Exercise 9.1

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

2. 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

3. 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

4. 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

5. 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

6. 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

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

8. 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

9. 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

10. 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

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