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Chapter 19 Profit Maximization • In both product market and factor market, the firm in concern is a price taker. Suppose it produces n outputs (y1, y2, ..., yn) and uses m inputs (x1, x2, ..., xm) where the output prices and the input prices are (p1, p2, ..., pn) and (w1, w2, ..., wm) respectively, then the goal of the firm is to max =p1y1+ p2y2+…+ pnyn-(w1x1+ w2x2+…+ wmxm), subject to the constraint that (y1, y2, ..., yn, x1, x2, ..., xm) is in the production set.
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. It is clear that it should reflect the opportunity costs (market prices), not the historical costs. Moreover, it is measured in flows and is the price that you rent something for some time.
Short-run profit maximization: maxx1=pf(x1,k)-w1x1-w2k. (fixed factors, quasi-fixed factors) FOC: pMP1(x1*,k)=w1, the value of the marginal product of a factor should equal to its rental price. If pMP1(x1*,k)>w1, should use more of this factor. On the other hand, if pMP1(x1*,k)<w1, should use less of this factor. w1=pMP1(x1,k) is the SR inverse factor demand. • Graphically, draw the isoprofit and the production function.
On the x1-y plane, =py-w1x1-w2k. Thus an isoprofit curve, say =py-w1x1-w2k=c would be a straight line with the slope w1/p because y=c/p+(w1/p)x1+w2k/p. • 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.
If w2 increases, the slope stays the same, so x1 stays the same. So what will change? • The intercept on the y axis is c/p+w2k/p, so if w2 increases, given a same line, the profit level c decreases. • Long-run profit maximization: maxx1, x2=pf(x1, x2)-w1x1-w2 x2. FOC: pMP1(x1*, x2*)=w1 and pMP2(x1*, x2*)=w2. • 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. 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. • 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. • 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∆x20.
∆p∆y-∆w1∆x1-∆w2∆x20: ∆w1=∆w2=0, then ∆p∆y0 so the supply slopes upwards. Similarly, ∆p=∆w2=0, ∆w1∆x1≤0, so the factor demand slopes downwards. • 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* (max ).
