From the two FOCs we can solve the demand functions for capital and labor

# From the two focs we can solve the demand functions

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° From the two FOCs, we can solve the demand functions for capital and labor. ° Second order condition.. Chapter 12 Competitive Markets ° The Partial Equilibrium Competitive Model ° Focus on the market for one good only. ° Not a complete economy! ° The market/aggregate demand for good X is the sum of all individual demands. ° Market demand function: X ( p X ) = P n i =1 x i ( p X ; p Y ; m i ) ; where x i ( p X ; p Y ; m i ) is individual i ³s demand for good X . 34
° X ( p X ) obviously depends p X ( endogenous variable). X ( p X ) also depends on other prices and incomes ( exogenous) . ° It is the horizontal sum of all individual demand curves. ° Market demand curve shifts when any exogenous variable changes. ° For simplicity, denote the market demand as Q D ( p ) = X ( p X ) . ° Elasticities are de²ned similarly ° Price elasticity: e Q;p = @Q D ( p ) @p p Q D ( p ) : ° In general, market demand cannot be written as a function of aggregate income. ° In short run, the market supply is the sum of all individual ²rm³s supply functions. ° Short-run market supply: Q S ( p ) = P m i =1 q i ( p; v; w ) , where the number of ²rms, m , is ²xed, and use SR supply functions. ° Example 12.2 A Short-run Supply Function ° there are 100 identical ²rms, each has supply function q i ( p ) = 10 3 p . ° The market supply function is Q S ( p ) = 100 10 3 p = 1 ; 000 3 p . ° Equilibrium price determination: Q D ( p ° ) = Q S ( p ° ) . ° In long run, each ²rm mostly follows its long-run supply function. ° If the number of ²rms is ²xed exogenously, then the analysis is the same as in short run (non-negative pro²t). ° If ²rms are free to enter/exist the market, then the number of ²rms is deter- mined endogenously ! ° When market price is higher than a ²rm³s min average cost, then a typical ²rm will have positive pro²t, and more ²rm will enter the market, the market price decreases, entry stops when it is no longer pro²table. ° Additional to price and quantity, the number of ²rms varies! ° Example 12.4 In²nitely Elastic Long-run Supply (simpli²ed) ° Market demand: Q D ( p ) = 25 ; 000 ´ 3 p: ° A representative ²rm³s cost function: C ( q ) = 40 q 2 + 16 ; 000 . ° Average cost function: AC ( q ) = 40 q + 16 ; 000 q : ° Marginal cost function: MC ( q ) = 80 q: 35
° To ²nd min AC, set MC ( q ) = AC ( q ) : 80 q = 40 q + 16 ; 000 q ) q = 20 . So the min of AC is AC (20) = 1 ; 600 . ° A ²rm³s supply function is q ( p ) = ³ p 80 when p ¸ 1 ; 600 ; 0 when p · 1 ; 600 : ° When there are n ²rms, the market supply function is Q S ( p ) = n q ( p ) . ° To ²nd the competitive equilibrium, set Q S ( p ) = Q D ( p ) n p 80 = 25 ; 000 ´ 3 p: ° Find the market price: p ( n ) = 2 ; 000 ; 000 n + 240 ° To ²nd the number of ²rms, p ( n ) ¸ 1600 > p ( n + 1) 2 ; 000 ; 000 n +240 ¸ 1 ; 600 ) n = 1 ; 010 If n = 1 ; 011 , then p (1 ; 011) = 1 ; 599 < 1 ; 600 : ° A good approximation of the market supply function is the in²nitely elastic supply function p = min AC ( q ) : Chapter 13 General Equilibrium and Welfare ° A general equilibrium model describes a complete economy. ° Every consumer maximizes his/her utility. Every ²rm maximizes its pro²t. ° At a general equilibrium , demand = supply in every market.

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• Spring '19