Frank Cowell: Microeconomics Exercise 9.6 MICROECONOMICS Principles and Analysis Frank Cowell February 2007

Frank Cowell: Microeconomics Ex 9.6(1): Question 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 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  method: find price-taking optimising demands for each of the two types, use these to compute the excess demand function and solve for 

Frank Cowell: Microeconomics Ex 9.6(1): budget constraints Use commodity 2 as numéraire Use commodity 2 as numéraire  price of good 1 is   price of good 2 is 1 Evaluate incomes for the two types, given their resources: Evaluate incomes for the two types, given their resources:  type a has endowment (30, k)  therefore y a = 30  + k  type b has endowment (60, 210  k)  therefore y b = 60  + [210  k] Budget constraints for the two types are therefore: Budget constraints for the two types are therefore:   x 1 a + x 2 a ≤ 30  + k   x 1 b + x 2 b ≤ 60  + [210  k]

Frank Cowell: Microeconomics Ex 9.6(1): optimisation We could jump straight to a solution We could jump straight to a solution  utility functions are simple…  …so we can draw on known results Cobb-Douglas preferences imply Cobb-Douglas preferences imply  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) In this case (for type a)  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… Jump to “equilibrium price”

Frank Cowell: Microeconomics Ex 9.6(1): optimisation, type a The Lagrangean is: The Lagrangean is:  2log x 1 a + log x 2 a + a [y a   x 1 a  x 2 a ]  where a is the Lagrange multiplier  and y a is 30  + k FOC for an interior solution FOC for an interior solution  2/x 1 a  a  = 0  1/x 2 a  a  = 0  y a   x 1 a  x 2 a = 0 Eliminating a from these three equations, demands are: Eliminating a from these three equations, demands are:  x 1 a  = ⅔ y a /   x 2 a  = ⅓ y a

Frank Cowell: Microeconomics Ex 9.6(1): optimisation, type b The Lagrangean is: The Lagrangean is:  log x 1 b + 2log x 2 b + b [y b   x 1 b  x 2 b ]  where b is the Lagrange multiplier  and y b is 60   k FOC for an interior solution FOC for an interior solution  1/x 1 b  b  = 0  2/x 2 b  b  = 0  y b   x 1 b  x 2 b = 0 Eliminating b from these three equations, demands are: Eliminating b from these three equations, demands are:  x 1 b  = ⅓ y b /   x 2 b  = ⅔y b

Frank Cowell: Microeconomics Ex 9.6(1): equilibrium price Take demand equations for the two types Take demand equations for the two types  substitute in the values for income  type-a demand becomes  type-b demand becomes Excess demand for commodity 2: Excess demand for commodity 2:  [10  + ⅓k]+[40  +140 − ⅔k] − 210  which simplifies to 50  − ⅓k − 70 Set excess demand to 0 for equilibrium: Set excess demand to 0 for equilibrium:  equilibrium price must be:   = [210 + k] / 150

Frank Cowell: Microeconomics Ex 9.6(2): Question and solution Incomes for the two types are resources: Incomes for the two types are resources:  y a = 30  + k  y b = 60  + [210  k] The equilibrium price is: The equilibrium price is:   = [210 + k] / 150 So we can solve for incomes as: So we can solve for incomes as:  y a = [ k] / 5  y b = [1470  3k] / 5 Equivalently we can write y a and y b in terms of  as Equivalently we can write y a and y b in terms of  as  y a = 180   210  y b = 420  90 

Frank Cowell: Microeconomics Ex 9.6(3): Question purpose: to use the outcome of the GE model to plot the “income- possibility” set purpose: to use the outcome of the GE model to plot the “income- possibility” set method: plot incomes corresponding to extremes of allocating commodity 2, namely k = 0 and k = 210. Then fill in the gaps. method: plot incomes corresponding to extremes of allocating commodity 2, namely k = 0 and k = 210. Then fill in the gaps.

Frank Cowell: Microeconomics Income possibility set yaya ybyb (42, 294) (294, 168)   incomes for k = 0   incomes for k = 210   incomes for intermediate values of k   attainable set if income can be thrown away   y b = 315  ½y a

Frank Cowell: Microeconomics Ex 9.6(4): Question purpose: find a welfare optimum subject to the “income-possibility” set purpose: find a welfare optimum subject to the “income-possibility” set method: plot contours for the function W on the previous diagram. method: plot contours for the function W on the previous diagram.

Frank Cowell: Microeconomics Welfare optimum: first case yaya ybyb   income possibility set   Contours of W = log y a + log y b   Maximisation of W over income- possibility set   W is maximised at corner   incomes are (294, 168)   here k = 210   so optimum is where all of resource 2 is allocated to type a

Frank Cowell: Microeconomics Ex 9.6(5): Question purpose: as in part 4 purpose: as in part 4 method: as in part 4 method: as in part 4

Frank Cowell: Microeconomics Welfare optimum: second case yaya ybyb   income possibility set   Contours of W = y a + y b   Maximisation of W over income- possibility set   again W is maximised at corner   …where k = 210   so optimum is where all of resource 2 is allocated to type a

Frank Cowell: Microeconomics Ex 9.6: Points to note Applying GE methods gives the feasible set Limits to redistribution   natural bounds on k   asymmetric attainable set Must take account of corners Get the same W-maximising solution   where society is averse to inequality   where society is indifferent to inequality