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

Transcription

1 Chapter 9 Welfare Exercise 9. In a two-commodity exchange economy there are two large equalsized groups of traders. Each trader in group a has an endowment of 300 units of commodity ; each person in group b has an endowment of 200 units of commodity 2. Each a-type person has preferences given by the utility function U a (x a ) = x a x a 2 and each b-type person s utility can be written as U b (x b ) = xb x b 2 x a where x h i means the consumption of good i by an h-type person.. Find the competitive equilibrium allocation 2. Explain why the competitive equilibrium is ine cient. 3. Suggest a means whereby a benevolent government could achieved an e - cient allocation.. Notice that the term x a that appears in the b-type s utility function is a negative externality: the more that a-people consume of commodity, the lower is every b-person s utility. This is the reason that the competitive equilibrium will be ine cient. To derive the competitive equilibrium, notice that the term x a is virtually irrelevant to the b-people s behaviour (they cannot do anything about it). Both a-people and b-people thus have Cobb-Douglas utility functions, and we know that their demands will be given by: x h i = y h ; h = a; b; i = ; 2: 2 p i Incomes are y a = 300p, y b = 200p 2. Using this information we can see that total demand for commodity is N x a + x b = N

2 Microeconomics CHAPTER 9. WELFARE where N is the large unknown number of traders in each group and := p =p 2 (Notice that only the price ratio matters in the solution). Clearly there are 300N units of commodity available, so the excess demand function for commodity is: E = N By Walras Law we know that if E = 0 then E 2 = 0 also. Clearly E = 0 when = 2=3; this is the equilibrium price ratio. Using the demand functions we nd x a x a = = (9.) 00 x b x b = 2 200= = (9.2) 00 This is the competitive equilibrium allocation. 2. To verify that this allocation is ine cient consider the following. Since there is a negative externality, it is likely that in the competitive equilibrium the a-people are consuming too much of commodity. So let us change the allocation in such a way that the a-people consume less of commodity ( x a < 0) but where the a-people s utility remains unchanged: this means that their consumption of good 2 must be increased, by an amount x a 2 = x a > 0 (remember that in equilibrium equals the marginal rate of substitution). Now since there is a xed total amount of each commodity, the b-people s consumptions must move in exactly the opposite direction; so x b = x a > 0 and x b 2 = x a 2 < 0. The e ect on their utility can be computed thus: log(u b ) = x b x b + x b x b 2 2 = = 50 xa > 0 x a x a 50 x a So, as we expected, it is possible to move away from the competitive equilibrium in such a way that some people s utility is increased, and no-one else s utility decreases. 3. We might think that for e ciency the relative price of commodity should just be increased to the a-people, relative to that facing the b-people. But this will not work since their income is determined by p, and, as the demand function reveals, their resulting consumption of commodity is independent of price. A rationing scheme may (in this case) be simpler. cfrank Cowell

3 Microeconomics Exercise 9.2 Consider a constitution based on a system of rank-order voting whereby the worst alternative gets point, the next worst, 2,... and so on, and the state that is awarded the most points by the citizens is the one selected. Alf s ranking of social states changes during the week. Bill s stays the same: Monday: Tuesday: Alf Bill Alf Bill What is the social ordering on Monday? What is it on Tuesday? How does this constitution violate the IIA Axiom? Use to denote strict preference and ~ to denote indi erence. The voting points scored for the three states ; 0 ; 00 are as follows; Monday: [5,5,2], Tuesday: [5,4,3]. So the social ordering changes from ~ 0 00 to Notice that the rankings by Alf and Bill of and 0 remain unchanged; yet their reranking of 00 with reference to other alternatives changes the social ordering of and 0 from indi erence to strict preference. cfrank Cowell

4 Microeconomics CHAPTER 9. WELFARE Exercise 9.3 Consider an economy that consists of just three individuals, fa; b; cg and four possible social states of the world. Each state-of-the-world is characterised by a monetary payo y h thus: a b c Suppose that person h has a utility function U h = log(y h ).. Show that if individuals know the payo s that will accrue to them under each state-of-the-world, then majority voting will produce a cyclic decision rule. 2. Show that the above conditions can rank unequal states over perfect equality. 3. Show that if people did not know which one of the identities fa; b; cg they were to have before they vote, if they regard any one of these three identities as equally likely and if they are concerned to maximise expected utility, then majority voting will rank the states strictly in the order of the distribution of the payo s. 4. A group of identical schoolchildren are to be endowed at lunch time with an allocation of pie. When they look through the dining hall window in the morning they can see the slices of pie lying on the plates: the only problem is that no child knows which plate he or she will receive. Taking the space of all possible pie distributions as a complete description of all the possible social states for these schoolchildren, and assuming that ex ante there are equal chances of any one child receiving any one of the plates discuss how a von Neumann-Morgenstern utility function may be used as a simple social-welfare function. 5. What determines the degree of inequality aversion of this social-welfare function? 6. Consider the possible problems in using this approach as a general method of specifying a social-welfare function.. We nd that majority voting produces the ranking but also ; i.e. the states f; 0 ; 000 g form a cycle. (ii) 0 would be strictly preferred to the state of perfect equality. (iii) Since the probability of being assigned any one of the three identities is 3, the utility payo s are: cfrank Cowell

5 Microeconomics a b c Expected Utility log(27) log(6) log(5) log(2) 2. Now consider ordering the states by income inequality Distribution Expected Utility [3,3,3] 3 log(27) 0 [,4,4] 3 log(6) 00 [,3,5] 3 log(5) 000 [,2,6] 3 log(2) 3. Concavity of the utility function implies (in a primordial ignorance about identity) a preference for equality. 4. Let there be n h schoolchildren, and let the allocation of pie received by child h be y h ; h = ; 2; :::n h. The set of all possible pie distributions is ( ) y ; y 2 Xn h ; :::; y nh : y h 0; y h If each child perceives an equal chance of being assigned any plate the expected utility of the perceived pie distribution is Xn h n h h= u y h where u is an increasing function. If u is concave then this utility function will rank more equal distributions as being preferable to less equal distributions. h= 5. Inequality aversion is identical to risk aversion. 6. In practice the problem of receiving a particular pie allocation will not be the same individuals; personal risk-aversion may not be an appropriate basis for inequality aversion with reference to life chances. cfrank Cowell

6 Microeconomics CHAPTER 9. WELFARE Exercise 9.4 Table 9. shows the preferences over three social states for two groups of voters; the row marked # gives the number of voters with each set of preferences; preferences are listed in row order, most preferred at the top.. Find the Condorcet winner among right-handed voters only. 2. Show that there is a cycle among left-handed voters only. 3. Suppose that the cycle among the left-handed voters is broken by ignoring the vote that has the smallest winner. Show that the winner is then the same as that among the right-handed voters. 4. Show that if the two groups are merged there is a Condorcet winner but is di erent from the winners found for the left-handers and the right-handers separately! 5. Would the above paradox occur if one used de Borda voting? Left-handers Right-handers # Table 9.: Left-handed and right-handed voters Use to denote beats in a bilateral vote.. Clearly, from the right-hand part of the table 0 and 00, both by a majority of 8 to 7. So is unambiguously the Condorcet winner among the right-hand voters 2. Consider pairwise votes among the left-hand group: versus 0. There are 0+2 votes for against 6+6 votes for 0. So 0 by 22 to 2 0 versus 00. There are votes for 0 against 2 votes for 00.So 0 00 by 22 to 2. versus 00. There are 0+6 votes for against 6+2 votes for 00. So 00 by 8 to 6 So there is a cycle The weakest link in the above cycle is 00 ; here 00 wins by only two votes as against the margin of 0 votes in the other two cases. If we remove this link it is then clear that the winner is as with the right-hand voters cfrank Cowell

7 Microeconomics 4. For the two groups together: versus 0. There are votes for against 6+6 votes for 0. So 0 by 57 to 2 0 versus 00. There are votes for 0 against votes for 00.So 00 0 by 47 to 22. versus 00. There are votes for against votes for 00. So 00 by 35 to 34 So now 00 0 unambiguously. 00 is the clear winner! 5. Borda voting where 3 is the score given to the best alternative, to the worst. Votes are as in Table 9.2. wins in each subgroup and overall. Left-handers Right-handers Both groups tot 8 7 tot Table 9.2: Borda votes cfrank Cowell

8 Microeconomics CHAPTER 9. WELFARE Exercise 9.5 Suppose social welfare is related to individual incomes y h thus: where () has the form Xn h W = (y h ) h= (x) = x and is a non-negative parameter.. What form does take for =? [Hint, use l Hôpital s rule]. 2. What is relative inequality aversion for this W? 3. Draw the contours of the social welfare function for the cases =,! 0,!. What is equally-distributed-equivalent income in each case? 4. If, instead of a nite population f; ::; n h g, there is a continuum of individuals distributed on R with density at income y given by f(y) write down the equivalent form of the social welfare function W in general and in the particular cases cited in part 3.. Clearly the denominator and the numerator of x both vanish as!. L Hôpital s rule implies that the limiting value can be found by taking the ratio of the rst derivatives of the denominator and numerator. Evaluating the derivatives we have 2. Di erentiating we get and so (x) =. x x log x lim = lim = log x.!! d(x) dx = x d 2 (x) dx 2 = x 3. For the case = contours will be rectangular hyperbolas and ede-income is the geometric mean. For = 0 contours will be diagonal straight lines and ede-income is the arithmetic mean. For = contours will be L- shapes and ede-income is the smallest of the incomes y h. 4. W = = = Z Z Z (y)f(y)dy yf(y)dy, if = 0 log yf(y)dy, if = = inffy : f(y) > 0g, if = cfrank Cowell

9 Microeconomics Exercise 9.6 In a two-commodity exchange economy there are two groups of people: type a have the utility function 2 log(x a ) + log(x a 2) and an endowment of 30 units of commodity and k units of commodity 2; type b have the utility function log(x b ) + 2 log(x b 2) and an endowment of 60 units of commodity and 20 k units of commodity 2.. Show that the equilibrium price,, of good in terms of good 2 is 20+k 50. [Hint: use the answer to Exercise 7.4]. 2. What are the individuals incomes (y a ; y b ) in equilibrium as a function of k? As a function of? 3. Suppose it is possible for the government to carry out lump-sum transfers of commodity 2, but impossible to transfer commodity. Use the previous answer to show the set of income distributions that can be achieved through such transfers. Draw this in a diagram. 4. If the government has the social welfare function W (y a ; y b ) = log(y a ) + log(y b ) nd the optimal distribution of income using the transfers mentioned in part 3. [hint use the diagram constructed earlier]. 5. If instead the government has the social welfare function W (y a ; y b ) = y a + y b nd the optimal distribution of income using transfers. Comment on the result.. Let us use commodity 2 as numéraire. From the details in the question we nd that incomes for the two types of people are: y a = 30 + k (9.3) y b = 60 + [20 k] (9.4) where is the price of commodity in terms of commodity 2. Again we have a Cobb-Douglas utility function (Cf the answer to Exercise 7.4), and so the demands by a for the two commodities are k 3 [30 + k] and for b the demands are k [ k] This means that the excess demand for commodity 2 at price must be k k 20 cfrank Cowell

10 Microeconomics CHAPTER 9. WELFARE The equilibrium value of is found by setting this equal to 0 (remember that Walras Law will ensure that the other market also has zero excess demand). Doing this we get the value speci ed in the question. 2. Substituting back into (9.3) and (9.4) we get: y a = y b = k k 5 or, using the formula for the equilibrium price, we have equivalently: (9.5) (9.6) y a = (9.7) y b = (9.8) 3. Equations (9.5) and (9.6) imply that there is a straight-line frontier on the set of income pairs (y a ; y b ) mapped out by letting k vary from 0 to 20 (or by using the price equations), in other words a straight line from (42,294) to (294,68). The equation of this line segment is y b = 35 2 ya : This is depicted as the solid line segment in Figure 9. are the incomes that the government could generate by choosing k in e ect a lump-sum transfer between the two persons. The shaded area gives all the possible combination of incomes if income can be thrown away. y b 300 (42,294) 200 (294,68) y a Figure 9.: Income possibility set cfrank Cowell

11 Microeconomics 4. If the government tries to maximise W = log(y a ) + log(y b ) subject to the (truncated) straight line given in gure 9., the best it can achieve is a corner solution giving all of resource 2 to person a. 5. But this of course is exactly what happens with ^W = y a + y b. So even though the SWF W exhibits inequality aversion, you get the same outcome as with the SWF ^W which ignores distributional issues and just seeks to maximise total income. cfrank Cowell

12 Microeconomics CHAPTER 9. WELFARE Exercise 9.7 This is an example of rent seeking. In a certain industry it is known that monopoly pro ts are available. There are N rms that are lobbying to get the right to run this monopoly. Firm f spends an amount c f on lobbying; the probability that rm f is successful in its lobbying activity is given by f := c f P N j= cj (9.9). Suppose rm f makes the same assumptions about other rms activities as in Exercise 3.2. It chooses c f so as to maximise expected returns to lobbying assuming the other rms lobbying expenditures are given. What is the rst-order condition for a maximum? 2. If the rms are identical show that the total lobbying costs chosen by the rms must be given by Nc = N 3. If lobbying costs are considered to contribute nothing to society what is the implication for the measurement of waste attributable to monopoly?. Making the behavioural assumption of Exercise 3.2 the probability of lobbying success (9.9) becomes where f = K := cf K + c f (9.0) NX j=;j6=f c j : Firm f maximises expected returns max f choose lobbying expenditure c f to maximise: The FOC for a maximum is K + c f c f K + c f c f c f K + c f c f so that the problem is, = 0: (9.) 2. If the rms are identical then the optimal lobbying expenditure is the same c for each rm f. So K + c f is just Nc and (9.) becomes c Nc Nc = 0: (9.2) This implies Nc = : (9.3) N 3. Clearly there is an additional component to the waste generated by a monopoly over and above deadweight loss; the additional component is given by (9.3). cfrank Cowell

13 Microeconomics Exercise 9.8 In an economy there are n commodities and n h individuals, and there is uncertainty: each individual may have good or poor health. The state of health is an independently distributed random variable for each individual and occurs after the allocation of goods has taken place. Individual h gets the following utility in state-of-the-world!: u h x h ;! := a h x h ;! + nx i=2 b h x h i where x h := x h ; x h 2; :::; x h n, x h i is the amount of commodity i consumed by h, the functions a h,b h are increasing and concave in consumption, and! takes one of the two values poor health or good health for each individual; good is health-care services.. The government estimates that for each individual the probability of stateof-the-world! is!. If aggregate production possibilities are described by the production constraint (x) = 0 (where x := (x ; x 2 ; :::; x n ) and x i is the aggregate consumption of commodity i) and the government has a social-welfare function Xn h h=! X! u h x h ;! nd the rst-order conditions for a social optimum. 2. The government also has the ability to tax or subsidise commodities at di erent rates for di erent individuals: so individual h faces a price p h i for commodity i. If the person has an income y h and estimates that the probability of state-of-the-world! is h!, and if he maximises expected utility, write down the rst-order conditions for a maximum. 3. Show that the solutions in parts and 2 can only coincide if p h p h j p h i p h j = (x) j (x) = i (x) ; i; j = 2; :::; n j (x) "P! h!a h x h ;! # P!!a h xh ;! ; j = 2; :::; n Is there a case for subsidising health-care? Is there a case for subsidising any other commodity?. The Lagrangean for the social optimum is L x ; x 2 ; :::; x m ; mx X := (!) h x h ;! h=!! mx x h h= cfrank Cowell

14 Microeconomics CHAPTER 9. WELFARE so that the rst-order conditions for the social optimum are X (!) h x h ;! = (x) (9.4)! h i x h i = i (x) ; i = 2; :::; n (9.5) where the subscripts on and denote derivatives with respect to consumption. 2. The Lagrangean for the consumer s optimum is " L h x h ; := X h (!) h x h ;! +! M h n X i= p h i x h i # so that the rst-order conditions for the consumer s optimum are X h (!) h x h ;! = p h (9.6) 3. From the (9.5) and (9.7) we get! i (x) j (x) = h i h j h i x h i = p h i (9.7) x h i = ph i p h ; i; j = 2; :::; n j x h j and from the (9.4) and (9.6) we get (x) j (x) p h p h j = = P! (!)h x h ;! h ; j = 2; :::; n j x h j P! h (!) h x h ;! h j x h j ; j = 2; :::; n from which we have = h j x h j = h j x h j = (x) j (x) = ph p h j P! (!)h xh ;! P! h (!) h xh ;! so that the relationship in part 3 holds. If personal perceptions of the risk of poor health di er from the government s assessment of risk the price of health-care may need to be adjusted for that individual. This does not arise with reference to pairs of goods that do not involve health-care: there the price ratio can be the same for everyone. cfrank Cowell

15 Microeconomics Exercise 9.9 Revisit the economy of San Serrife (Exercise 4.) Heterogeneity amongst the inhabitants of San Serrife was ignored in Exercise 4.. However, it is now known that although all San Serrife residents have preferences of form A in Exercise 4.2 they di er in their tastes: Northern San Serrifeans spend 34% of their budget on milk and only 2% on wine, while Southern San Serrifeans spend just 4% of their budget on milk and 32% on wine. The question of entry to the EU is to be reviewed; the consequences for the prices of milk and wine of entry to the EU are as in Exercise 4... Assume that there are eight times as many Southerners as Northerners in the San Serrife population, but that the average income of a Northerner is four times that of a Southerner. On the basis of the potential-superiority criterion, should San Serrife enter the EU? 2. Suppose Northerners and Southerners had equal incomes. Should San Serrife enter the EU? 3. What would be the outcome of a straight vote on entry to the EU?. Given that the i have the interpretation of expenditure shares it is clear that Northerners would lose by EU entry and Southerners would gain. So what should happen? The total CV (summed over all San Serrife) is X CV h (p! ^p) = [ N ]n N y N + [ S ]n S y S (9.8) h where the subscripts refer to North and South, n j is the population in region j and y j is the income in region j. Given the information in the question we nd that N = 2 30:34 0:02 = 2 and S = 2 30:04 0:32 = 2 0:2 and that (9.8) is proportional to 4[ N ] + 8[ S ] = :2 = 4[ 2 0:8 ] < 0 There appears to be no potential Pareto gain in entering the EU - the total gain to the poor Southerners is less than the loss experienced by the rich Northerners. A conventional cost-bene t rule would thus indicate that San Serrife should not enter. 2. It is interesting to note that had Northerners and Southerners had the same percapita income then the conclusion would have been the opposite: + 8[ 2 0:04 ] = 7 2 2:96 > 0: 3. Simple voting would have given the opposite answer to the potential Pareto gain rule. cfrank Cowell