Cost and Supply Revised: October 1, 2001 We continue our analysis of market conditions, this time focusing on supply. Our plan is to construct the market supply curve for a competitive market. Along the way, we answer these questions: Should a firm stay in a market or leave? If it stays, how much should it produce at each price? If a market has many producers, how much would we expect them to produce altogether in the short run (ie, given capacity)? In the long run (ie, when capacity can adjust to market conditions)? Cost Concepts Some basic cost concepts:  Fixed Cost (FC). A cost that must be paid even if no output is produced and which does not depend on the output level.  Variable Cost (VC). That cost which would be zero if the output level were zero.  Total Cost (TC). Sum of fixed cost and variable cost.  Average Cost (AC) (also "unit cost"). Total cost divided by output level.  Marginal Cost (MC). The derivative of the cost function, which we approximate by the cost of one additional unit: the total cost of producing q+1 units minus total cost of producing q units of output. We can express the same ideas mathematically. Suppose we represent the total cost of producing output q by C(q). The total cost has the following expression: C(q) = FC + VC(q). Note that the fixed cost does not depend on q. The definitions give us FC = C(0) so VC(q) = C(q) – C(0). Average cost is simply the total cost divided by the number of units produced: AC(q) = C(q)/q = FC/q + VC(q)/q. Marginal cost is the derivative (ie, we’re using calculus now) of total cost: MC(q) = C´(q) = VC´(q) ≈ C(q+1)-C(q). Here F´(x) = dF/dx indicates the derivative of the function F with respect to the variable x and ≈ means “is approximately equal to.” Firms and MarketsLecture NotesCost and Supply Page 2 T-Shirt Factory: Costs To illustrate these cost concepts, consider a stylized t-shirt factory. To produce t-shirts, a manager leases one machine at the rate of $20 per week. For now, let us say that the manager has recently entered into a one-year agreement on the machine and is therefore stuck with the weekly cost of $20. A machine needs one worker to produce output. The hourly wage paid to the worker is $1 on weekdays (up to 40 hours), $2 on Saturdays (up to 8 hours), and $3 on Sundays (up to 8 hours). Finally, the machine—operated by the worker—produces one t-shirt per hour. Let’s translate this into our terms using an output of q=40 (shirts): •= The fixed cost is given by the weekly lease on the machine. We thus have FC=$20. •= The variable cost is the labor cost of one t-shirt. At q=40, this is $1 per hour times 40 hours, or $40. •= The total cost is the sum of these two, or $60. •= The average cost is (20+40)/40 = $1.50 per shirt. •= The marginal cost is the cost of producing an extra shirt. To produce a 41st shirt, the worker needs to work on Saturday, which costs $2/hour, so the marginal cost is $2. We chose Saturday because Sunday is more expensive. These costs were computed for a particular value of output. However, both average cost and marginal cost depend on output. By computing the values of marginal cost and average cost for each value of output, we get the marginal cost and average cost curves. In this example, the marginal cost is $1 for output up to 40, $2 for output between 41 and 48, and $3 for output between 49 and 56. We will take this pattern of increasing marginal cost as standard, although most of what follows can be adapted for other patterns. T-Shirt Factory: How Much to Produce Marginal cost is the least obvious concept here, but it plays a critical role in helping us decide how much to produce. Consider: suppose Benetton, the sole buyer of t-shirts from our small factory, is offering a price of p=$1.8 per shirt. (Benetton is well-known for its large network of small suppliers.) Moreover, Benetton is willing to buy as many shirts as the factory wants to sell at that price. Given this offer, should the factory operate on Saturday? At an output of q=40 shirts a week, we have seen that average cost is $1.5. So at p=$1.8, the factory is making money. It might seem, for this reason, that it is worth it to operate on Saturdays as well: "if you are making money at the current output level, produce more and you will make more money.'' As it turns out, this is wrong. What is relevant for the decision of whether or not to operate on Saturdays is the comparison between price and marginal cost, not the comparison between price and average cost. And sinceCost and Supply Page 3 the marginal cost of operating on Saturday is $2, it is not worth doing if shirts sell for $1.8. In other words, although the factory is making money at output level q=40 (because price is greater than marginal cost), profits would be lower if output were increased (because price is lower than marginal cost); the factory would lose money “at the margin.” (By "lose money at the margin'' we mean lose money by producing an additional “marginal” unit of output.) We can verify this by computing the revenue, cost, and profit at q=40 and q=41. At q=40, (total) cost is 40+20=60, revenue is 40 x 1.8 = 72, profit is 12. At q=41, cost is 62, revenue is 73.8, and profit is 11.8: less! We can apply similar logic at other prices. If Benetton offered $2.50 a shirt, the factory should produce 48 shirts, since the marginal cost of running the factory on Saturdays is less than the price. And if Benetton offered $3.50, it would be worth operating even on Sundays (q=56). In short, a comparison of price and marginal cost tells you how much to produce. There’s a lesson here. If we are producing, then we decide how much to produce by comparing price and marginal cost. If p>MC, then we’re making money (at the margin) from additional units. In short, the MC curve tells us how much to supply at each price. It’s the factory’s supply curve. Cost and Supply: Mathematical Version The t-shirt factory example is a bit special for lots of reasons. We will generally assume that marginal cost and average cost functions are continuous functions, that fixed costs are positive, and marginal cost increases with output. We also assume that the firm is competitive: it produces the same homogeneous product as other firms in the industry and is too small