Frank Cowell: Microeconomics Exercise 9.1 MICROECONOMICS Principles and Analysis Frank Cowell March 2007

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

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

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

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 ⋅ 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 = [ /  ] 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

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

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

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

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

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

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