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

Ex 9.1(1): Question purpose: Analyse consumption externality and efficiency method: Solve for equilibrium prices and allocation using standard GE. Then examine source of inefficiency

Ex 9.1(1): incomes and demands The term x1a is irrelevant to b-people's behaviour they cannot do anything about it… …although it affects their utility Incomes are ya = 300 p1 ya = 200 p2 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 x1*a = ½ ya / p1 , x2*a = ½ ya / p2 x1*b = ½ yb / p1 , x2*b = ½ yb / p2 Skip Lagrangean

Ex 9.1(1): Lagrangean method Lagrangean for either type can be written kx1h x2h + n[yh  p1x1h  p2 x2h ] where n is a Lagrange multiplier k is a constant (k =1 for type a , k =1/ x1a for type b) FOC for an interior maximum kx2h  np1 = 0 kx1h  np2 = 0 yh  p1x1h  p2 x2h = 0 Substitute from FOC1, FOC2 into FOC3 to find n yh  p1[np2 /k]  p2[np1 /k] = 0 n = ½kyh /p1p2 Substitute this value of n back into FOC2, FOC1 to get the demands: x1*h = ½ yh / p1 x2*h = ½ yh / p2

Ex 9.1(1): Equilibrium price ratio Total demand for commodity 1 is N [ x1*a + x1*b ] = N [ ½ ⋅300 + ½ ⋅ 200/r ] where N is the large unknown number of traders and r := p1 / p2 only the price ratio matters in the solution There are 300N units of commodity 1 So the excess demand function for commodity 1 is E1 = [150 + 100/r ] N  300 N = [100/r  150] N To find equilibrium sufficient to put E1 = 0 if E1 = 0 then E2 = 0 also by Walras' Law Clearly E1 = 0 exactly where r = ⅔ the equilibrium price ratio

Ex 9.1(1): Equilibrium allocation Take the equilibrium price ratio r = ⅔ Then, using the demand functions we find x1*a = ½ ⋅ 300 = 150 x2*a = ½ ⋅ 300r = 100 x1*b = ½ ⋅ 200 / r = 150 x2*b = ½ ⋅ 200 = 100 This is the equilibrium allocation

Ex 9.1(2): Question method: Verify that CE allocation is inefficient by finding a perturbation that will produce a Pareto improvement

Ex 9.1(2): Source of inefficeincy 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 that the a-people consume less of good 1 Dx1a < 0 but where the a-people's utility remains unchanged The means that their consumption of good 2 must increase given that, in equilibrium, r = MRS, required adjustment is Dx2a = −rDx1a >0

Ex 9.1(2): Pareto-improving adjustment b-people's consumptions move in the opposite direction (there is a fixed total amount of each good) Dx1b = −Dx1a > 0 Dx2b = −Dx2a < 0 Effect on their utility can be computed thus: Dlog Ub = Dx1b / x1b + Dx2b / x2b − Dx1a /x1a = [ − 1/150 + ⅔(1/100) − 1/150] Dx1a = − Dx1a / 150 >0 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

Ex 9.1(3): Question and answer 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 a-people’s income is also determined by p1 … …and their resulting consumption of commodity 1 is independent of price A rationing scheme may work

Ex 9.1: Points to remember Be careful to model what is under each agent’s control Use common-sense to spot Pareto improvements