Public Economics Ben Heijdra Chapter 2: Taxation and the Supply of Labour

Transcription

1 Public Economics: Chapter 2 1 Public Economics Ben Heijdra Chapter 2: Taxation and the Supply of Labour

2 Public Economics: Chapter 2 2 Overview Theoretical insights static / dynamic models [dynamics treated in Chapter 3] linear / nonlinear tax system hours / participation decision individual / family decision certainty / uncertainty [uncertainty treated in Chapter 4] Empirical evidence Warning: selective overview only

3 Basic Model of Labour Supply Key assumptions: choice of hours certainty static linear tax system individualistic the utility function is: Public Economics: Chapter 2 3 U = U(C, L L) with U C > 0, U L L > 0, U CC < 0, U L L, L L < 0, and indifference curves bulge toward origin: U CC U L L, L L U 2 > C, L L 0.

4 the budget restriction is: the tax function is linear: Public Economics: Chapter 2 4 P(1 + t C )C = M + WL T The Lagrangian is: T = T 0 + t L WL L U(C, L L) + λ [M T 0 + (1 t L )WL P(1 + t C )C] first-order conditions: L C = U C λp(1 + t C ) = 0 L L = U L L + λw(1 t L ) = 0

5 Public Economics: Chapter 2 5 eliminate Lagrange multiplier from first-order conditions: λ = U C P(1 + t C ) = U L L W(1 t L ) where w is the after-tax real wage rate: ( 1 tl U L L U C ) = w w W P 1 + t C the marginal rate of substitution between leisure and consumption depends on the after-tax real wage rate facing households (both t L and t C feature) graphical representation in Figure 2.1. Optimum: slope of indifference curve equal to slope of budget line.

6 budget line can be written as: Public Economics: Chapter 2 6 C + w ( L L ) = M T 0 + (1 t L )W L P(1 + t C ) Y 0 which is downward sloping in ( C, L L ) space

7 Public Economics: Chapter 2 7 C A U = U 0 L > 0 L < 0 C * E 0 B 0 - (LL) * C L - - L L Figure 2.1: Optimal Consumption Leisure Choice

8 Tax Effects in Basic Model three different tax rates in the model: Public Economics: Chapter 2 8 T 0, the lump-sum tax (or lump-sum transfer, if T 0 < 0) t L, the labour income tax rate t C, the consumption tax rate graphical analysis of tax effects for the very special (but often used) case of homothetic preferences: U(C, L L) can be written as G [ Ū(C, L L) ] with G [ ] strictly increasing and Ū(C, L L) homogeneous of degree one (linear homogeneous) in its arguments U L L and U C both homogeneous of degree zero (depends on C/ ( L L ) only) so also...

9 Public Economics: Chapter 2 9 U L L /U C unique function of C/ ( L ) L Engel curve linear through the origin A rise in the lump-sum tax leads to a parallel shift downward of the budget line. There is only an income effect. See Figure 2.2 optimum shifts from E 0 to E 1 both C and L L fall (both are so-called normal goods) utility falls from U 0 to U 1 (even lump-sum taxes hurt)

10 Public Economics: Chapter 2 10 C A U = U 1 U = U 0 Income Expansion Path AN E 1 E 0 B BN 0 C L - - L L Figure 2.2: Increasing the Lump-Sum Tax [homothetic case]

11 Public Economics: Chapter 2 11 A rise in the labour income tax or consumption tax leads to both a shift and a rotation of the budget line. There are both income effects (as before) and substitution effects. See Figure 2.3 for an increase in t L budget line rotates counter-clockwise around point B optimum shifts from E 0 to E 1 C falls but L L rises (case drawn has dominant substitution effect in labour supply) utility falls: indifference curve tangent to E 1 associated with U 1 < U 0 Pure substitution effect always positive: move from E 0 to E Income effect negative: move from E to E 1

12 Public Economics: Chapter 2 12 C A U = U 0 IEP 0 AN EO E 0 E 1 EN IEP 1 B 0 C L - - L L Figure 2.3: Increasing the Labour Income Tax [homothetic case, dominant SE]

13 Public Economics: Chapter 2 13 It is quite possible for the labour supply curve to be backward bending (IE dominates SE). E.g. if U ( ) is Leontief there is no pure SE and the new optimum would be at E More formally the famous Slutsky equation can be used: L w = [ L w ] U=U 0 + L L m 0, where m 0 M T 0 P(1+t C ) is real non-labour income.

14 Public Economics: Chapter 2 14 Immediate insight: if leisure is a normal good then IE is relatively important for poor folks. The labour income tax affects the poor more than the rich. See Figure 2.4 for the Leontief case (σ = 0) Poor has no non-labour income (m 0 = 0) Rich wants to consume L units of leisure (m 0 very high) Increase in t L does not affect the rich. The poor experiences a large income effect

15 Public Economics: Chapter 2 15 C A AN D DN E 1 P E 0 P U 1 P U 0 P E 0 R U 0 R IEP 0 0 C L - - L L Figure 2.4: Increasing the Labour Income Tax: The Rich and the Poor

16 Public Economics: Chapter 2 16 Digression on the Expenditure Function [TOOL] Some very useful tools of duality theory which will be used time and again [see Diamond & McFadden (J Pub Econ 1974) for details] Focus on two-good case for exposition purposes: X 1 and X 2 are the goods, P 1 and P 2 are the respective prices, and Y 0 is lump-sum income Expenditure function: minimum level of lump-sum income needed to attain a given level of utility, U 0, when faced with the consumer prices P 1 and P 2 : E (P 1,P 2,U 0 ) min {X 1,X 2 } P 1X 1 + P 2 X 2 subject to: U (X 1,X 2 ) = U 0 Indirect utility function: maximum achievable utility at given prices P 1 and P 2 and lump-sum income Y 0 : V (P 1,P 2,Y 0 ) max U (X 1,X 2 ) subject to: Y 0 = P 1 X 1 + P 2 X 2 {X 1,X 2 }

17 Key properties: Public Economics: Chapter 2 17 under local non-satiation V ( ) is strictly increasing in Y 0 and we can find E ( ) by inverting V ( ) E (P 1,P 2,U 0 ) is homogeneous of degree one in prices E (P 1,P 2,U 0 ) is concave in prices E (P 1,P 2,U 0 ) is strictly increasing in U 0 and non-decreasing in prices Compensated [Hicksian] demand curves are given by: X H i (P 1,P 2,U 0 ) = E (P 1,P 2,U 0 ) P i (SL) where the superscript H stands for Hicksian

18 Public Economics: Chapter 2 18 Uncompensated [Marshallian] demand curves are given by Roy s Identity: X M i (P 1,P 2,Y 0 ) = V (P 1,P 2,Y 0 ) P i V (P 1,P 2,Y 0 ) Y 0 (RI) where the superscript M stands for Marshallian Obviously it is identically true that: X M i P 1,P 2,E (P 1,P 2,U 0 ) Xi H (P }{{} 1,P 2,U 0 ) (A) =Y 0 Quick & Dirty derivation of Slutsky equation: differentiate (A) with respect to P j : X M i P j + XM i Y 0 E (P 1,P 2,U 0 ) P j = XH i P j (B)

19 use (SL) and (RI) in (B) to get: Public Economics: Chapter 2 19 X M i P j = XH i P j X M j X M i Y 0

20 Labour Supply Application Public Economics: Chapter 2 20 Define leisure as H L L and note that the budget equation in real terms is 1C + w H = Y 0, where Y 0 is defined as: Y 0 m 0 + w L (a) where m 0 M T 0 P(1+t C ) and w W(1 t L) P(1+t C ). Definitions: E (1,w,U 0 ) V (1,w,Y 0 ) min 1C + w H subject to: U (C,H) = U 0 {C,H} maxu (C,H) subject to: Y 0 = 1C + w H {C,H}

21 Hicksian and Marshallian demands: C H (1,w,U 0 ) = E (1,w,U 0 ) 1 C M (1,w,Y 0 ) = Public Economics: Chapter 2 21, H H (1,w,U 0 ) = E (1,w,U 0 ) w V (1,w,Y 0 ) 1 V (1,w,Y 0 ), H M (1,w,Y 0 ) = Y 0 V (1,w,Y 0 ) w V (1,w,Y 0 ) Y 0 Differentiating the Marshallian demand for leisure we get: H M w = = [ H M ] w [ ] H M w Y 0 constant Y 0 constant + HM Y 0 Y 0 w + L HM Y 0, (b) where we have used (a) in the second step.

22 Public Economics: Chapter 2 22 Identity [for leisure]: H H (1,w,U 0 ) = H M (1,w,E (1,w,U 0 )) Differentiating this expression we get: H H w = H H w = [ H M ] w [ ] H M w Y 0 constant Y 0 constant + HM E (1,w,U 0 ) Y 0 w + H M HM Y 0 (c) By combining (b) and (c) (and noting that H H = H M = L L M ) we obtain the Slutsky equation for leisure demand: H M w = HH w HM + LM Y 0 (d)

23 In terms of labour supply we obtain: Public Economics: Chapter 2 23 L M w = LH LM + LM w Y 0 which is the Slutsky equation for labour supply.

24 Public Economics: Chapter 2 24 Progressive Taxes in the Basic Model We now make the following extensions: Most (labour) income tax systems are progressive, in the sense that the tax rate rises with the tax base Pursue quantitative-mathematical (rather than qualitative-graphical) analysis of tax effects [will be used again in lecture on tax incidence Chapters 6-7] We augment the basic model by specifying the general tax function T(WL) and continue to assume that non-labour income is untaxed. Definitions: Marginal tax rate t M is defined as t M dt(wl)/d(wl) Average tax rate t A is defined as t A T(WL)/(WL) Note that earlier tax schedule is progressive if T 0 is negative

25 Rest of the model unchanged: Public Economics: Chapter 2 25 utility function: U = U(C, L L) budget restriction: P(1 + t C )C = M + WL T(WL) M + (1 t A )WL (BC)

26 Public Economics: Chapter 2 26 Solution: Lagrangian expression: L U(C, L L) + λ [M + (1 t A )WL P(1 + t C )C] first-order conditions: L C = U C λp(1 + t C ) = 0 L L = U L L + λw [ (1 t A ) L ( dta dl )] = 0

27 Public Economics: Chapter 2 27 Since t A T(WL) WL ( ) dta L dl = L = L we have that: ( (WL) dt(wl) dl ( (WL) dt(wl) d(wl) T (WL) dwl dl (WL) 2 dwl dl (WL) 2 ) = dwl T (WL) dl ) = t M t A

28 Hence the expansion path is: Public Economics: Chapter 2 28 U L L U C U C λ = P(1 + t C ) = U L L W(1 t M ) ( ) 1 tm = w 1 + t C (FOC) where w W/P is the gross real wage. The marginal rate of substitution between leisure and consumption depends on the marginal (and not on the average) tax rate facing households

29 Public Economics: Chapter 2 29 We assume that the utility function is homothetic and define the substitution elasticity between consumption and leisure as follows: σ = %ge change in C/( L L) %ge change in U L L /U C d ln(c/( L L)) d ln(u L L /U C ) 0 where σ measures how easy it is (in utility terms) for the household to substitute consumption for leisure. if σ is low: substitution very difficult [sharp kinks in its indifference curves] if σ is high: substitution very easy [flat indifference curves] Quantitative analysis makes use of linearized expressions: suitable for marginal tax changes.

30 Public Economics: Chapter 2 30 Linearization of (FOC) yields: d ln ( U L L U C ) ( ) [ ] 1 = w t M t C = C ( L L) σ C + (1/ω L ) L = σ [ ] w t M t C (1) with: w dw w C dc C t M dt M 1 t M ω L L L L L dl L t C dt C 1+t C

31 Public Economics: Chapter 2 31 Linearization of (BC) yields: C + t C = ω M m + (1 ω M ) [ w + L t A ] (2) where m M/P is real non-labour income, ω M m m+(1 t A )wl is the initial share of m in total income [ω M is low or zero for poor folks], and t A dt A /(1 t A ). It follows that the average labour income tax rate influences the budget restriction of the household.

32 Public Economics: Chapter 2 32 By combining (1) and (2) we find: 1/ω L 1 (1 ω M ) 1 L C = σ [ w t M t C ] ω M m + (1 ω M ) [ w t A ] t C Inverting the matrix on the left-hand side we obtain the solution: L C = ω L ω L (1 ω M ) (1 ω M ) 1/ω L σ [ ] w t M t C ω M m + (1 ω M ) [ ] w t A t C

33 Public Economics: Chapter 2 33 Focus on the expression for labour supply: L = ω L [ σ ( w t M t C ) ωm m (1 ω M ) ( w t A ) + t C 1 + ω L (1 ω M ) ] The compensated [Hicksian] wage elasticity is: ε H w σω L 1 + ω L (1 ω M ) > 0 The uncompensated [Marshallian] wage elasticity is: ε M w ω L [σ (1 ω M )] 1 + ω L (1 ω M ) 0 which depends on interplay between SE and IE [note the role of ω M which is high for rich folks] The marginal labour income tax rate isolates the pure SE: t M > 0 causes L < 0 The average labour income tax rate isolates the IE: t A > 0 causes L > 0

34 Public Economics: Chapter 2 34 The consumption tax has both SE and IE: net effect depends on sign of σ 1 Leisure is a normal good: m > 0 causes L < 0 In Figure 2.5 we illustrate the effect of an increase in the marginal tax rate t M holding constant the average tax rate t A. For convenience we assume in that diagram: there is a linear progressive tax schedule: T = t M WL Pz 0 where z 0 > 0 is the lump-sum transfer (in real terms) the tax on consumption is zero (t C = 0) there is no non-labour income (m = 0)

35 The budget line can be written as follows: which is the line BC in the figure But it can also be written as: Public Economics: Chapter 2 35 C = z 0 + (1 t M )w L (1 t M )w ( L L ) C = (1 t A ) wl = (1 t A ) w L (1 t A ) w ( L L ) which is line EF in the figure. (EF steeper than BC as t M > t A )

36 Public Economics: Chapter 2 36 Now t M rises but t A is held constant. initial equilibrium is at E 0 : tangency of indifference curve with line BC in that point. t M rises so budget line BC rotates counter-clockwise to BD t A unchanged so EF stays put in the absence of compensation measures new equilibrium would be at A. But there t A would be too high. To keep t A unchanged z 0 must rise so that new equilibrium is at E 1 (parallel shift in BD) C falls but L L rises

37 Public Economics: Chapter 2 37 C G - w(1 - t A )L E H - z 0 + w(1 - t M )L C E 0 D A E 1 z 0 B 0 F L - - L L Figure 2.5: Increasing the Marginal Tax Rate [Constant Average Tax Rate]

38 Public Economics: Chapter 2 38 Some Cautionary Remarks we have seen that a progressive tax system can be easily handled. Choice set remains convex so that the optimum is unique many features of actual tax system may make choice set non-convex (e.g. means-tested transfer programs). In that case: there may be multiple tangencies standard comparative static effects (e.g. Slutsky decomposition) no longer valid econometric testing is much more complicated

39 Public Economics: Chapter 2 39 Means-Tested Benefits and Labour Supply Suppose that transfers received from the government are means-tested, i.e. they depend on the income of the recipient For simplicity assume the following transfer scheme (in real terms): z = z 0 + t Z w (L MIN L) z 0 for 0 < L L MIN for L > L MIN (A) L MIN is the policy-determined critical number of hours. If L L MIN then the household received additional transfers t Z is the means-testing parameter the more the household works, the lower transfers get. It operates as an effective tax

40 Public Economics: Chapter 2 40 The tax system is T = t P MwL and the household budget constraint (in real terms, abstracting from t C ) is: C = m + z + wl t M wl Using (A) we find for 0 < L L MIN : C = [m + z 0 + t Z wl MIN ] + (1 t M t Z ) wl and for L > L MIN : C = [m + z 0 ] + (1 t M )wl

41 Public Economics: Chapter 2 41 In terms of Figure 2.6, the budget line features a kink at point A (where leisure is L L MIN ) for low labour supply the means-testing of the transfers implies that the effective tax rate is higher than for high levels of labour supply in the case drawn there are two tangencies between the non-convex budget line and the indifference curve (at E 0 and E 1 ) standard comparative statics invalid as small change may produce inframarginal jumps, e.g. tiny decrease in t Z rotates AB to AB and shifts equilibrium discretely from E 1 to E 0

42 Public Economics: Chapter 2 42 C U = U 0 E 0 A E 1 B BN C 0 - L L MIN L - - L L Figure 2.6: Means-Tested Transfer System

43 Public Economics: Chapter 2 43 Matters are even more complex if the tax system features increasing marginal tax rates (rather than one single one), e.g.: T P = t 1 M wl for 0 < L L 1 t 1 M wl 1 + t 2 M w (L L 1) for L > L 1 with t 2 M > t1 M and L 1 > L MIN ). Figure 2.7 shows that the budget line features two kinks in that case (one at point A because of the benefit system and the other at point E 0 because of the tax progression). Again there may be multiple local optima. We have drawn one tangency optimum (at E 1 ) and one corner solution (at E 0 ) To determine the global optimum we must check all local optima and determine which one features the highest utility. (In the figure U 0 > U 1 ) We must know the form of the individual s utility function (not just local curvature)

44 Public Economics: Chapter 2 44 C E 0 A E 1 U = U 0 B U = U 1 C 0 - L L 1 - L L MIN L - - L L Figure 2.7: Means-Tested Transfers and Marginal Tax Rate Progression

45 Labour Force Participation Public Economics: Chapter 2 45 Up to now we have assumed that choice of hours can be made freely, i.e. L can take on any value between 0 and 1 (the time endowment). This is rather unrealistic In practice the major decision is either to work full time (L = L F ) or not to work at all (L = 0) Note: obviously part-time work can be easily introduced (L = L P where 0 < L P < L F ) Different people may have different attitude toward work

46 Public Economics: Chapter 2 46 A Simple Model Utility function of person i: U i (C, 1 L) C α (1 L) β i where α > 0 and β i 0. There exists a frequency distribution for β i across the population. Budget constraint of working household: C = wl F (1 t) where w W/P is the real wage and we abstract from non-labour income. Budget constraint of non-working household: C = b where b is the real unemployment transfer (untaxed)

47 Public Economics: Chapter 2 47 Rudimentary benefit system: benefits linked to after-tax wage income: b = γwl F (1 t), 0 < γ < 1 where γ is the replacement rate. Labour participation is now a yes-no decision. For household i: utility when working (L = L F and C = wl F (1 t)) is: U i worker (wl F (1 t)) α (1 L F ) β i utility when not working (L = 0 and C = b) is: U i unemployed b α hence: U i worker U i unemployed = (wl F(1 t)) α (1 L F ) β i [γwl F (1 t)] α = (1 L F) βi γ α

48 the labour supply choice is thus: Public Economics: Chapter 2 48 L i = 0 if γ α (1 L F ) β i < 1 L F if γ α (1 L F ) β i > 1 The marginal household is indifferent between working and not working, i.e. it has a β i = β M such that γ α (1 L F ) β M = 1. By taking logarithms on both sides of this expression we can solve for β M : α ln γ + β M ln(1 L F ) = 0 β M = α ln γ ln(1 L F ) > 0 the sign follows from the fact that 0 < γ < 1 and 0 < L F < 1 (so that ln γ < 0 and ln(1 L F ) < 0).

49 Public Economics: Chapter 2 49 Households whose β i exceeds β M prefer not to work (they like leisure too much ) whereas households with a β i smaller than β M choose to work. (Someone with β i = 0 is the proverbial workaholic.) Assume that the β i s are distributed uniformly over the interval [0,β max ]. The frequency distribution is drawn in Figure 2.8. Assume that the population size is Z. All households with a β i β M are workers whereas all households with a β i > β M are loungers. There are thus (β max β M )Z/β max loungers and β M Z/β max workers (who each work L F hours). Aggregate labour supply is thus: L S = β MZL F β max

50 Public Economics: Chapter 2 50 The macroeconomic labour supply curve is drawn in Figure 2.9. Note that this aggregate labour supply curve is vertical because β M does not depend on the wage rate or the income tax (due to the fact that unemployment benefits are linked to after-tax wage income). The replacement rate exerts a negative influence on aggregate labour supply in this economy: β M γ = α γ ln(1 L F ) < 0, L S γ = ZL F β max β M γ < 0, where the signs follow from the fact that ln(1 L F ) < 0. The reduction in β M causes the aggregate labour supply curve to shift to the left, as is indicated in Figure 2.9.

51 Public Economics: Chapter /$ max workers loungers 0 $ M $ max $ i Figure 2.8: Frequency Distribution of β i Coefficients

52 Public Economics: Chapter 2 52 w L S $ M NL F /$ max L S Figure 2.9: Aggregate Labour Supply

53 A Different Benefit System Public Economics: Chapter 2 53 Details of the benefit system matter a lot Now we assume that the unemployment benefits are linked to the gross wage: b = γwl F and we assume that γ < 1 t Reworking the earlier steps we find: utility when unemployed: U i unemployed = b α = (γwl F ) α = ( ) α γ (wl F (1 t)) α 1 t

54 Public Economics: Chapter 2 54 utility comparison: U i worker U i unemployed ( γ 1 t = (wl F(1 t)) α (1 L F ) β i ) α [wlf (1 t)] α = ( ) α 1 t (1 L F ) β i γ critical value: α [ln(1 t) ln γ] + β M ln(1 L F ) = 0 β M = α [ln γ ln(1 t)] ln(1 L F ) > 0 where the sign follows from the fact that ln(1 L F ) < 0 and the assumption that γ < 1 t. now an increase in the tax rate leads to an increase in γ/(1 t) and thus to an increase in the effective replacement rate. This implies that β M falls so that aggregate labour supply falls.

55 Other Theoretical Approaches Public Economics: Chapter 2 55 mentioned only briefly. Mandatory for RM students only. See Section 2.3 and Atkinson and Stiglitz (pp ). Literature is vast household production theory. Key idea: household uses its time endowment for three activities: leisure labour supply to the labour market labour used for home production of useful goods or services (e.g. cooking, child rearing, etc.) households may experience enjoyment of work. Utility function is changed to: U(C, L L,L)

56 Public Economics: Chapter 2 56 family decision making. Key issue: household consists of different individuals who may have conflicting interest. What objective function? Samuelson s Household Welfare Function approach (correct intra-generational transfers) dictatorial joint utility maximization by head of household Nash bargaining between household members [including possibility of household dissolution] in all these models the effects of taxes are different from those obtained with the basic model

57 Public Economics: Chapter 2 57 A Brief Overview of the Empirical Evidence Focus on static evidence Good surveys for men by Pencavel (1986 Handbook of Labor Economics), Killingsworth and Heckman (1986 HLE), and Blundell and MaCurdy (1999 HLE) Actual tax and welfare system very complex Non-convex case probably relevant for at least some [Blundell & MaCurdy cite 89% effective tax for California households in the income bracket $750 - $1500] Both labour supply (hours) and participation behaviour differ for men and women Table 1 reports on some non-linear budget constraint models for men [source: Blundell & MaCurdy (1999)] Table 2 does the same for married women

58 Public Economics: Chapter 2 58 Conclusions: uncompensated wage elasticity is near-zero for men but positive for married women income elasticity negative: leisure normal good (both men and married women)

59 Public Economics: Chapter 2 59 Study Country Variable Uncompensated Income Wage Elasticity Elasticity Blomquist SWE annual hours Bourguignon FRA weekly hours Blundell UK weekly hours Flood SWE annual hours Hausman USA annual hours 0.00 to to Kaiser GER annual hours MaCurdy USA annual hours Triest USA annual hours van Soest NED weekly hours Table 1. Nonlinear Budget Constraint Models: Men

60 Public Economics: Chapter 2 60 Study Country Variable Uncompensated Income Wage Elasticity Elasticity Arellano UK weekly hours 0.29 to to Arrufat UK weekly hours Blomquist SWE annual hours Bourguignon FRA weekly hours Blundell UK weekly hours Colombino Turin, ITA annual hours 1.18 to Hausman USA annual hours Kaiser GER annual hours Kuismanen FIN annual hours Triest USA annual hours van Soest NED weekly hours Table 2. Nonlinear Budget Constraint Models: Married Women

61 Punchlines Public Economics: Chapter 2 61 consumption-leisure choice of vital importance in public economics concepts: homothetic preferences, expenditure function (Shephard s Lemma), indirect utility function (Roy s Identity) income and substitution effects: Slutsky decomposition progressive income taxes easy to handle: choice set convex concepts: substitution elasticity method: linearization and comparative statics actual tax systems may make choice set non-convex labour force participation decision and the role of unemployment benefits empirical evidence suggests low elasticity for men, higher for married women