Public Economics Ben Heijdra Chapter 9: Introduction to Normative Public Economics

Transcription

1 Public Economics: Chapter 9 1 Public Economics Ben Heijdra Chapter 9: Introduction to Normative Public Economics

2 Objectives of this chapter Public Economics: Chapter 9 2 Read Atkinson & Stiglitz (1980, ch. 11) for your interest In the previous chapters we have pursued positive economics how does a household determine consumption and labour supply in the presence of a tax system? how are household savings decisions affected by the tax system? what is the effect of taxation when the household faces uncertainty? what is the effect of the tax system on the firm s real and financial decisions? how do taxes affect the economy when general equilibrium repercussions are taken into account? how does tax incidence depend on market structures?

3 Public Economics: Chapter 9 3 In the next set of chapters we look at normative economic questions such as: how should commodities be taxed? how should the income tax system be designed? what is the appropriate way to provide public goods? Central in normative economics is the so-called Social Welfare Function (SWF) proposed by Bergson (QJE 1938) and Samuelson (1947) A typical example of an individualistic SWF is: SW = Ψ [ U 1, U 2,, U H] (SWF) SW is an indicator for social welfare U h is the utility level of individual household h [h = 1, 2,, H] Ψ [ ] is some function featuring positive partial derivatives, i.e. Ψ h Ψ U h > 0 for all h

4 Public Economics: Chapter 9 4 Remarks on the SWF we call the SWF individualistic because its only arguments are the individual utility functions: comparison between alternatives states only involves individual utility levels in those states the SWF confers social (or ethical) weights on the different individuals since Ψ h > 0, social welfare rankings honour the Pareto Principle: if one individual s utility level increases (decreases) and all other individual utility levels are held constant, then SW increases (decreases). the SWF goes beyond the Pareto Principle because it assumes that gains and losses can be compared in that sense it goes beyond the New Welfare Economics of the 1930s which denied the validity of making interpersonal comparisons defense: the Pareto Principle does not provide sufficient guidance for public policy decisions [inherent bias to maintain the status quo]

5 Public Economics: Chapter 9 5 Different types of SWF have been used in the literature: Benthamite SWF: social welfare is the sum of individual utility levels (or a positive linear transformation of this sum): SW = H U h (BSWF) h=1 [Jeremy Bentham ( ): the greatest happiness, independent of how it is distributed] Rawlsian SWF: social welfare is the welfare of the worst-off individual ( maxi-min ): SW = min h [ U 1, U 2,, U H] (RSWF) [John Rawls ( ): decision made under a veil of ignorance (about one s own position)]

6 Generalized iso-elastic SWF: Public Economics: Chapter 9 6 SW = H h=1 ( ) U h 1 1/ζ 1, ζ 0 (GSWF) 1 1/ζ where ζ measures the degree of substitutability between the U h s. Special cases: for ζ = 0 we obtain (RSWF) for ζ = 1 we obtain a logarithmic SWF for ζ we obtain (BSWF)

7 Public Economics: Chapter 9 7 Brief Overview of Welfare Economics We use the general equilibrium model of Samuelson (1947) as presented by Tresch (2002, ch. 2) there are H households there are F production factors there are G goods Key elements: individual preferences production technologies market clearance

8 Public Economics: Chapter 9 8 Individual preferences of household h: U h = U h [ X h 1,, X h G, V h 1,, V h F ] (1) U h is utility of household h [h = 1, 2,, H] X h g V h f is consumption of good g by household h [g = 1, 2,, G] is the supply of production factor f by household h [f = 1, 2,, F ] Production technology for good g: Y g = F g [Z1, g, Z g F ] (2) Y g is aggregate production of good g F g [ ] is the production function for good g Z g f is factor f used in the production of good g

9 Public Economics: Chapter 9 9 Market clearing: goods markets (g = 1, 2,, G): Y g = Note: G equations. factor markets (f = 1, 2,, F ): H h=1 X h g (3) H h=1 V h f = G g=1 Z g f (4) Note: F equations The general equilibrium model consists of equations (1)-(4).

10 Public Economics: Chapter 9 10 Efficiency: Pareto Optimality Pareto optimality: allocation of resources such that no one consumer can be made better off by a reallocation of resources without... at the same time making at least one other consumer worse off The locus of Pareto-optimal allocations defines the Utility Possibility Frontier (UPF). In Figure 9.1 such a UPF is illustrated for the 2 household case (H = 2) UPF may have a very irregular shape even if individual utility functions are well-shaped: it is a frontier of so-called Point UPF s [that are associated with a particular point of the Production Possibility Frontier, i.e. a particular bundle of goods] point A is a Pareto optimal point [as are all other points on the UPF] points within the UPF are not Pareto optimal [e.g. point C] points beyond the UPF are unattainable [e.g. point B]

11 Public Economics: Chapter 9 11 U 2 D B U 2 A A C U 1 A E U 1 Figure 9.1: The Utility Possibility Frontier

12 Public Economics: Chapter 9 12 Formal definition of the set of all Pareto optimal allocations: focus on arbitrary household, say household h = 1 hold every other household s utility constant, i.e. U h = U h 0 for h = 2,, H maximize household 1 s utility subject to the restrictions, i.e. the social planner chooses X h g, V h f, Zg f, Y g in order to maximize: U 1 = U 1 [ X 1 1,, X 1 G, V 1 1,, V 1 F ] (OF)

13 subject to: H h=1 Public Economics: Chapter 9 13 U h 0 = U h [ X h 1,, X h G, V h 1,, V h F ], (for h = 2,, H) (C1) Y g = F g [ Z g 1,, ] Zg F, (for g = 1,, G) (C2) H Y g = Xg h, (for g = 1,, G) (C3) V h f = h=1 G g=1 Z g f, (for f = 1,, F ) (C4)

14 The Lagrangian for this problem is: Public Economics: Chapter 9 14 L U [ ] 1 X1, 1, XG, 1 V1 1,, VF 1 H [ + λ [ h U h X1 h,, XG, h V1 h,, VF h h=2 G [ µ g Yg F g [Z1, g, Z g F ] ] g=1 [ G H ] ν g Xg h Y g g=1 h=1 [ F H ξ f f=1 h=1 V h f G g=1 Z g f ] ] U h 0 ] where the Lagrange multipliers are λ h (for h = 2,, H), µ g (for g = 1,, G),

15 Public Economics: Chapter 9 15 ν g (for g = 1,, G), and ξ f (for f = 1,, F ), i.e. there are (H 1) + 2G + F Lagrange multipliers in all. The first-order necessary conditions (assuming interior solutions) are the constraints (of course) and: for the good demands [G H equations]: L X 1 g L X h g = U 1 X 1 g = λ h U h X h g + ν g = 0 (F1) + ν g = 0, h 1 (F2)

16 Public Economics: Chapter 9 16 for the factor supplies [H F equations]: L V 1 f L V h f = U 1 V 1 f = λ h U h V h f + ξ f = 0 (F3) + ξ f = 0, h 1 (F4) for the output decisions [G equations]: L Y g = µ g ν g = 0 (F5) for the factor demands [F G equations]: L Z g f = µ g F g Z g f ξ f = 0 (F6)

17 Public Economics: Chapter 9 17 Although these conditions look complex, we can eliminate the various Lagrange multipliers and derive the condensed statement of the necessary conditions for the Paretian optimum. We look at the following pairings. For arbitrary household h we derive: from (F1) or (F2): ν g1 ν g2 = U h / X h g 1 U h / X h g 2 i.e. the MRS between any two consumption goods g 1 and g 2 is the same for all households h. See Figure 9.2. from (F3) or (F4): ξ f1 ξ f2 = U h / V h f 1 U h / V h f 2 i.e. the MRS between any two supplied factors f 1 and f 2 is the same for all households h. (P1) (P2)

18 Public Economics: Chapter (X 2 ) A O 2 X 1 (U 1 ) D D 1 (X 1 ) A E A C (U 1 ) C (U2 ) D 2 (X 1 ) A B (U 1 ) A (U 2 ) C (U 1 ) B (U 2 ) A O 1 (U 2 ) B 1 (X 2 ) A X 2 Figure 9.2: Efficient Exchange

19 from (F1)&(F3) [or (F2)&(F4)]: Public Economics: Chapter 9 19 ξ f = U h / Vf h ν g U h / Xg h (P3) i.e. the MRS between any supplied factor f and any consumption good g is the same for all households h. On the production side we derive: from (F6) and (F5): 1 = µ g 1 F g1 / Z g1 f µ g2 F g 2 / Z g 2 f = ν g 1 F g1 / Z g1 f ν g2 F g 2 / Z g 2 f (P4) i.e. the social value of the marginal product of factor f is the same for any two goods g 1 and g 2

20 Public Economics: Chapter 9 20 from (F6): ξ f1 ξ f2 = F g / Z g f 1 F g / Z g f 2 (P5) i.e. the MRS between any two factors f 1 and f 2 is the same for all goods g. See Figure 9.3 or Figure 9.4. We can now combine even further: from (P1) and (P4) we find: [ ] νg1 = ν g2 U h / X h g 1 U h / X h g 2 = F g2 / Z g2 f F g 1 / Z g 1 f (P6) i.e. the MRS between g 1 and g 2 [left-hand side] must be equated to the marginal rate of transformation (MRT) between these two goods [right-hand side]. See Figure 9.5.

21 Public Economics: Chapter 9 21 K (L 2 ) A O 2 D (Y 1 ) D (K 1 ) A E A C (Y 1 ) C (Y 2 ) D (K 2 ) A B (Y 1 ) A (Y 2 ) C O 1 (Y 2 ) B (Y 1 ) B (L 1 ) A (Y 2 ) A L Figure 9.3: Efficient Production

22 Public Economics: Chapter 9 22 Y 1 D (Y 1 ) D (Y 1 ) B B (Y 2 ) D (Y 2 ) B Y 2 Figure 9.4: Transformation Curve

23 Public Economics: Chapter 9 23 Y 1 (Y 1 ) A 2 (X 2 ) A A P 1 (X 1 ) A A C 2 (X 1 ) A O 1 1 (X 2 ) A (Y 2 ) A Y 2 Figure 9.5: Efficient Exchange and Production

24 Public Economics: Chapter 9 24 from (P2) and (P5) we find: [ ] ξf1 = ξ f2 U h / V h f 1 U h / V h f 2 = F g g / Zf 1 F g / Z g f 2 (P7) i.e. for any good g, the common MRS between any two supplied factors f 1 and f 2 must be equated to the marginal rate of technical substitution (MRTS) between these two factors in production from (P3) and (F5)-(F6) we find: [ ] ξf = ν g U h / V h f U h / X h g = F g Z g f (P8) i.e. the common MRS between any consumed good g and any supplied factor f is equal to the marginal product of that factor f in producing that good g

25 Public Economics: Chapter 9 25 Concluding remarks: the first-order necessary conditions stated above solve for a single point on the UPF, conditional upon the utility levels held constant for households h = 2,, H. Hence... by changing one of these household s utility level we can rederive the Pareto optimal allocation and obtain another point of the UPF [assuming feasibility etcetera]. But... there are infinitely many ways to do this so what we end up with is all the points on the UPF. the planning problem does not single out a single optimal point on the UPF: all allocations on the UPF are Pareto optimal Pareto optimality is too weak a criterion for public decision making: it can t even decide between points like D and A (or E and A) in Figure 9.1.

26 Public Economics: Chapter 9 26 Equity: SWF and the Optimal Distribution By postulating a SWF the social planning problem is able to solve both efficiency and distributional choices Recall that the SWF is written in general format as: SW = Ψ [ U 1, U 2,, U H] (OF) U h is utility of household h [h = 1, 2,, H] Ψ [ ] is some function featuring positive partial derivatives, i.e. Ψ h Ψ U h > 0 for all h The social planner now chooses X h g, V h f, Zg f, Y g in order to maximize social welfare

27 Public Economics: Chapter 9 27 (OF) subject to: H h=1 U h = U h [ X h 1,, X h G, V h 1,, V h F ] (C1) Y g = F g [Z1, g, Z g F ] (C2) H Y g = (C3) V h f = h=1 G g=1 X h g Z g f (C4) for f = 1,, F, g = 1,, G, and h = 1,, H.

28 The Lagrangian for this problem is: Public Economics: Chapter 9 28 [ L Ψ U [ ] 1 X1, 1, XG, 1 V1 1,, VF 1, U [ ] 2 X1, 2, XG, 2 V1 2,, VF 2,, U [ ] ] H X1 H,, XG H, V1 H,, VF H G [ + µ g Yg F g [Z1, g, Z g F ] ] + + g=1 [ G H ] ν g Xg h Y g g=1 h=1 [ F H ξ f f=1 h=1 V h f where the Lagrange multipliers are µ g (for g = 1,, G), ν g (for g = 1,, G), G g=1 Z g f ]

29 Public Economics: Chapter 9 29 and ξ f (for f = 1,, F ), i.e. there are 2G + F Lagrange multipliers in all. The first-order necessary conditions (assuming interior solutions) are the constraints (of course) and: for the good demands [G H equations]: L X h g = Ψ U h U h X h g + ν g = 0 (FF1) for the factor supplies [H F equations]: L V h f = Ψ U h U h V h f + ξ f = 0 (FF2) for the output decisions [G equations]: L Y g = µ g ν g = 0 (FF3)

30 Public Economics: Chapter 9 30 for the factor demands [F G equations]: L Z g f = µ g F g Z g f ξ f = 0 (FF4) In principle we can redo the different pairings leading to (P1)-(P8). This is not necessary because (P1)-(P8) still hold, i.e. the social planner selects a point on the UPF First-best analysis. The only interpersonal equity conditions are obtained by using (FF1) and (FF2) and comparing different households, say h 1 and h 2 from (FF1) we find for any good g: [ ν g =] Ψ U h 1 U h 1 X h 1 g = Ψ U h 2 U h 2 X h 2 g Hence, interpersonal equity is achieved if all goods are distributed such that on the margin the increase in social welfare is the same no matter who consumes (P9)

31 the last unit of the good Public Economics: Chapter 9 31 from (FF2) we find that for any factor f : [ ξf = ] Ψ U h 1 U h 1 V h 1 f = Ψ U h 2 U h 2 V h 2 f Hence, all factor supplies should be set such that on the margin the increase in social welfare is the same no matter who supplies the last unit of the factor In Figure 9.6 the determination of the first-best social optimum is illustrated for the two household case. suppose that minimal-government solution would be at point A, where the distribution of welfare is uneven. The associated level of social welfare is SW A although point A is Pareto-optimal, the social planner can improve the social outcome by choosing a tangency between the UPF and a SWF. This tangency occurs at point E, where social welfare is SW E (greater than SW A ) (P10)

32 Public Economics: Chapter 9 32 how is the redistribution engineered? The only way it can be done is if the social planner has access to lump-sum redistribution instruments (i.e. taxes/subsidies which do not introduce inefficiencies into the economy and thus keep the economy on the UPF )

33 Public Economics: Chapter 9 33 U 2 U 2 A A U 2 E E SW E SW A U 1 A U 1 E U 1 Figure 9.6: The First-Best Social Optimum

34 Public Economics: Chapter 9 34 Basic Theorems of Welfare Economics The basic theorems of welfare economics provide important guidance concerning the link between the notion of Pareto efficiency and the outcome produced in the decentralized market economy. See Atkinson & Stiglitz (1980, p. 343): First Theorem of Welfare Economics: If (i) households and firms are perfect competitors and thus take prices for all goods and factor as given, (ii) there is a full set of markets, and (iii) there is perfect information, then a competitive equilibrium (if it exists) is Pareto efficient. Second Theorem of Welfare Economics: If (i) household indifference maps and firm production sets are convex, (ii) there is a full set of markets, (iii) there is perfect information, and (iv) lump-sum taxes/transfers can be carried out costlessly, then any Pareto-efficient allocation can be achieved as a competitive equilibrium with appropriate lump-sum transfers and taxes.

35 Public Economics: Chapter 9 35 The importance of the availability of lump-sum taxes and transfers cannot be overstated. Indeed if such taxes and transfers are not available (or not sufficiently flexible) then: one enters the much more complex (and more realistic) realm of Second-Best Welfare Economics there is generally a conflict between achieving efficiency (reaching the UPF) and equity (achieving a fair distribution of welfare) Figure 9.7 shows in heuristic terms what is going on. Initial position is at point A If the policy maker has the required lump-sum instruments it can ensure that the first-best equilibrium at E can be attained If the policy maker does not have the (sufficiently flexible) lump-sum instruments, the effective set of allocations that can be reached is represented by the shaded area. Of course point A is in that area but point E is not

36 Public Economics: Chapter 9 36 The second-best social optimum is then represented by point B, i.e. the point that can be reached from the initial situation A and has a higher social welfare level than at point A (SW B >SW A but SW B <SW E ) much of actual policy making is concerned with second-best situations Student exercise: the decentralized economy and Pareto efficiency Assume that household h is characterized by the following utility function and budget restriction: G g=1 U h = U [ h X1 h,, XG, h V1 h,, VF h P g Xg h = F W f Vf h f=1 where P g is the market price of good g and W f is the market price of factor f ]

37 Public Economics: Chapter 9 37 Assume that firm g maximizes profit Π g (taking P g and W f as given): F Π g P g Y g W f Z g f f=1 Y g = F g [Z g 1,, Z g F ] Assume furthermore that all markets clear. Show that the decentralized optimizing decisions by households and firms give rise to the first-order conditions (P1)-(P8) stated above.

38 Public Economics: Chapter 9 38 U 2 U 2 A A U 2 B U 2 E B E SW B SW E SW A U 1 A U 1 B U 1 E U 1 Figure 9.7: The Second-Best Social Optimum