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Unformatted text preview: QY Solving these two equations for KY & LY we get the demands by producer Y. KY1/2LY1/2 = _r_ KY1/2LY1/2 w LY = _r_ KY w KY = LYw r LY = KYr w sub KY into production function: QY = [LY(w / r)]1/2LY1/2 QY = LY(w / r)1/2 LX = QX(r / w)1/2 Similarly, we can find that... KX = QX(w / r)1/2 Constant Returns to Scale Since f(KX,LX) is a constant returns to scale production function, we can divide factor demands KX,LX by the total output level QX in order to get the factor demands on a per unit of output basis as follows: KX = kX = (w / r)1/2 QX LX = lX = (r / w)1/2 QX Marginal Cost MCX = r kX + w lX Perfect Competition Under perfect competition, producer X must satisfy the zero profit condition: PX = MCX (r,w) PX = r kX + w lX PX = r (w / r)1/2 + w (r / w)1/2 PX = (wr2 / r)1/2 + (rw2 / w)1/2 PX = 2(wr)1/2 MARKET FOR GOOD X Substituting these output prices into individual consumer demands... 254 LY = QY(r / w)1/2 Similarly, we can find that... KY = QY(w / r)1/2 Constant Returns to Scale Since g(KY,LY) is a constant returns to scale production function, we can divide factor demands KY,LY by the total output level QY in order to get the factor demands on a per unit of output basis as follows: KY = kY = (w / r)1/2 QY LY = lY = (r / w)1/2 QY Marginal Cost MCY = r kY + w lY Perfect Competition Under perfect competition, producer Y must satisfy the zero profit condition: PY = MCY (r,w) PY = r kY + w lY PY = r (w / r)1/2 + w (r / w)1/2 PY = (wr2 / r)1/2 + w (rw2 / w)1/2 PY = 2(wr)1/2 MARKET FOR GOOD Y Substituting these output prices into individual consumer demands... XA = r + 3w = r + 3w 10PX 20(wr)1/2 XB = 4r + 2w = 4r + 2w 10PX 20(wr)1/2 and aggregate demand for good X X = r + 3w + 20(wr)1/2 X = 5r + 5w 20(wr)1/2 X=r+w 4(wr)1/2 X = X(r , w) On the supply side, producer X provides QX as the market supply of good X. At market equilibrium, the output supply of good X must equal the aggregate demand of good X. QX = X(r , w) = r + w 4(wr)1/2 MARKET FOR CAPITAL Substituting this output supply, QX, into the producer demands for capital, we get... KX = QX(w / r)1/2 w1/2 KX = r + w 1/2 4(wr) r1/2 KX = r + w 4r 255 4r + 2w 20(wr)1/2 YA = r + 3w = r + 3w 10PY 20(wr)1/2 YB = 4r + 2w = 4r + 2w 10PY 20(wr)1/2 and aggregate demand for good Y Y = r + 3w + 20(wr)1/2 Y = 5r + 5w 20(wr)1/2 Y=r+w 4(wr)1/2 Y = Y(r, w) On the supply side, producer Y provides QY as the marke...
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 Spring '10
 sning
 Economics

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