1. 1 Intermediate Microeconomic Theory Cost Curves

2. 2 Cost Functions We have solved the first part of the problem: given factor prices, what is cheapest way to produce q units of output? Given by conditional factor demands for each input i, x i (w 1,…, w n, q) However, this is only half the problem. To model behavior of the firm, we also have to derive how much output a firm will find optimal to produce, given input and output prices.

3. 3 Cost functions Key to firm’s output decision is the firm’s cost function Cost of producing a given amount of output, given some input prices and assuming the firm acts optimally (i.e. cost minimizes). Suppose a firm used n inputs for production, with conditional demand functions for each input given by: x 1 (w 1,…, w n, q) : x n (w 1,…, w n, q) What would be the generic form for its cost function?

4. 4 Cost functions: Example Consider a firm with Cobb-Douglas technology of the form q = f(x 1, x 2 ) = x 1 0.25 x 2 0.25 Recall a cost function will be of the form: C(q) = Σw i x 1 (w 1,,…, w n, q) What will be conditional factor demands for this technology? So what will be this firm’s cost function if w 1 = 4 and w 2 = 1?

5. 5 Short run vs. Long(er) run It is often important to distinguish between the Short-Run (SR) and Long(er)-Run (LR) when considering costs. Short-run: some factors of production are fixed (i.e. can’t be adjusted). Long(er)-run: previously fixed factors of production can be adjusted.

6. 6 Short Run Cost Curve The key aspect of a fixed factor of production is that it will mean there will be some component of cost that is the same regardless of how much output (if any) is produced (in short run). What might be some production processes that have fixed inputs the short run?

7. 7 Short Run Cost Curve Short-run cost function where x 2 fixed at x 2 f (and only two inputs) C SR (q) = w 1 x 1 (w 1,w 2,q|x 2 = x 2 f ) + w 2 x 2 f Short run cost function where there are n inputs, where inputs 1 to k are variable and k to n are fixed: So, short-run cost function can be written C SR (q) = c v (q) + F

8. 8 Short Run vs. Long Run Costs Analytically Consider again a firm where: q = f(L, K) = L 0.25 K 0.25, w L = 4, w K = 1. From before, we know long-run cost function in this case will be C(q) = 4q 2 Suppose in the short-run Captial (K) is fixed at 16 machine hrs. What is short-run cost function?

9. 9 Short-Run Cost function Given how cost functions are derived, will the cost of producing any given level of output be greater in the short-run or the longer run?

10. 10 Cost functions with Opportunity Costs Suppose I want to start a ski instructor school where I am the only teacher. To teach one lesson I need a pair of skis. Each lesson takes 2 hrs. of my time (i.e. q = f(L) = L/2, where q is number of ski lessons and L is hours of my time) What is my cost function if skis cost $400 (but I can re-sell them for $200 at the end of the year), my current job pays $20/hr., and I can save at a rate of 10%? What if I can rent skis for $5/hr. How will this change my (long-run) cost function?

11. 11 Returns-to-Scale Returns-to-Scale is a concept describing how costs change as scale of operation (i.e. quantity produced) changes. What do people usually mean when they use this term? (for example in NYT article on Whirlpool and Maytag)

12. 12 Cost functions and Returns-to-Scale To describe Returns-to-scale technically, consider first the “unit cost function” or the cost minimizing way of producing one unit of output (given input prices), as given by C(1). If C(q) = q C(1), then technology exhibits Constant-Returns-to-Scale (CRS) If C(q) > q C(1), then technology exhibits Decreasing-Returns-to-Scale (DRS) If C(q) < q C(1), then technology exhibits Increasing-Returns-to-Scale (IRS)

13. 13 Cost functions and Returns-to-Scale Consider Cobb-Douglas production function f(x 1, x 2 ) = x 1 0.25 x 2 0.25, with w 1 = 4 and w 2 = 1 Recall that the (Long-Run) cost function for this technology was C LR (q) = 4q 2 Does this exhibit CRS, DRS, or IRS? What about when C LR (q) = 4q? What about when C LR (q) = 4q 0.5 ?

14. 14 Cost functions and Returns-to-Scale What about C SR (q) = q 4 /4+ 16 (i.e. the short-run cost function for f(x 1, x 2 ) = x 1 0.25 x 2 0.25, with w 1 = 4 and w 2 = 1, and x 2 fixed at 16)

15. 15 Cost functions and Returns-to-Scale From now on, we will generally be considering relatively Short-run (i.e. at least one factor fixed), so cost functions will generally exhibit both IRS and DRS.

16. 16 Cost Curves In modeling optimal firm behavior, it will often be helpful to think of costs graphically via “cost curves”. Average Cost Curve – AC(q) Denotes the average cost of producing each unit, given q units are produced. AC(q) = C(q)/q Average Variable Cost Curve – AVC(q) As discussed above, we can often think of our SR cost function as: C(q) = c v (q) + F So AVC(q) = c v (q)/q Marginal Cost Curve – MC(q) Denotes the cost of producing a “little bit” more, given you have already produced q units So MC(q) ≈ [C(q+Δq) – C(q)]/Δq = [c v (q+Δq) – c v (q)]/Δq Actually rate of change, however, so consider when Δq goes to zero, So

17. 17 Cost Curves Consider our example, C(q) = q 4 /4 + 16 What is equation for AC(q)? What is Avg. Cost per unit for producing 2 units? How about 4 units? What is equation for AVC(q)? What is Avg. Variable Cost per unit for producing 2 units? How about 4 units? What is equation for MC(q)? What is Marginal Cost for producing 2nd unit? How about 4th?

18. 18 Cost Curves How about, C(q) = q 3 /3 – 5q 2 + 60q + 20 What is equation for AC(q)? What is equation for AVC(q)? What is equation for MC(q)?

19. 19 Marginal Costs Graphically More generally, we can get an idea of what the MC(q) curve looks like from the cost curve and vice versa. C(q) q MC(q) $ $ q

20. 20 Cost Curves (cont.) How do these curves relate to each other? First note: MC(q) ≈ [C(q) – C(q-1)]/1 Next, recall: AVC(q) = c v (q)/q = [C(q)-F]/q (noting that c v (q) = C(q)- F) = [C(q)-C(0)]/q (noting that C(0) = F) = [(C(q)-C(q-1) + (C(q-1)-C(q-2))+…+(C(1)-C(0))]/q So AVC(q) ≈ [MC(q) + MC(q-1) + …+ MC(1)]/q

21. 21 Cost Curves (cont) Given AVC(q) ≈ [MC(q) + MC(q-1) + …+ MC(1)]/q, AVC is essentially the average Marginal cost of producing each unit, given firm has produced q units. Therefore, If MC(q) < AVC(q) over some range of q, then AVC(q) must be decreasing over that range (if you continually add something below the average, average will go down) Alternatively, if MC(q) > AVC(q) over some range of q, then AVC(q) must be increasing over that range (if you continually add something above the average, average will go up) So MC(q) must intersect AVC(q) at the q with the minimum Average Variable cost (call it q*)

22. 22 MC(q) and AVC(q) q*q $ MC(q) AVC(q)

23. 23 Cost Curves (cont) Now, recall AC(q) = C(q)/q = [c v (q) + F]/q = AVC(q) + F/q So AC(q) - AVC(q) = F/q (difference between AC(q) and AVC(q) decreases as q increases) Also, AC(q) ≈ [MC(q) + MC(q-1) + …+ MC(1)]/q + F/q Therefore, if MC(q) < AC(q) over some range of q, then AC(q) must be decreasing over that range. Alternatively, if MC(q) > AC(q) over some range of q, then AC(q) must be increasing over that range. So MC(q) also intersects AC(q) at the q with the minimum Average Cost (call it q**).

24. 24 MC(q) and AVC(q) q* q**q $ F MC(q) AVC(q) AC(q)

25. 25 Long-run vs. Short run AC & MC curves Recall our discussion of long-run vs. short-run. In SR, at least one factor is fixed. For example, consider a firm deciding how large of a plant to build. Suppose there are three possible size plants. Each plant size will be associated with its own cost curve and therefore: Each plant size will have its own AC curve. Each plant size will have its own MC curve.