U b (x b ) = xb 1x b 2 x a 1. means the consumption of good i by an h-type person.

Transcription

1 Capter 9 Welfare Eercise 9. In a two-commodity ecange economy tere are two large equalsized groups of traders. Eac trader in group a as an endowment of 300 units of commodity ; eac person in group b as an endowment of 200 units of commodity 2. Eac a-type person as preferences given by te utility function U a ( a ) = a a 2 and eac b-type person s utility can be written as U b ( b ) = b b 2 a were i means te consumption of good i by an -type person.. Find te competitive equilibrium allocation 2. Eplain wy te competitive equilibrium is ineffi cient. 3. Suggest a means wereby a benevolent government could acieved an effi - cient allocation.. Notice tat te term a tat appears in te b-type s utility function is a negative eternality: te more tat a-people consume of commodity, te lower is every b-person s utility. Tis is te reason tat te competitive equilibrium will be ineffi cient. To derive te competitive equilibrium, notice tat te term a is virtually irrelevant to te b-people s beaviour (tey cannot do anyting about it). Bot a-people and b-people tus ave Cobb-Douglas utility functions, and we know tat teir demands will be given by: i = y, = a, b; i =, 2. 2 p i Incomes are y a = 300p, y b = 200p 2. Using tis information we can see tat total demand for commodity is N [ [ a + b ] = N ] ρ 37

2 Microeconomics CHAPTER 9. WELFARE were N is te large unknown number of traders in eac group and ρ := p /p 2 (Notice tat only te price ratio matters in te solution). Clearly tere are 300N units of commodity available, so te ecess demand function for commodity is: [ E = N ] ρ 300 By Walras Law we know tat if E = 0 ten E 2 = 0 also. Clearly E = 0 wen ρ = 2/3; tis is te equilibrium price ratio. Using te demand functions we find [ ] [ a a = ] [ ] ρ = (9.) 00 [ ] [ b b = 2 200/ρ ] [ ] = (9.2) 00 Tis is te competitive equilibrium allocation. 2. To verify tat tis allocation is ineffi cient consider te following. Since tere is a negative eternality, it is likely tat in te competitive equilibrium te a-people are consuming too muc of commodity. So let us cange te allocation in suc a way tat te a-people consume less of commodity ( a < 0) but were te a-people s utility remains uncanged: tis means tat teir consumption of good 2 must be increased, by an amount a 2 = ρ a > 0 (remember tat in equilibrium ρ equals te marginal rate of substitution). Now since tere is a fied total amount of eac commodity, te b-people s consumptions must move in eactly te opposite direction; so b = a > 0 and b 2 = a 2 < 0. Te effect on teir utility can be computed tus: log(u b ) = b b + b b 2 2 a a [ = ] a 50 = 50 a > 0 So, as we epected, it is possible to move away from te competitive equilibrium in suc a way tat some people s utility is increased, and no-one else s utility decreases. 3. We migt tink tat for effi ciency te relative price of commodity sould just be increased to te a-people, relative to tat facing te b-people. But tis will not work since teir income is determined by p, and, as te demand function reveals, teir resulting consumption of commodity is independent of price. A rationing sceme may (in tis case) be simpler. c Frank Cowell

3 Microeconomics Eercise 9.2 Consider a constitution Σ based on a system of rank-order voting wereby te worst alternative gets point, te net worst, 2,... and so on, and te state tat is awarded te most points by te citizens is te one selected. Alf s ranking of social states canges during te week. Bill s stays te same: Monday: Tuesday: Alf Bill Alf Bill θ θ θ θ θ θ θ θ θ θ θ θ Wat is te social ordering on Monday? Wat is it on Tuesday? How does tis constitution violate te IIA Aiom? Use to denote strict preference and to denote indifference. Te voting points scored for te tree states [ θ, θ, θ ] are as follows; Monday: [5,5,2], Tuesday: [5,4,3]. So te social ordering canges from θ θ θ to θ θ θ. Notice tat te rankings by Alf and Bill of θ and θ remain uncanged; yet teir reranking of θ wit reference to oter alternatives canges te social ordering of θ and θ from indifference to strict preference. c Frank Cowell

4 Microeconomics CHAPTER 9. WELFARE Eercise 9.3 Consider an economy tat consists of just tree individuals, {a, b, c} and four possible social states of te world. Eac state-of-te-world is caracterised by a monetary payoff y tus: a b c θ θ 4 4 θ 5 3 θ 2 6 Suppose tat person as a utility function U = log(y ).. Sow tat if individuals know te payoffs tat will accrue to tem under eac state-of-te-world, ten majority voting will produce a cyclic decision rule. 2. Sow tat te above conditions can rank unequal states over perfect equality. 3. Sow tat if people did not know wic one of te identities {a, b, c} tey were to ave before tey vote, if tey regard any one of tese tree identities as equally likely and if tey are concerned to maimise epected utility, ten majority voting will rank te states strictly in te order of te distribution of te payoffs. 4. A group of identical scoolcildren are to be endowed at lunc time wit an allocation of pie. Wen tey look troug te dining all window in te morning tey can see te slices of pie lying on te plates: te only problem is tat no cild knows wic plate e or se will receive. Taking te space of all possible pie distributions as a complete description of all te possible social states for tese scoolcildren, and assuming tat e ante tere are equal cances of any one cild receiving any one of te plates discuss ow a von Neumann-Morgenstern utility function may be used as a simple social-welfare function. 5. Wat determines te degree of inequality aversion of tis social-welfare function? 6. Consider te possible problems in using tis approac as a general metod of specifying a social-welfare function.. We find tat majority voting produces te ranking θ θ θ but also θ θ ; i.e. te states {θ, θ, θ } form a cycle. (ii) θ would be strictly preferred to te state of perfect equality θ. (iii) Since te probability of being assigned any one of te tree identities is 3, te utility payoffs are: c Frank Cowell

5 Microeconomics a b c Epected Utility θ log(27) θ log(6) θ log(5) θ log(2) 2. Now consider ordering te states by income inequality Distribution Epected Utility θ [3,3,3] 3 log(27) θ [,4,4] 3 log(6) θ [,3,5] 3 log(5) θ [,2,6] 3 log(2) 3. Concavity of te utility function implies (in a primordial ignorance about identity) a preference for equality. 4. Let tere be n scoolcildren, and let te allocation of pie received by cild be y, =, 2,...n. Te set of all possible pie distributions is { } [y, y 2 ] n,..., y n : y 0, y If eac cild perceives an equal cance of being assigned any plate te epected utility of te perceived pie distribution is n n = u ( y ) were u is an increasing function. If u is concave ten tis utility function will rank more equal distributions as being preferable to less equal distributions. = 5. Inequality aversion is identical to risk aversion. 6. In practice te problem of receiving a particular pie allocation will not be te same individuals; personal risk-aversion may not be an appropriate basis for inequality aversion wit reference to life cances. c Frank Cowell

6 Microeconomics CHAPTER 9. WELFARE Eercise 9.4 Table 9. sows te preferences over tree social states for two groups of voters; te row marked # gives te number of voters wit eac set of preferences; preferences are listed in row order, most preferred at te top.. Find te Condorcet winner among rigt-anded voters only. 2. Sow tat tere is a cycle among left-anded voters only. 3. Suppose tat te cycle among te left-anded voters is broken by ignoring te vote tat as te smallest winner. Sow tat te winner is ten te same as tat among te rigt-anded voters. 4. Sow tat if te two groups are merged tere is a Condorcet winner but is diff erent from te winners found for te left-anders and te rigt-anders separately! 5. Would te above parado occur if one used de Borda voting? Left-anders Rigt-anders # θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ Table 9.: Left-anded and rigt-anded voters Use to denote beats in a bilateral vote.. Clearly, from te rigt-and part of te table θ θ and θ θ, bot by a majority of 8 to 7. So θ is unambiguously te Condorcet winner among te rigt-and voters 2. Consider pairwise votes among te left-and group: θ versus θ. Tere are 0+2 votes for θ against 6+6 votes for θ. So θ θ by 22 to 2 θ versus θ. Tere are votes for θ against 2 votes for θ.so θ θ by 22 to 2. θ versus θ. Tere are 0+6 votes for θ against 6+2 votes for θ. So θ θ by 8 to 6 So tere is a cycle θ θ θ θ. 3. Te weakest link in te above cycle is θ θ; ere θ wins by only two votes as against te margin of 0 votes in te oter two cases. If we remove tis link it is ten clear tat te winner is θ as wit te rigt-and voters c Frank Cowell

7 Microeconomics 4. For te two groups togeter: θ versus θ. Tere are votes for θ against 6+6 votes for θ. So θ θ by 57 to 2 θ versus θ. Tere are votes for θ against votes for θ.so θ θ by 47 to 22. θ versus θ. Tere are votes for θ against votes for θ. So θ θ by 35 to 34 So now θ θ θ unambiguously. θ is te clear winner! 5. Borda voting were 3 is te score given to te best alternative, to te worst. Votes are as in Table 9.2. θ wins in eac subgroup and overall. Left-anders Rigt-anders Bot groups tot 8 7 tot θ θ θ Table 9.2: Borda votes c Frank Cowell

8 Microeconomics CHAPTER 9. WELFARE Eercise 9.5 Suppose social welfare is related to individual incomes y tus: were ζ( ) as te form n W = ζ(y ) = ζ() = ɛ ɛ and ɛ is a non-negative parameter.. Wat form does ζ take for ɛ =? [Hint, use l Hôpital s rule]. 2. Wat is relative inequality aversion for tis W? 3. Draw te contours of te social welfare function for te cases ɛ =, ɛ 0, ɛ. Wat is equally-distributed-equivalent income in eac case? 4. If, instead of a finite population {,.., n }, tere is a continuum of individuals distributed on R wit density at income y given by f(y) write down te equivalent form of te social welfare function W in general and in te particular cases cited in part 3.. Clearly te denominator and te numerator of ɛ ɛ bot vanis as ɛ. L Hôpital s rule implies tat te limiting value can be found by taking te ratio of te first derivatives of te denominator and numerator. Evaluating te derivatives we ave 2. Differentiating ζ we get and so ι() = ɛ. ɛ ɛ log lim = lim = log. ɛ ɛ ɛ dζ() d = ɛ d 2 ζ() d 2 = ɛ ɛ 3. For te case ɛ = contours will be rectangular yperbolas and ede-income is te geometric mean. For ɛ = 0 contours will be diagonal straigt lines and ede-income is te aritmetic mean. For ɛ = contours will be L- sapes and ede-income is te smallest of te incomes { y }. 4. W = = = ζ(y)f(y)dy yf(y)dy, if ɛ = 0 log yf(y)dy, if ɛ = = inf{y : f(y) > 0}, if ɛ = c Frank Cowell

9 Microeconomics Eercise 9.6 In a two-commodity ecange economy tere are two groups of people: type a ave te utility function 2 log( a ) + log( a 2) and an endowment of 30 units of commodity and k units of commodity 2; type b ave te utility function log( b ) + 2 log( b 2) and an endowment of 60 units of commodity and 20 k units of commodity 2.. Sow tat te equilibrium price, ρ, of good in terms of good 2 is 20+k 50. [Hint: use te answer to Eercise 7.4]. 2. Wat are te individuals incomes (y a, y b ) in equilibrium as a function of k? As a function of ρ? 3. Suppose it is possible for te government to carry out lump-sum transfers of commodity 2, but impossible to transfer commodity. Use te previous answer to sow te set of income distributions tat can be acieved troug suc transfers. Draw tis in a diagram. 4. If te government as te social welfare function W (y a, y b ) = log(y a ) + log(y b ) find te optimal distribution of income using te transfers mentioned in part 3. [int use te diagram constructed earlier]. 5. If instead te government as te social welfare function W (y a, y b ) = y a + y b find te optimal distribution of income using transfers. Comment on te result.. Let us use commodity 2 as numéraire. From te details in te question we find tat incomes for te two types of people are: y a = 30ρ + k (9.3) y b = 60ρ + [20 k] (9.4) were ρ is te price of commodity in terms of commodity 2. Again we ave a Cobb-Douglas utility function (Cf te answer to Eercise 7.4), and so te demands by a for te two commodities are [ ρ+k ρ 3 [30ρ + k] and for b te demands are [ 3 60ρ+20 k ρ 2 3 ] [60ρ + 20 k] Tis means tat te ecess demand for commodity 2 at price must ρ be [0ρ + 3 ] k + [40ρ ] k 20 ] c Frank Cowell

10 Microeconomics CHAPTER 9. WELFARE Te equilibrium value of ρ is found by setting tis equal to 0 (remember tat Walras Law will ensure tat te oter market also as zero ecess demand). Doing tis we get te value specified in te question. 2. Substituting back into (9.3) and (9.4) we get: y a = y b = k k 5 or, using te formula for te equilibrium price, we ave equivalently: (9.5) (9.6) y a = 80ρ 20 (9.7) y b = ρ (9.8) 3. Equations (9.5) and (9.6) imply tat tere is a straigt-line frontier on te set of income pairs (y a, y b ) mapped out by letting k vary from 0 to 20 (or by using te price equations), in oter words a straigt line from (42,294) to (294,68). Te equation of tis line segment is y b = 35 2 ya. Tis is depicted as te solid line segment in Figure 9. are te incomes tat te government could generate by coosing k in effect a lump-sum transfer between te two persons. Te saded area gives all te possible combination of incomes if income can be trown away. y b 300 (42,294) 200 (294,68) y a Figure 9.: Income possibility set c Frank Cowell

11 Microeconomics 4. If te government tries to maimise W = log(y a ) + log(y b ) subject to te (truncated) straigt line given in figure 9., te best it can acieve is a corner solution giving all of resource 2 to person a. 5. But tis of course is eactly wat appens wit Ŵ = ya + y b. So even toug te SWF W eibits inequality aversion, you get te same outcome as wit te SWF Ŵ wic ignores distributional issues and just seeks to maimise total income. c Frank Cowell

12 Microeconomics CHAPTER 9. WELFARE Eercise 9.7 Tis is an eample of rent seeking. In a certain industry it is known tat monopoly profits Π are available. Tere are N firms tat are lobbying to get te rigt to run tis monopoly. Firm f spends an amount c f on lobbying; te probability tat firm f is successful in its lobbying activity is given by π f := c f N j= cj (9.9). Suppose firm f makes te same assumptions about oter firms activities as in Eercise 3.2. It cooses c f so as to maimise epected returns to lobbying assuming te oter firms lobbying ependitures are given. Wat is te first-order condition for a maimum? 2. If te firms are identical sow tat te total lobbying costs cosen by te firms must be given by [ Nc = Π ] N 3. If lobbying costs are considered to contribute noting to society wat is te implication for te measurement of waste attributable to monopoly?. Making te beavioural assumption of Eercise 3.2 te probability of lobbying success (9.9) becomes were π f = K := cf K + c f (9.0) N j=,j f Firm f maimises epected returns ma π f Π c f so tat te problem is, coose lobbying ependiture c f to maimise: Te FOC for a maimum is Π K + c f Πc f K + c f cf c j. ] [ Πcf K + c f = 0. (9.) 2. If te firms are identical ten te optimal lobbying ependiture is te same c for eac firm f. So K + c f is just Nc and (9.) becomes ] Π [ Nc Πc Nc = 0. (9.2) Tis implies [ Nc = Π Π ]. (9.3) N 3. Clearly tere is an additional component to te waste generated by a monopoly over and above deadweigt loss; te additional component is given by (9.3). c Frank Cowell

13 Microeconomics Eercise 9.8 In an economy tere are n commodities and n individuals, and tere is uncertainty: eac individual may ave good or poor ealt. Te state of ealt is an independently distributed random variable for eac individual and occurs after te allocation of goods as taken place. Individual gets te following utility in state-of-te-world ω: u (, ω ) := a (, ω ) + n i=2 b ( ) i were := (, 2,..., n), i is te amount of commodity i consumed by, te functions a,b are increasing and concave in consumption, and ω takes one of te two values poor ealt or good ealt for eac individual; good is ealt-care services.. Te government estimates tat for eac individual te probability of stateof-te-world ω is π ω. If aggregate production possibilities are described by te production constraint Φ() = 0 (were := (, 2,..., n ) and i is te aggregate consumption of commodity i) and te government as a social-welfare function n = π ω u (, ω ) find te first-order conditions for a social optimum. ω 2. Te government also as te ability to ta or subsidise commodities at different rates for different individuals: so individual faces a price p i for commodity i. If te person as an income y and estimates tat te probability of state-of-te-world ω is π ω, and if e maimises epected utility, write down te first-order conditions for a maimum. 3. Sow tat te solutions in parts and 2 can only coincide if p p j p i p j = Φ () Φ j () = Φ i (), i, j = 2,..., n Φ j () [ ( ω π ωa, ω ) ] ω π ( ωa, ω ), j = 2,..., n Is tere a case for subsidising ealt-care? Is tere a case for subsidising any oter commodity?. Te Lagrangean for te social optimum is L (, 2,..., m, λ ) m := π(ω)υ (, ω ) ( m ) λφ c Frank Cowell = ω =

14 Microeconomics CHAPTER 9. WELFARE so tat te first-order conditions for te social optimum are π(ω)α (, ω ) = λφ () (9.4) ω β i ( i ) = λφ i (), i = 2,..., n (9.5) were te subscripts on α and β denote derivatives wit respect to consumption. 2. Te Lagrangean for te consumer s optimum is [ L (, µ ) := ω π (ω)υ (, ω ) + µ M ] n p i i so tat te first-order conditions for te consumer s optimum are π (ω)α (, ω ) = µp (9.6) 3. From te (9.5) and (9.7) we get ω β i i= ( i ) = µp i (9.7) ( ) Φ i () Φ j () = β i ( i ) = p i j p, i, j = 2,..., n j β j and from te (9.4) and (9.6) we get ( Φ () ω = π(ω)α, ω ) Φ j () β ( ), j = 2,..., n j j (, ω ) p p j = ω π (ω)α β j ( j ), j = 2,..., n from wic we ave /β ( ) j j /β ( ) j j = Φ () Φ j () = p p j ω π(ω)α ω π (ω)α (, ω ) (, ω ) so tat te relationsip in part 3 olds. If personal perceptions of te risk of poor ealt differ from te government s assessment of risk te price of ealt-care may need to be adjusted for tat individual. Tis does not arise wit reference to pairs of goods tat do not involve ealt-care: tere te price ratio can be te same for everyone. c Frank Cowell

15 Microeconomics Eercise 9.9 Revisit te economy of San Serrife (Eercise 4.) Heterogeneity amongst te inabitants of San Serrife was ignored in Eercise 4.. However, it is now known tat altoug all San Serrife residents ave preferences of form A in Eercise 4.2 tey diff er in teir tastes: Nortern San Serrifeans spend 34% of teir budget on milk and only 2% on wine, wile Soutern San Serrifeans spend just 4% of teir budget on milk and 32% on wine. Te question of entry to te EU is to be reviewed; te consequences for te prices of milk and wine of entry to te EU are as in Eercise 4... Assume tat tere are eigt times as many Souterners as Norterners in te San Serrife population, but tat te average income of a Norterner is four times tat of a Souterner. On te basis of te potential-superiority criterion, sould San Serrife enter te EU? 2. Suppose Norterners and Souterners ad equal incomes. Sould San Serrife enter te EU? 3. Wat would be te outcome of a straigt vote on entry to te EU?. Given tat te α i ave te interpretation of ependiture sares it is clear tat Norterners would lose by EU entry and Souterners would gain. So wat sould appen? Te total CV (summed over all San Serrife) is CV (p ˆp) = [ θ N ]n N y N + [ θ S ]n S y S (9.8) were te subscripts refer to Nort and Sout, n j is te population in region j and y j is te income in region j. Given te information in te question we find tat θ N = = 2 and θ S = = and tat (9.8) is proportional to 4[ θ N ] + 8[ θ S ] = = 4[ ] < 0 Tere appears to be no potential Pareto gain in entering te EU - te total gain to te poor Souterners is less tan te loss eperienced by te ric Norterners. A conventional cost-benefit rule would tus indicate tat San Serrife sould not enter. 2. It is interesting to note tat ad Norterners and Souterners ad te same percapita income ten te conclusion would ave been te opposite: + 8[ ] = > Simple voting would ave given te opposite answer to te potential Pareto gain rule. c Frank Cowell