Chapter 19 Profit Maximization

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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.

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• 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.

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• 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)gtw1, should use more
  of this factor. On the other hand, if
  pMP1(x1,k)ltw1, should use less of this factor.
  w1pMP1(x1,k) is the SR inverse factor demand.
• Graphically, draw the isoprofit and the
  production function.

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• On the x1-y plane, ?py-w1x1-w2k. Thus an
  isoprofit curve, say ?py-w1x1-w2kc would be a
  straight line with the slope w1/p because
  yc/p(w1/p)x1w2k/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.

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Fig. 19.1
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Fig. 19.2
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• 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/pw2k/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.

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• 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.

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• 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
  ?zzt-zs. We have ?p?y-?w1?x1-?w2?x2?0.

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• ?p?y-?w1?x1-?w2?x2?0 ?w1?w20, then ?p?y?0 so
  the supply slopes upwards. Similarly, ?p?w20,
  ?w1?x10, 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 ?).

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Fig. 19.4
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Fig. 19.5