Profit Maximization in Chapter 20: Key Concept FOC

1. Chapter 20 Profit Maximization • Key Concept: FOC • pMP1(x1*, x2*)=w1 • pMP2(x1*, x2*)=w2 • Revealed profitability tells us about the slope of the supply curve.

2. Chapter 20 Profit Maximization • In both product market and factor market, the firm in concern is a price taker. • It produces n outputs (y1, y2, ..., yn). • It uses m inputs (x1, x2, ..., xm). • Output prices are (p1, p2, ..., pn). • Input prices are (w1, w2, ..., wm).

3. The goal of the firm is to • max  = (p1y1+ p2y2+…+ pnyn ) – (w1x1+ w2x2+…+ wmxm) • s.t. (y1, y2, ..., yn, x1, x2, ..., xm) is in the production set.

4. Note that all factors should be valued at their market rental prices. • For instance, the imputed wage of the owner should be included, the imputed rental rate of the machine should be included (>0). • It should reflect the opportunity costs (market prices), not the historical costs.

5.  • Moreover, it is measured in flows and is the price that you rent something for some time.

6. Short-run profit maximization: • maxx1=pf(x1,k)-w1x1-w2k. • FOC: pMP1(x1*,k)=w1, • value of the marginal product of a factor = its rental price (intuition) • pMP1(x1*,k)>w1, use more of this factor pMP1(x1*,k)<w1, use less of this factor

9. Fig. 19.1

10. Can do some comparative statics. • If w1 increases, the slope increases. With diminishing MP1, x1 decreases. The short-run factor demand slopes downwards. • Similarly, if p increases, the slope decreases. With diminishing MP1, x1 increases. The short-run supply slopes upwards.

11. Fig. 19.2

13. Long-run profit maximization: • maxx1, x2=pf(x1, x2)-w1x1-w2 x2 • FOC • pMP1(x1*, x2*)=w1 • pMP2(x1*, x2*)=w2

14. One relationship between CRS and the LR profit. • In a perfect competitive market, a CRS firm is getting 0 profit in the LR. • In the LR, cannot have a negative profit for if it does, it should go out of business.

15. What about a positive profit? *=pf(x1*,x2*)-w1x1*-w2 x2* • Double the inputs you get • pf(2x1*,2x2*)-2w1x1*-2w2x2* • = 2pf(x1*,x2*)-2w1x1*-2w2 x2* • = 2* • a contradiction.

16. The only reasonable LR profit of a CRS firm is 0! • Firms are trying to max profit. How can it be that they only get 0 profit in the LR?

17. If the firm expands indefinitely… • Getting so large that cannot operate efficiently, so CRS may not be valid. • Getting so large that dominates the market. Competitiveness may not be valid. • If a firm can do it, other firms with the same technology may also do it. This will push down the price of output.

18. Revealed profitability: when a profit maximizing firm makes its choice of inputs and outputs, it reveals two things. • First, the input-output bundle is feasible or in the production set. • Second, this choice is more profitable than any other feasible choice in the production set.

19. Fig. 19.4

20. Fig. 19.5

21. t: (pt, w1t,w2t) (yt, x1t,x2t) • s: (ps, w1s,w2s) (ys, x1s,x2s) • Then: pt yt-w1tx1t-w2tx2t pt ys-w1tx1s-w2tx2s and ps ys-w1sx1s-w2sx2s ps yt-w1sx1t-w2sx2t. Let ∆z=zt-zs. • We have ∆p∆y-∆w1∆x1-∆w2∆x20.

22. ∆p∆y-∆w1∆x1-∆w2∆x20: • ∆w1=∆w2=0, then ∆p∆y0 so the supply slopes upwards. • Similarly, ∆p=∆w2=0, ∆w1∆x1≤0, so the factor demand slopes downwards. • (mention SR)

23. As in WARP, WAPM (weak axiom of profit maximization) can help us recover the production function. • A profit maximizing firm must be cost minimizing. • Convenient to break  max to • 1) cost min for every y, then • 2) choose the optimal y*.

24. Chapter 20 Profit Maximization • Key Concept: FOC • pMP1(x1*, x2*)=w1 • pMP2(x1*, x2*)=w2 • Revealed profitability tells us about the slope of the supply curve.