General Equilibrium Tax Model

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1 Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20 General Equilibrium Tax Model Further Economic Analysis Dr. Keshab Bhattarai Hull Univ. Business School April 4, 2011

2 Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20 Simplest General Equilibrium Tax Model: Demand Side Households problem max U = C L (1) Subject to p (1 + t) C + L = L (2) Lagrangian for houshold optimisation L (C, L, λ) = C L + λ p (1 + t) C + L L (3)

3 Household Optimisation L (C, L, λ) C = L + λp (1 + t) = 0 (4) L (C, L, λ) L = C + λ = 0 (5) L (C, L, λ) = p (1 + t) C + L L = 0 (6) λ Above three FOC equations (4) - (6) can be solved for three variables : MRS CL = L(C,L,λ) C L(C,L,λ) L =) L C = p (1 + t) (7) Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20

- (6) can be solved for three variables : MRS CL = L(C,L,λ) C L(C,L,λ) L =) L C =

4 Household Optimisation Putting (8) into (6) L = p (1 + t) C (8) p (1 + t) c + L L = p (1 + t) C + L = C = 1 L 2 p (1 + t) p (1 + t) p (1 + t) C = p (1 + t) C L = 0 (9) (10) 1 L 2 p (1 + t) = 1 2 L (11) Demand for goods is lo ith higher taxes and prices, high ith higher age rate and labour endoment; high ith the higher share of spending on goods and services.given these preferences the demand for leisure is half of the labour endoment. Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20

high ith higher age rate and labour endoment; high ith the higher share of spending on goods and services.

5 Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20 Supply Side of the General Equilibrium Model Firms pro t maximisatin problem Subject to max π = p.y.ls (12) Y = p r 1 LS = 2 L (13) Consumers pay tax not the producers. In no tax case, given this production technology and demand side derivations, labour demand equals labour supply Labour market clearing Goods market clearing L + LS = L (14) C = Y (15)

maximisatin problem Subject to max π = p.y.

6 Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20 Numerical Example of the General Equilibrium Model Let total labour endoment L be 200. Then labour supply is 1 LS = L L = L 2 L = 1 L = 100 (16) 2 Given this labour supply the level of output ill be Y = p r 1 LS = 2 L = p 100 = 10 (17) From the zero pro t condition required equilibrium π = p.y.ls = 0 and setting the numeraire p = 1 p.y =.LS (18) = Y LS = = 1 10 (19)

7 Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20 Numerical Example of the General Equilibrium Model Given the equilibrium relative age rate of = 1 market clear hen t = 0, demand for good eqauls supply as: 10 both goods and labour C = 1 L 2 p (1 + t) = (200) = 10 (20) 10 Similarly the demand for labour and leisure equal total endoment of labour L + LS = = 200 = L (21) Labour market clears. Therefore this is a general equilibrium; at these prices both goods and labour market clear, household maximise utility and rms maximise pro t.

market clear hen t = 0, demand for good eqauls supply as: 10 both goods and labour C = 1 L 2 p (1 + t) = 1 2 1 (200) = 10 (20) 10 Similarly the

8 Numerical Example of the General Equilibrium Model ith Tax When t = 0.2 then C = 1 L 2 p (1 + t) = Government revenue and spending: Markets clear in this case too Houshold s elfare before tax Welfare after tax (200) 1.2 = 8.33 (22) R = p.t.c = = 1.67 = G (23) C + G = = 10 = Y (24) U = C L = = 1000 (25) U = C L = = 833 (26) Thus 20 percent tax has reduced the household elfare by 16.7 percent. Can utility from public spending of compensate for this lost elfare? Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20

tax (200) 1.2 = 8.33 (22) R = p.t.c = 1 0.2 8.33 = 1.67 = G (23) C + G = 8.33 + 1.67 = 10 = Y (24) U = C L = 10 100 = 1000 (25) U = C L = 8.

9 Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20 Household Problem: Maximise Utility Household gets utility from consuming goods and leisure Subject to Lagrangian optimisation: Max c,l U = C α L (1 α) (27) p.c +.L h = L (28) L (C, L h, λ) = C α L (1 α) + λ L p.c.l h Optimal demand for goods C.solving the rst order conditions C = αl p = αl p Households buy more hen goods are cheaper and hen they have more income (29) (30)

consuming goods and leisure Subject to Lagrangian optimisation: Max c,l U = C α L (1 α) (27) p.c +.

10 Firms Problem: maximise pro t Optimal demand for leisure L h (1 α) L.L h = = (1 α) L (31) if L = 1600 and α = 0.4 then.l h = = 960. Π = PY L f (32) Y = L β f (33) L f = 1 βp 1 β (34) Let β = 0.5 Y = β βp 1 β (35) Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20

11 Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20 Clearing Goods and Labour Markets: Real Wage Rate Y = C (36) L f + L h = L; L f = = 640 (37) Y = C = αl p β βp 1 β = = (38) = p = y = (39) p = = (40) if = 1 set as numeraire labour market clears as L f + L h = = 1600 = L (41)

Rate Y = C (36) L f + L h = L; L f = 1600 960 = 640 (37) Y = C = αl p β βp 1 β = 640 0.

12 Parameters and shado prices Table: Parameters of the General Equilibrium Model 0.6 λ = α Lh C p Shado price in tax scenario λ T = α Parameters Value α 0.4 β 0.5 L 1600 (normalised) Lh C p = = = 0.12 (42) = (43) This is the change in utility associated to unit change in income. Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20

5 L 1600 (normalised) 1 0.6 Lh C p = 0.4 640 0.6 25.29 25.29 = 0.4 480 0.6 21.90 21.90 = 0.12 (42) = 0.

13 Allocations and Prices in Equilibrium Table: General Equilibrium Solutions Variable Base No Tax Solution Tax Solution output (Y ) Consumption(C ) Leisure(L h ) Labour demand(l f ) Utility(U) Relative price p Shado Price Welfare loss to households from the government = ( ) / = = 20.56%. E ective labour tax = 400/1600=0.25= 25%. True if households do not get utility of from public spending. Ho far this is true depends on the e ciency of the public sector. Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20

116 Welfare loss to households from the government = (224.19 178.09) /224.19 = 0.2056 = 20.56%. E ective labour tax = 400/1600=0.25= 25%.

14 Decomposition of income and substitution e ects of tax changes Subject to Lagrangian optimisation: Max U = X X (44) p 1.X 1 + p 2.X 2 = 150 (45) L (X 1, X 2, λ) = X X λ [150 p 1.X 1 p 2.X 2 ] (46) For base equilibrium assume that p 1 = 3 and p 2 = 2. Optimal demand for goods X (150) X 1 = = 60 p 1 3 = 20; X 0.6 (150) 2 = = 90 = 45 (47) p 2 2 U 0 = X X = (20) 0.4 (45) 0.6 = (48) No assume that there is a subsidy in X 1 of 1 and price reduces from 3 to 2; p 1 = 2. Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20

Optimal demand for goods X 1 0.4 (150) X 1 = = 60 p 1 3 = 20; X 0.6 (150) 2 = = 90 = 45 (47) p 2 2 U 0 = X 0.4 1 X 0.6 2 = (20) 0.4 (45) 0.6 = 32.

15 Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20 Equivalent Variation What is the Hicksian Equivalent and compensating variations of price change? What are the income and substitution e ects of this price change? First nd out ho much money is required at ne prices to guarantee the original utility by solving 0.4 (m U 0 = 0 ) (m 0 ) 0.6 = (49) (m U 0 = 0 ) (m 0 ) 0.6 ; m 0 = 2 (32.53) = (50) Equivalent variation (money to be taken aay hen prices fall) EV = = (51)

change? What are the income and substitution e ects of this price change?

16 Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20 Compensating Variation For compensating variation rst compute the demand in ne prices and utility X 1 = 0.4 (150) = 60 p 1 2 = 30; X 0.6 (150) 2 = = 90 = 45 (52) p 2 2 U 1 = U 1 = X X = (30) 0.4 (45) 0.6 = (53) m A 0.6 (m 00 ) 0.6 = (54) 2 m 00 = 2 (38.26) = (55) CV = = (56)

the demand in ne prices and utility X 1 = 0.4 (150) = 60 p 1 2 = 30; X 0.

17 Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20 Summarising the Money Metric Utility Changes Due to Taxes Table: Summary of Equivalent and Compensating Variation Fall in Price Rise in Price EV + - CV - + Fall in Price Basis of evaluation Ne Price-Old Utility OLD Price- Ne Utility Substitution E ect : 2.5 +(10-7.5); Income e ect:7.6= (22.5/3) and total e ect: 10. This is partial equilibrium result - general equilibrium impacts must take interaction ith all other markets. Ultimate impact can be much higher or much loer than this. It need to bring production, income distribution sides into account.

Fall in Price Rise in Price EV + - CV - + Fall in Price Basis of evaluation 22.51 Ne Price-Old Utility -26.39 OLD Price- Ne Utility Substitution E ect : 2.

18 Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20 Burden of Taxes in Partial Equilibrium Analysis (it depends on elasticities) Consider linear demand and supply model D = 150 3P (57) S = P (58) Equilibrium D =S implies P=24 and Q = 78. No there is tax in commodity so that consumers pay more and suppliers get less. P D = P S + t (59) here t is tax imposed per unit. Let t = 2. D = 150 3P D = P S + 2 (60)

elasticities) Consider linear demand and supply model D = 150 3P (57) S = 30 + 2P (58) Equilibrium D =S

19 Burden of Taxes in Partial Equilibrium Analysis (it depends on elasticities) D = 150 3P D = P S + 2 (61) S = P S (62) P D = 24.8 P S = 22.8 Q= 75.6 Deadeight loss of taxes =loss of consumer surplus+loss of producer surplus= = 0.5(0.82.4) ( ) = = 2.4 Elasticity of demand = = 0.92 Elasticity of supply = = Thus more burden is taken by producers. General equilibrium impacts are much higher than the partial equilibrium impacts. You can compute a general equilibrium tax model using GAMS for more realistic economy using input-output tables presented last eek. Demo version can solve only small models but not the large ones. Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20

92 Elasticity of supply = 2 24 78 78 = 0.61. Thus more burden is taken by producers. General equilibrium impacts are much higher than the partial equilibrium impacts.

20 References Atkinson A. B.and N. H. Stern (1974) Pigou, Taxation and Public Goods The Revie of Economic Studies, 41:1: Bhattarai K (2010) Taxes, public spending and groth in OECD countries, Journal of Perspective and Management, 1/2010. Bhattarai K and J. Whalley (2009) Redistribution E ects of Transfers, Economica 76:3: July. Blundell R (2010) Empirical Evidence and Tax Policy Design: Lessons from the Mirrlee s Revie, Institute of Fiscal Studies. Darling A. Chancellor of Exchequer, HM Treasury (2009), Securing the Recovery: Groth and Opportunity, Pre-Budget Report, December, Feldstein M (1974) Incidence of Capital Income Tax in a Groth Economy ith Varying Saving Rates, Revie of Economic Studies, 41:4: Fullerton, D., J. Shoven and J. Whalley (1983) Dynamic General Equilibrium Impacts of Replacing the US Income tax ith a Progressive Consumption Tax, Journal of Public Economics 38, Meade J (1978) Structure of Direct Taxation, Institute of Fiscal Studies, London. Mirlees, J.A. (1971) An exploration in the theory of optimum income taxation,revie of Economic Studies, 38: Perroni, C. (1995), Assessing the Dynamic E ciency Gains of Tax Reform When Human Capital is Endogenous, International Economic Revie 36, Main budget: Green Budget: Dr. Bhattarai (Hull Univ. Business School) GE Tax April 4, / 20

Whalley (2009) Redistribution E ects of Transfers, Economica 76:3:413-431 July.

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