Frank Cowell Microeconomics February 2007 Exercise 9 6

Frank Cowell: Microeconomics February 2007 Exercise 9. 6 MICROECONOMICS Principles and Analysis Frank Cowell

Ex 9. 6(1): Question Frank Cowell: Microeconomics n n purpose: to derive equilibrium prices and incomes as a function of endowment. To show the limits to redistribution within the GE model for a alternative SWFs method: find price-taking optimising demands for each of the two types, use these to compute the excess demand function and solve for r

Ex 9. 6(1): budget constraints Frank Cowell: Microeconomics n Use commodity 2 as numéraire u u n Evaluate incomes for the two types, given their resources: u u n price of good 1 is r price of good 2 is 1 type a has endowment (30, k) therefore ya = 30 r + k type b has endowment (60, 210 k) therefore yb = 60 r + [210 k] Budget constraints for the two types are therefore: u u rx 1 a + x 2 a ≤ 30 r + k rx 1 b + x 2 b ≤ 60 r + [210 k]

Ex 9. 6(1): optimisation Frank Cowell: Microeconomics Jump to “equilibrium price” n We could jump straight to a solution u u n Cobb-Douglas preferences imply u u u n utility functions are simple… …so we can draw on known results indifference curves do not touch the origin… …so we need consider only interior solutions also demand functions for the two commodities exhibit constant expenditure shares In this case (for type a) u u coefficients of Cobb-Douglas are 2 and 1 so expenditure shares are ⅔ and ⅓ (and for b they will be ⅓ and ⅔ ) gives the optimal demands immediately…

Ex 9. 6(1): optimisation, type a Frank Cowell: Microeconomics n The Lagrangean is: u u u n FOC for an interior solution u u u n 2 log x 1 a + log x 2 a + na[ya rx 1 a x 2 a ] where na is the Lagrange multiplier and ya is 30 r + k 2/x 1 a nar = 0 1/x 2 a na = 0 ya rx 1 a x 2 a = 0 Eliminating na from these three equations, demands are: u u x 1 a = ⅔ ya / r x 2 a = ⅓ ya

Ex 9. 6(1): optimisation, type b Frank Cowell: Microeconomics n The Lagrangean is: u u u n FOC for an interior solution u u u n log x 1 b + 2 log x 2 b + nb[yb rx 1 b x 2 b ] where nb is the Lagrange multiplier and yb is 60 r + 210 k 1/x 1 b nbr = 0 2/x 2 b nb = 0 yb rx 1 b x 2 b = 0 Eliminating nb from these three equations, demands are: u x 1 b = ⅓ yb / r u x 2 b = ⅔yb

Ex 9. 6(1): equilibrium price Frank Cowell: Microeconomics n Take demand equations for the two types u substitute in the values for income type-a demand becomes u type-b demand becomes u n Excess demand for commodity 2: u u n [10 r + ⅓k]+[40 r +140 − ⅔k] − 210 which simplifies to 50 r − ⅓k − 70 Set excess demand to 0 for equilibrium: u u equilibrium price must be: r = [210 + k] / 150

Ex 9. 6(2): Question and solution Frank Cowell: Microeconomics n Incomes for the two types are resources: u u n The equilibrium price is: u n r = [210 + k] / 150 So we can solve for incomes as: u u n ya = 30 r + k yb = 60 r + [210 k] ya = [210 + 6 k] / 5 yb = [1470 3 k] / 5 Equivalently we can write ya and yb in terms of r as u u ya = 180 r 210 yb = 420 90 r

Ex 9. 6(3): Question Frank Cowell: Microeconomics n n purpose: to use the outcome of the GE model to plot the “incomepossibility” set method: plot incomes corresponding to extremes of allocating commodity 2, namely k = 0 and k = 210. Then fill in the gaps.

Income possibility set Frank Cowell: Microeconomics yb §incomes for k = 0 §incomes for k = 210 §incomes for intermediate values of k 300 • §attainable set if income can be thrown away (42, 294) § yb = 315 ½ya 200 • (294, 168) 100 ya 0 100 200 300

Ex 9. 6(4): Question Frank Cowell: Microeconomics n n purpose: find a welfare optimum subject to the “income-possibility” set method: plot contours for the function W on the previous diagram.

Welfare optimum: first case Frank Cowell: Microeconomics yb §income possibility set §Contours of W = log ya + log yb §Maximisation of W over incomepossibility set 300 § W is maximised at 200 corner • § incomes are (294, 168) § here k = 210 100 § so optimum is where all of resource 2 is allocated to type a ya 0 100 200 300

Ex 9. 6(5): Question Frank Cowell: Microeconomics n n purpose: as in part 4 method: as in part 4

Welfare optimum: second case Frank Cowell: Microeconomics yb §income possibility set §Contours of W = ya + yb §Maximisation of W over incomepossibility set 300 § again W is maximised 200 at corner • § …where k = 210 § so optimum is where all 100 of resource 2 is allocated to type a ya 0 100 200 300

Ex 9. 6: Points to note Frank Cowell: Microeconomics n n Applying GE methods gives the feasible set Limits to redistribution u u n n natural bounds on k asymmetric attainable set Must take account of corners Get the same W-maximising solution u u where society is averse to inequality where society is indifferent to inequality