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Unformatted text preview: om Walras Law (once all other markets are in equilibrium). In other words, we can eliminate one equation from the solution by using Walras Law. In other words, we can eliminate one unknown from the solution by using relative prices (numeraire). Thus, using both Walras Law and labour as the numeraire (w 1), we can further reduce the two equations (1), (2) to one equation of capital market equilibrium in one unknown capital price K(r, w) = K 136 This is the simplest reduction of the problem: solving one equation for one unknown. We cannot simplify the production economy any further. How about an example... Consider the "square root economy" with square root functions for both consumers and producers UA = XA1/2YA1/2 UB = XB1/2YB1/2 QX = KX1/2LX1/2 QY = KY1/2LY1/2 and the following initial factor endowment distribution: Consumer A Consumer B Total Capital (K) KA = 0.2 KB = 0.8 KT = 1 Labour (L) LA = 0.6 LB = 0.4 LT = 1 We want to find the equilibrium prices (PX, PY, r, w) that clear all markets simultaneously. CONSUMER A Consumer A has the following data on utility function and endowment income: MRSA = YA XA MA = r KA + w LA = 0.2 r + 0.6 w Utility Maximization maximize UA = XA1/2YA1/2 subject to PX XA + PY YA = MA Consumer Equilibrium Analytically, the two conditions for consumer equilibrium must be satisfied: CONSUMER B Consumer B has the following data on utility function and endowment income: MRSB = YB XB MB = r KB + w LB = 0.8 r + 0.4 w Utility Maximization maximize UB = XB1/2YB1/2 subject to PX XB + PY YB = MB Consumer Equillibrium Analytically, the two conditions for consumer equilibrium must be satisfied: 137 MRSA = PX PY PX XA + PY YA = MA Solving these two equations for XA & YA we get the demands by consumer A. XA = M A 2PX XA = 0.2r + 0.6w 2PX XA = r + 3w 10PX YA = M A 2PY YA = 0.2r + 0.6w 2PY YA = r + 3w 10PY PRODUCER OF GOOD X The producer of good X has the following MRTS: MRTSX = LX KX Cost Minimization minimize cost r KX + w LX subject to KX1/2LX1/2 = QX Producer Equilibrium Analytically, the two conditions for producer equilibrium must be satisfied: MRSB = PX PY PX XB + PY YB = M...
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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.
- Spring '10