Download presentation
Presentation is loading. Please wait.
Published byImogene Gray Modified over 9 years ago
1
Frank Cowell: Microeconomics Exercise 9.1 MICROECONOMICS Principles and Analysis Frank Cowell March 2007
2
Frank Cowell: Microeconomics Ex 9.1(1): Question purpose: Analyse consumption externality and efficiency purpose: Analyse consumption externality and efficiency method: Solve for equilibrium prices and allocation using standard GE. Then examine source of inefficiency method: Solve for equilibrium prices and allocation using standard GE. Then examine source of inefficiency
3
Frank Cowell: Microeconomics Ex 9.1(1): incomes and demands The term x 1 a is irrelevant to b-people's behaviour The term x 1 a is irrelevant to b-people's behaviour they cannot do anything about it… …although it affects their utility Incomes are Incomes are y a = 300 p 1 y a = 200 p 2 Both types have Cobb-Douglas utility functions Both types have Cobb-Douglas utility functions so we could jump straight to demand functions… …skip the Lagrangean step We know that their demands will be given by We know that their demands will be given by x 1 *a = ½ y a / p 1, x 2 *a = ½ y a / p 2 x 1 *b = ½ y b / p 1, x 2 *b = ½ y b / p 2 Skip Lagrangean
4
Frank Cowell: Microeconomics Ex 9.1(1): Lagrangean method Lagrangean for either type can be written Lagrangean for either type can be written kx 1 h x 2 h + [y h p 1 x 1 h p 2 x 2 h ] where is a Lagrange multiplier k is a constant (k =1 for type a, k =1/ x 1 a for type b) FOC for an interior maximum FOC for an interior maximum kx 2 h p 1 = 0 kx 1 h p 2 = 0 y h p 1 x 1 h p 2 x 2 h = 0 Substitute from FOC1, FOC2 into FOC3 to find Substitute from FOC1, FOC2 into FOC3 to find y h p 1 [ p 2 /k] p 2 [ p 1 /k] = 0 = ½ky h /p 1 p 2 Substitute this value of back into FOC2, FOC1 to get the demands: Substitute this value of back into FOC2, FOC1 to get the demands: x 1 *h = ½ y h / p 1 x 2 *h = ½ y h / p 2
5
Frank Cowell: Microeconomics Ex 9.1(1): Equilibrium price ratio Total demand for commodity 1 is Total demand for commodity 1 is N [ x 1 *a + x 1 *b ] = N [ ½ ⋅ 300 + ½ ⋅ 200/ ] where N is the large unknown number of traders and := p 1 / p 2 only the price ratio matters in the solution There are 300N units of commodity 1 There are 300N units of commodity 1 So the excess demand function for commodity 1 is So the excess demand function for commodity 1 is E 1 = [150 + 100/ ] N 300 N = [100/ 150] N To find equilibrium sufficient to put E 1 = 0 To find equilibrium sufficient to put E 1 = 0 if E 1 = 0 then E 2 = 0 also by Walras' Law Clearly E 1 = 0 exactly where = ⅔ Clearly E 1 = 0 exactly where = ⅔ the equilibrium price ratio
6
Frank Cowell: Microeconomics Ex 9.1(1): Equilibrium allocation Take the equilibrium price ratio = ⅔ Take the equilibrium price ratio = ⅔ Then, using the demand functions we find Then, using the demand functions we find x 1 *a = ½ ⋅ 300= 150 x 2 *a = ½ ⋅ 300 = 100 x 1 *b = ½ ⋅ 200 / = 150 x 2 *b = ½ ⋅ 200= 100 This is the equilibrium allocation This is the equilibrium allocation
7
Frank Cowell: Microeconomics Ex 9.1(2): Question method: Verify that CE allocation is inefficient by finding a perturbation that will produce a Pareto improvement Verify that CE allocation is inefficient by finding a perturbation that will produce a Pareto improvement
8
Frank Cowell: Microeconomics Ex 9.1(2): Source of inefficeincy It is likely that the a-people are consuming “too much” of good 1 It is likely that the a-people are consuming “too much” of good 1 there is a negative externality in the CE this is ignored So try changing the allocation So try changing the allocation so that the a-people consume less of good 1 x 1 a < 0 but where the a-people's utility remains unchanged The means that their consumption of good 2 must increase The means that their consumption of good 2 must increase given that, in equilibrium, = MRS, required adjustment is x 2 a = − x 1 a >0
9
Frank Cowell: Microeconomics Ex 9.1(2): Pareto-improving adjustment b-people's consumptions move in the opposite direction b-people's consumptions move in the opposite direction (there is a fixed total amount of each good) x 1 b = − x 1 a > 0 x 2 b = − x 2 a < 0 Effect on their utility can be computed thus: Effect on their utility can be computed thus: log U b = x 1 b / x 1 b + x 2 b / x 2 b − x 1 a /x 1 a = [ − 1/150 + ⅔(1/100) − 1/150] x 1 a = − x 1 a / 150 >0 So it is possible to make a Pareto-improving perturbation So it is possible to make a Pareto-improving perturbation move away from the CE in such a way that some people's utility is increased no-one else's utility decreases
10
Frank Cowell: Microeconomics Ex 9.1(3): Question and answer Can this be done by just tweaking prices? Can this be done by just tweaking prices? increase relative price of commodity 1 for the a-people… …relative to that facing the b-people? This will not work This will not work a-people’s income is also determined by p 1 … …and their resulting consumption of commodity 1 is independent of price A rationing scheme may work A rationing scheme may work
11
Frank Cowell: Microeconomics Ex 9.1: Points to remember Be careful to model what is under each agent’s control Be careful to model what is under each agent’s control Use common-sense to spot Pareto improvements Use common-sense to spot Pareto improvements
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.