John Riley 5 Setember 0 NSWERS T DD NUMERED EXERCISES IN CHPTER 3 SECTIN 3: Equilibrium and Efficiency Exercise 3-: Prices with Quasi-linear references (a) Since references are convex, an allocation is a Walrasian Equilibrium if and only if it is Pareto Efficient We begin by characterizing the Pareto Efficient allocations and then solve for the no-trade equilibrium rices given that an endowment is Pareto Efficient gent h has a MRS( x, x ) ( x ) h h h / = Fig D3-: Efficient allocations For an efficient allocation in the interior of the Edgeworth ox, the two agents must have the same MRS hence the equilibrium rice ratio / = 0 (a) long the line segment, x = x = 00 long the line the MRS is 0 Thus MRS( x, x ) = (00 + α) > (00 α) = MRS( x, x ) / / Thus agent s indifference curve is steeer than that of agent and so all the allocations on are Pareto Efficient long the line segment agent s indifference curve is steeer so these allocations are also Pareto Efficient nswers Ch 3 age
John Riley 5 Setember 0 Consider the line segment Since these oints are in the interior of agent s consumtion set, the equilibrium rice line must be tangential That is, / = MRS( x, x ) = (00 + α) / Thus the rice ratio rises along from 0 to 0 rguing symmetrically, the rice ratio again rises from 0 to 0 along Exercise 3-3: Walrasian Equilibrium (a) Each igg has symmetric Cobb-Douglas references with endowment (4,8) Thus a igg sends half his income 4 + 8on each commodity His demand for commodity is therefore x = r+ 4, where r = / Define ˆ x = x + a, where a= (4,6) and ˆ ω = ω + a = (4,6) Then each ittle s otimization roblem can be rewritten as follows Max{ln xˆ + ln xˆ xˆ = ( x + a) ˆ ω} xˆ ˆ ˆ This is again the symmetric Cobb-Douglas case so x = ω Hence xˆ = r + 8 and so x = r+ Summing, the market demand for commodity by each igg-ittle air is x = 4r + 6 Suly is 8 Equating suly and demand, the equilibrium rice is / (b) t an interior PE allocation the marginal rates of substitution must be equal Thus MRS 6 + x x = = = MRS 4 + x x 6+ x x 6+ x 6+ ω 4 lying the Ratio Rule, MRS = = = = = = 4+ x x 4+ x 4+ ω 48 (c) Note that the MRS is constant for all interior PE allocations so that a change in the distribution of the total endowment has no effect on the equilibrium rice unless the new equilibrium is no longer in the interior of the Edgeworth ox nswers Ch 3 age
John Riley 5 Setember 0 C Fig D3-: Equilibrium Price Ratios (d) This is the endowment oint C Since it is Pareto Efficient there is no trade and the equilibrium rice ratio is ½ (e) For any endowment in the unshaded region the Walrasian equilibrium rice is ½ For any endowment in the shaded region the equilibrium allocation is on the boundary of the Edgeworth ox Since (4,8) is a Pareto Efficient allocation in the interior of iggs consumtion set, MRS x 8 3 ( x, x) = x = = = 4 7 Thus the new Walrasian equilibrium rice is 3/7 Exercise 3-5: More on the iggs and ittles Since the ittles get no satisfaction from commodity an efficient allocation gives all of this commodity to the iggs Similarly, all of commodity must be allocated to the ittles Thus the only PE allocation is the oint in the South East corner of the Edgeworth ox nswers Ch 3 age 3
John Riley 5 Setember 0 (4,8) 0 Fig D3-3: Unique Equilibrium llocation The sloe of the line segment connecting the endowment oint and the PE allocation is 8/0 = /5 Thus the Walrasian equilibrium rice is /5 iggs sells his 8 units of commodity and urchases 0 units of commodity Exercise 3-5: Market Excess Demand (a) While not necessary, it roves useful to define excess of (,), that is y = x j j y to be lex s consumtion in Then lex s budget constraint, x + x (7 + α) + ( α), can be rewritten as follows y + y (5 + α) ( α) lex then solves the following standard Cobb-Douglas otimization roblem Max{ U = y y y + y (5 + α) ( + α)} y 4 + The consumtion set is the set of all y Thus to be able to urchase a vector in the consumtion set it is necessary and sufficient that (5 + α) ( α) 0 Hence nswers Ch 3 age 4
John Riley 5 Setember 0 α 5 + α rguing symmetrically, ev s otimization roblem can be written as follows Max{ U = y y y + y ( + α) + (5 + α)} y 5 Thus for ev to be able to make a urchase in her consumtion set, 5 + α α (b) We have reduced the otimization roblems to standard Cobb-Douglas otimization roblems Solving, Hence y = ( (5 + α) ( + α)) and y = ( ( + α) + (5 + α)) x 5 6 (5 α) ( α) 6 6 6 5 = + + + and x = ( + α) + (5 + α) 6 6 (c) dding these two exression and then subtracting aggregate suly, market excess demand is α e( ) = x + x 8= ( ) 3 (d) Excess demand for commodity thus increases with if and only if α > 0 Note that lex has a very large fraction of the aggregate endowment if α > 0 He is a net seller of commodity thus the income effect on his demand for commodity is large and ositive ev has a small endowment of commodity so her income effect is negative but small The market income effect is therefore ositive This ositive effect more than offsets the negative substitution effects for both consumers (e) For all rice rations satisfying the inequalities in (a), excess demand is zero Thus all these rice rations are WE rice ratios Thus we have shown by examle that there can be a continuum of equilibrium rice ratios! nswers Ch 3 age 5
John Riley 5 Setember 0 Exercise 3-7: Suorting and searating hyerlanes (a) Suose there exist y inty such that y = y 0 For δ > 0 sufficiently small ŷ y δ Y = + Then ŷ = y + δ = y 0 + δ > y 0 which contradicts (i) (b) If s or t is in the interior then y = s t int Y since there exists a neighborhood around y that is in Y, the same neighborhood around s or t that is in S or T Then y < 0, that is, s < t SECTIN 3: The Fundamental Welfare Theorems Exercise 3-: Second welfare theorem with identical homothetic references Consider the economy with a single reresentative agent et xˆ = arg Max{ U( x) x y+ ω, y Y } where Y is the aggregate roduction set, that xy, F f f f is, Y = { y y = y, y Y } The aggregate roduction set Y is convex since it is f = the sum of convex sets Since U is strictly increasing, xˆ = yˆ+ ω lies on the boundary of the aggregate roduction set Thus by the searating hyerlane theorem there exists a non-zero vector such that (i) y y y U( x) > U( x) x> x * Y and (ii) ˆ ˆ It is readily shown that, if (i) is true, then f f f f y Y y yˆ From (ii), nswers Ch 3 age 6
John Riley 5 Setember 0 xˆ = arg Max{ U( x) x xˆ = ( yˆ+ ω)} xy, h Finally MRS( θ xˆ) = MRS( xˆ) Thus for any { θ h } H, h h= θ =, { h ˆ } H θ x h= is Pareto efficient H h= nswers Ch 3 age 7