the first order condition, dAC dy = 0, and solve for the output at which average cost is minimized.]

(iii) Graph the marginal and average cost curves. Your graph should be neat and should be labeled as accurate as possible, but does not need to be perfectly to scale.

(iv) Explain why the marginal cost curve must pass through the minimum of the average cost curve.

(v) Does this firm have increasing, constant, or decreasing returns to scale? Explain how you know.

b) Now let’s consider the profit maximizing output decision of the firm. We’ll begin by considering the time scale in which the firm has to pay the fixed cost, whether it produces any output or not. (In our example this would be the time scale within which the firm is contractually obligated to pay rent on its building, whether it produces output or not.)

(i) Plug the cost function into the profit function of the firm and use profit maximization to derive the supply curve and the inverse supply curve of the firm. Explain why the inverse supply curve is the same as the marginal cost curve.

(ii) Suppose the market price of the firm’s output is $2000. What will be the profit-maximizing output of the firm? Mark this output level on your graph.

(iii) Compute the profit of the firm at this output level, using two different approaches. First approach: compute the firm’s per-unit profit margin and multiply by the number of units being produced and sold. Show the firm’s profit on your graph. Second approach: plug the profit maximizing output level into the firm’s profit function. Confirm that these two approaches give you the same answer.

(iv) Now suppose the market price of the firm’s output is $500. Once again compute the profit maximizing output of the firm and compute the firm’s profit at that quantity of output. (You can use either approach to compute profit, and you don’t need to graph it this time.) Confirm that operating at this output level is better for the firm than not producing at all, given the time scale we are looking at.

(c) Now let’s consider the time scale in which the firm can avoid paying its fixed costs. (In our example this would be a time scale long enough for the firm to reach the end of its rental contract, allowing it to stop paying rent and shut down completely.)

(i) Show that when the market price of the firm’s output is $500 it is better for the firm to shut down completely than to produce at the profit-maximizing output level that you solved for above.

(ii) Show that this will be true for any price below the minimum of the average cost curve.

(iii) Mark on your graph the inverse supply curve for this longer-term time scale in which the firm can shut down completely. Explain why we say that the firm’s inverse supply curve is the marginal cost curve above the average cost curve.

(d) [Harder] Finally, let’s modify the problem a bit to include an additional fixed cost that the firm only has to pay if they choose to produce non-zero output. (In our example this might be the cost of the electricity to light the building, and the cost of heating and cooling the building, both of which would be zero if the firm chose not to produce anything at all, even if they continued to stay in business and pay rent on the building.) Suppose this additional fixed cost is $10,125. Once again, consider the time scale in which the firm can choose to not produce any output (thus avoiding the cost of electricity, heating and cooling) but still has to pay for the rent on the building.

(i) What is the range of prices within which the firm would choose to produce non-zero output even though they earn negative profits? [Hint: There are two ways to approach this problem. The first is to think of the cost of labor and machinery, plus the cost of electricity and heating and cooling, as the “avoidable” cost of the firm over the given time scale, which is to say, the cost that would go away if they produced zero output but still had to pay rent. You can treat this avoidable cost exactly the same way Varian treats variable cost. The second way to approach the problem is to simply compute the profit function of the firm with the new cost included, and then solve for the price above which the firm’s profit is greater than the profit it would make if it produced zero output. To use this approach you will need to plug in the supply curve, y(p) wherever you find y in the profit function. The two approaches are about equal amounts of work.]

(ii) Explain why the firm would choose to do this over this time scale.

Answer 1

YSIS MC AC A= min (AC) loo quantity (iv) from the graph and the above calculations it is clear that Mc cluts AC at it's minimum point. The to ressan reason is all the points before A, AC>MC, and hence AC was falling because AC= FTC Now TC = EMC Since MC was below AC hence to AC was falling. Now the moment MCTAC, AC starts rising as stated earlier AC is indirectly read related to MC.