4. Suppose that a firm's inputs are labor (variable costs) and trucks (fixed costs) and the firm owns one truck and the cost function, in the short run, is C(q) = 50 + q + 1 2 q 2 .

(a) The marginal cost function is now MC(q) = 1 + q. Marginal cost is equal to the derivative of the cost function or C/q for very small q. The derivative of q is 1 and the derivative of 1 2 q 2 is q. What are the average total cost and the average variable cost functions? Note: I'm asking for functions of q, or AC(q) and AV C(q).

(b) Graph the three cost functions, MC(q), AC(q), and AV C(q). Recall that to graph a function, you pick 3 or 4 different value of q (on the horizontal axis) and find the value of the cost per unit (the vertical axis), then you plot the points, and then you connect them to draw the function.

(c) If the price is $15, how much output will the firm produce if it is maximizing profits? Assume the firm is a price-taker and its output decision does not impact the price.

(d) Suppose that in the long run the firm can purchase more trucks (and of course also hire more labor). If the firm expects the price to stay at $15, will the firm buy more trucks? Answer intuitively. The question is, is this business profitable so that if it doubled its labor and doubled its capital (trucks) it would make more money? Equivalently, you might ask would another firm want to enter this industry if it had the same costs as you? Hint: think about what quantity the firm produces and what is the firm's average costs when it produces this quantity. Extra: If you want to answer this mathematically, note that with two trucks the fixed costs rise to 100 and the average variable cost function becomes q + 1 4 q 2 .

(e) If the firm expects the price to drop to $9 and stay at $9, will the firm buy more trucks? Explain. Again, answer intuitively. Is its business profitable so that if it doubled its labor and doubled its capital (trucks) it would make more money? Hint: think about what quantity the firm produces and what the firm's average cost is when the firm produces this quantity.