Coursenotes_ECON301

# 8 r 04 w utility maximization maximize ub xb12yb12

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Unformatted text preview: B Solving these two equations for XB & YB we get the demands by consumer B. XB = MB 2PX XB = 0.8r + 0.4w 2PX XB = 4r + 2w 10PX YB = MB 2PY YB = 0.8r + 0.4w 2PY YB = 4r + 2w 10PY PRODUCER OF GOOD Y The producer of good Y has the following MRTS: MRTSY = LY KY Cost Minimization minimize cost r KY + w LY subject to KY1/2LY1/2 = QY Producer Equilibrium Analytically, the two conditions for producer equilibrium must be satisfied: 138 MRTSX = _r_ w KX1/2LX1/2 = QX Solving these two equations for KX & LX we get the demands by producer X. KX-1/2LX1/2 = _r_ KX1/2LX1/2 w LX = _r_ KX w KX = LXw r LX = KXr w sub KX into production function: QX = [LX(w / r)]1/2LX1/2 QX = LX(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 returns scale production function, we can candivide 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 MRTSY = _r_ w KY1/2LY1/2 = QY Solving these two equations for KY & LY we get the demands by producer Y. KY-1/2LY1/2 = _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 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 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 139 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... XA = r + 3w = r + 3w 10PX 20(wr)1/2 XB = 4r + 2w = 4r + 2w 20(wr)1/2 10PX 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)1/2 140 4r + 2w 20(wr)1/2 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... YA = r +...
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## This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.

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