Spring 2013 Econ 567 Project #2 Wei Zhang & Qing Tian. The study of the welfare effect of the income tax and the excise tax

Transcription

1 The study of the welfare effect of the income tax and the excise tax Wei Zhang Qing Tian April 16,

2 Table of Contents I. Background and Introduction.. 3 II. Methodology..4 III. Model Setup and Results. 5 1) Excise tax on single good ) Excise tax on good ) No tax imposed...6 2) Income tax on consumer...6 3) Excise tax on good 1 and good2...8 IV. Further Discussion... 9 V. Conclusion...11 VI. Reference...12 VII. Appendix

3 Background and Introduction The debate between the income tax and the excise tax has always been a hot topic. The excise tax and the income tax are used interchangeably within one country. The income tax is collected directly on personal income. Adding an income tax will decrease the disposable income and thus the budget constraint for the individual. The excise tax will be charged on the price paid by consumers and thus creating a gap between the price faced by consumer and the price face by producer. The tax revenue is the price difference times the quantity of goods sold. However, which tax will generate a higher utility for people while keeping the same tax revenue? The objective of this project is to simulate a model using GAMS to examine the welfare effect of adding an income tax and an excise tax under a one consumer s circumstance. Further, we provide some discussion as to how our model can be extended to explore the effect of these two taxes under a community with identical individuals. We first examine our model under the one-consumer case intuitively with a graph. Assume in a closed economy with two goods, good 1 and good 2. Good 1 are on the horizontal axis and 2 are on the vertical axis. Also, assume the price of good 2 and income are held constant. AB is the initial budget line. Q 1 is the initial equilibrium. We add a 50% excise tax on good 1. Then the new price will be twice than the old price. So the new budget constraint will be AC. C is on the middle point of OB. Then Q 2 is the new tangent point between indifference curve and budget line. Now, if we add an income tax on this individual, his or her personal income will decrease and thus the budget line declines. The budget line will shift parallel inward toward the origin because the prices of good 1 and good 2 are not changed. Q 3 will be the new equilibrium. 3

4 From the figure 1, we can see that Q 3 is on a higher indifference curve than Q 2. Therefore, we reach a conclusion that the income tax is more preferable than the excise tax in a one consumer circumstance. Figure 1 The budget constraint curves and the consumer s indifference curves in a one-agent situation Methodology The relative effect on welfare of an excise tax and an income tax is compared in a single-agent, two-good economy using computer modeling in GAMS. This model is based on the argument made by Milton Friedman in his 1953 paper: The Welfare Effects of an Income Tax and an Excise Tax. Our model supports his main conclusion that with equal tax revenue, an income tax is superior to an excise tax. Using the same set up, we also modified the model to examine how we could possibly fully reverse the welfare loss of an excise tax by imposing tax on both goods outputs. Specifically, the welfare of this economy under three policy scenarios will be analyzed: (1) A sudden excise tax imposed by government on good 1; no income tax. (2) Income tax imposed on consumer producing the same tax revenue as in (1); no excise tax. (3) Excise tax on both good 1 and good 2 producing the same tax revenue as in (1); no income tax. 4

5 General Model Setup and Results Assume a closed economy consists of a single consumer and the production of two goods. The preferences of the consumer satisfy the axioms of consumer choice, and it can be represented with the following Cobb-Douglas utility function: U = α C1 β C2 1 β ; with C1 and C2 represents the consumption of good 1 and good 2. Also, the consumer subjects to budget constraint: Y = p 1 C 1 + p 2 C 2 ; with p1 and p2 represents the pre-tax price of good 1 and good2. For simplicity, we assume that the pre-tax price of good 1 and 2 are equal, perfectly inelastic and normalized to 1: p1=p2=1. We also assume that the tax revenue collected by government is used to buyback the leftover production after consumption of private sector. In addition, for simplicity, we assume the government is keeping the tax revenue constant, i.e. government is not paying tax for buying goods, the price it faces is always the same. 1.1) Situation (1) Excise tax on good 1, no income tax: 1.A. Parameters and Initial Values Assigned. Table1. Parameters of good 1 excise tax model Name Description Initial Value Alpha Shift parameter in utility 2.89 [1] Beta Share parameter in utility 0.5 [2] Y Income (budget constraint) 100 P1 Price of good 1 faced by 1/(1-t) [3] consumer P2 Price of good 2 faced by 1 consumer U0 Initial utility level 100 C01 Initial consumption of 30 [4] good 1 C02 Initial consumption of 40 good 2 T tax rate 50% 5

6 Note: [1] Alpha is calculated (by GAMS) using the equation: Alpha = U0/ (C01 beta C01 1 beta ). The level of alpha does not affect the conclusion of our model. [2] Beta is 0.5 is assumed for simplicity, as well as the same pre-tax price (price facing supplier) of both goods. [3] Price facing consumer*(1-t) = Price facing supplier =1. [4] Initial consumptions of good1 and good 2 are constructed so that they satisfy the budget constraint. It only provides the GAMS with a start point to search for solution and their levels do not affect the conclusion of our model. 1.B. Variables and Equations for GAMS U: Utility level. C1: Consumption of good 1. C2: Consumption of good 2. R: Tax revenue. 1. Utility Function: U = alpha C1 beta C2 1 beta. 2. Budget Constraint: Y = P 1 C 1 + P 2 C Tax Revenue collected: R = t P 1 C 1. We use GAMS to maximize utility, i.e. objective utility function 1; subjected to budget constraint (function 2). We also want GAMS to calculate the excise tax revenue collected by government under the maximized utility condition, given the tax rate t. 1.C. Results Summary (for full output, please refer to appendix) Table 2. Utility Maximization Results of one good excise tax model Variable Solution value U C1 25 C2 50 R ) No tax: Under the same utility function (alpha and beta calculated in 1.1)), but with no tax imposed by government, the same program was ran to look at the maximized utility under no tax condition. Results are presented in the following table: 6

7 Table 3. Utility Maximization Results under No Tax condition. Variable Solution value U C1 50 C2 50 Without tax, under the same budget constraint and utility function, the maximum utility level is ; while with tax imposed on a single good, the maximum utility is There is a welfare loss due to the imposition of tax. 2) Situation (2) Income tax on consumer 2.A. Parameters and Initial Values Table 4. Parameters of Income Tax Model Name Description Initial Value Alpha Shift parameter in utility 2.89 Beta Share parameter in utility 0.5 Y Income (budget constraint) 100-R [5] P1 Price of good 1 faced by 1 consumer P2 Price of good 2 faced by 1 consumer U0 Initial utility level 100 C01 Initial consumption of 35 [6] good 1 C02 Initial consumption of 40 good 2 R tax revenue 25 [7] Note: [5]. Income tax imposed on consumer leads to an equal reduction in the amount of budget constraint for consumption. [6]. Initial consumption of good1 and good 2 are constructed so that they satisfy the new budget constraint and new prices faced by consumer. [7]. Tax revenue of income tax should equal to tax revenue of excise tax from situation (1). 2.B. Variables and Equations for GAMS U: Utility level. 7

8 C1: Consumption of good 1. C2: Consumption of good Utility Function: U = alpha C1 beta C2 1 beta. 2. Budget Constraint: Y = P 1 C 1 + P 2 C 2. We use GAMS to maximize utility of consumer, subjected to the new budget constraint (reduced by the amount of income tax revenue). Also note that since there is no excise tax, the price faced by consumer (P1 and P2) equals the price faced by supplier (pre-tax price p1 and p2). P1=P2=1. 2.C. Results summary (for full output, please refer to appendix) Table 5. Results of Income Tax Model Variable Solution value U C C Now we can compare the utility maximization results of situation (1)-single excise tax on good1 and situation (2)-income tax on consumer. When we assume the same consumer preference (the same utility function in terms of alpha and beta) and the same tax revenue collected, the maximized utility for single excise tax situation is ; while for income tax situation, it is Thus there is a larger social welfare loss when excise tax is imposed on a single output comparing to the same tax imposed on consumer income. 3) Situation (3) excise tax on good 1 and good2 3.A. Parameters and Initial Values Table 6. Parameters of Two-Excise Tax Model Name Description Initial Value Alpha Shift parameter in utility 2.89 Beta Share parameter in utility 0.5 Y Income (budget constraint) 100 [8] R tax revenue 25 [9] Note: [8]. Since there is no income tax, the budget constraint is restored to the original level. 8

9 [9]. The total tax revenue collected from good 1 and good 2 should equal to tax revenue in situation (1) and (2). This will be reflected in the tax revenue equation discussed below. 3.B. Variables and Equations for GAMS U: Utility level. C1: Consumption of good 1. C2: Consumption of good 2. P1: Price of good 1. P2: Price of good2. t1: tax rate on good 1. t2: tax rate on good Utility Function: U = alpha C1 beta C2 1 beta. 2. Budget Constraint: Y = P 1 C 1 + P 2 C Tax Revenue: R = t (P 1 C 1 + P 2 C 2 ). 4. Price of good 1: P1 = 1/(1 t1) 5. Price of good 2: P2 = 1/(1 t2) We use GAMS to maximize utility of consumer, subjected to both the budget constraint and the constraint that under t1 and t2, the total tax revenue collected from good 1 and good 2 should equal income tax revenue in previous cases. Price faced by consumer for good 1 is determined by t1; price faced by consumer for good 2 is determined by t2. 3.C. Results summary (for full output, please refer to appendix) Table 7. Results of Two Excise Tax model Variable Solution Value U C C P P t1 25% t2 25% 9

10 From this result, when excise tax is imposed on both outputs no income tax, the maximized utility condition is achieved when C1=C2 and t1=t2, which makes intuitive sense given the form of the utility function. Also, the maximum utility level under this two excise tax condition is now , which equals that of the income tax situation. The social welfare loss of excise tax is partially restored (to the level equal to that of the income tax situation) when tax revenue is evenly extracted from both outputs. Further Discussion The analysis applied to one consumer is adequate to say that the income tax is more preferable than the excise tax. However, the interpretation is not sufficient when we apply it to the community level. Assume we have a community with identical individuals who have same income and tastes. Also, we need to take into account of the technical possibilities. GH is the production possibility frontier. AB is the budget constraint and Q 1 is the initial consumption equilibrium. It is at the situation when the rate of substitution in consumption is equal to the rate of substitution in purchase and in production. If the income taxes collected are giving back to individuals as a per capita subsidy, then Q 1 in figure 2 remains unchanged. If the revenue collected is used to produce other good, say good 3, if we fix the amount of good 3 produced, the effect of an income tax and an excise tax will be the same because the production possibility frontier depends on the amount of good 3 not how the tax revenue is collected. When the excise tax is imposed, there is a difference between the price paid by consumers and price sold by producers. So the new equilibrium needs to be determined by not only the consumption indifference curve but also by the production possibility frontier. Then the new equilibrium will be on a lower indifference curve. From the figure 2, we can see that Q 4 is lower than Q 1. Therefore, similar to the one consumer scenario, at full competitive equilibrium, the income tax is more preferable than the excise tax. However, if the equilibrium is at Q 4, say some monopoly exists, imposing an excise 10

11 tax on good 1 will converge to the price level between consumers and producers, and thus reach the equilibrium at Q 1. That is, in this case, the excise tax is superior to the income tax. Further discussion is needed to validate this proof using more sophisticate programming in the future. Figure 2. The production possibility frontiers and indifference curves in a community with identical individuals Conclusion By using GAMS, we simulated three scenarios of the effect of the income tax and the excise tax on the welfare in a single agent economy. When collecting the same amount of tax revenue, the income tax is superior to a single excise tax in terms of maximized utility for consumer. This conclusion holds under the situation when the excise tax is only imposed on one good. When the excise tax is collected equally on two goods, then the welfare loss through the excise tax can be restored fully. However, when consider the model on a community basis with identical individuals; the proof may need further explanation. The income tax may not always be superior to the excise tax. When the economy starts from the full competitive equilibrium, the income tax will generate higher utility. On the other hand, if the initial starting point is not at the full competitive equilibrium, the income tax may not give the higher welfare. It would be interesting to further extend our model and discuss the effect of two types of taxes on social welfare at the community level. 11

12 Reference Gilbert, J., Tower, E. An Introduction to GAMS Modeling for International Trade Theory and Policy. Friedman, M. The Welfare Effects of an Income Tax and an Excise Tax. Journal of Political Economy Vol. 60, No. 1 (Feb., 1952), pp

13 Appendix: GAMS outputs for three policy scenarios: Situation (1) Exercise tax on good 1: GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/14/13 16:55:20 Page 1 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m C o m p i l a t i o n 1 *Situation 1 government suddenly impose an exercise tax on goods *Define the indexes for the problems 4 SET I GOODS /1,2/; 5 6 *Create names for parameters 7 PARAMETERS 8 ALPHA Shift parameter in utility 9 BETA(I) Share parameters in utility 10 Y Income (Budget Constraint) 11 P(I) Price of goods faced by consumer 12 U0 Initial utility level 13 C0(I) Initial Consumption level 14 t tax rate; *Assign values to tax rate 17 t=0.5; 18 P('1')=1/(1-t); 19 P ('2')=1; 20 Y=100; 21 U0=Y; 22 C0('1')=30; 23 C0('2')=40; 24 BETA(I)=1/2; 25 ALPHA=U0/PROD(I,C0(I)**BETA(I)); *Create names for variables 28 VARIABLES 29 U Utility level 30 C(I) Consumption levels 31 R Tax revenue; *Assign initial values to variables, and set lower bounds 34 U.L=U0; 35 C.L(I)=C0(I); 36 C.LO(I)=0; 13

14 37 R.L=10; *Create names for equations 40 EQUATIONS 41 UTILITY Utility Function 42 BUDGET Budget Constraint 43 TAX Tax Revenue; *Define equations 46 UTILITY..U=E=ALPHA*PROD(I,C(I)**BETA(I)); 47 BUDGET..Y=E=SUM(I,C(I)*P(I)); 48 TAX..R=E=t*P('1')*C('1'); *Define the model and solve 51 MODEL UMAX1 /ALL/; 52 SOLVE UMAX1 USING NLP MAXIMIZING U; 53 COMPILATION TIME = SECONDS 3 Mb WEX Feb 14,

15 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/14/13 16:55:20 Page 2 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Equation Listing SOLVE UMAX1 Using NLP From line UTILITY =E= Utility Function UTILITY.. U - ( )*C(1) - (1.25)*C(2) =E= 0 ; (LHS = 0) ---- BUDGET =E= Budget Constraint BUDGET.. - 2*C(1) - C(2) =E= -100 ; (LHS = -100) ---- TAX =E= Tax Revenue TAX.. - C(1) + R =E= 0 ; (LHS = -20, INFES = 20 ****) 15

16 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/14/13 16:55:20 Page 3 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Column Listing SOLVE UMAX1 Using NLP From line U Utility level U 1 UTILITY (.LO,.L,.UP,.M = -INF, 100, +INF, 0) ---- C Consumption levels C(1) (.LO,.L,.UP,.M = 0, 30, +INF, 0) ( ) UTILITY -2 BUDGET -1 TAX C(2) (-1.25) UTILITY -1 BUDGET (.LO,.L,.UP,.M = 0, 40, +INF, 0) ---- R Tax revenue R 1 TAX (.LO,.L,.UP,.M = -INF, 10, +INF, 0) 16

17 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/14/13 16:55:20 Page 4 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Model Statistics SOLVE UMAX1 Using NLP From line 52 MODEL STATISTICS BLOCKS OF EQUATIONS 3 SINGLE EQUATIONS 3 BLOCKS OF VARIABLES 3 SINGLE VARIABLES 4 NON ZERO ELEMENTS 7 NON LINEAR N-Z 2 DERIVATIVE POOL 10 CONSTANT POOL 17 CODE LENGTH 11 GENERATION TIME = SECONDS 4 Mb WEX Feb 14, 2013 EXECUTION TIME = SECONDS 4 Mb WEX Feb 14,

18 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/14/13 16:55:20 Page 5 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Solution Report SOLVE UMAX1 Using NLP From line 52 S O L V E S U M M A R Y MODEL UMAX1 OBJECTIVE U TYPE NLP DIRECTION MAXIMIZE SOLVER CONOPT FROM LINE 52 **** SOLVER STATUS 1 Normal Completion **** MODEL STATUS 2 Locally Optimal **** OBJECTIVE VALUE RESOURCE USAGE, LIMIT ITERATION COUNT, LIMIT EVALUATION ERRORS 0 0 CONOPT 3 Feb 14, WEX WEI x86_64/ms Windows C O N O P T 3 version 3.15I Copyright (C) ARKI Consulting and Development A/S Bagsvaerdvej 246 A DK-2880 Bagsvaerd, Denmark The model has 4 variables and 3 constraints with 7 Jacobian elements, 2 of which are nonlinear. The Hessian of the Lagrangian has 2 elements on the diagonal, 1 elements below the diagonal, and 2 nonlinear variables. ** Optimal solution. Reduced gradient less than tolerance. CONOPT time Total seconds of which: Function evaluations = 13.2% 1st Derivative evaluations = 0.0% LOWER LEVEL UPPER MARGINAL ---- EQU UTILITY EQU BUDGET

19 ---- EQU TAX... EPS UTILITY Utility Function BUDGET Budget Constraint TAX Tax Revenue LOWER LEVEL UPPER MARGINAL ---- VAR U -INF INF. U Utility level ---- VAR C Consumption levels LOWER LEVEL UPPER MARGINAL INF EPS INF. LOWER LEVEL UPPER MARGINAL ---- VAR R -INF INF. R Tax revenue **** REPORT SUMMARY : 0 NONOPT 0 INFEASIBLE 0 UNBOUNDED 0 ERRORS EXECUTION TIME = SECONDS 2 Mb WEX Feb 14, 2013 USER: GAMS Development Corporation, Washington, DC G871201/0000CA-ANY Free Demo, , sales@gams.com, DC0000 **** FILE SUMMARY Input Output C:\Users\Wei Zhang\Documents\gamsdir\projdir\Untitled_2.gms C:\Users\Wei Zhang\Documents\gamsdir\projdir\Untitled_2.lst 19

20 Situation (2) Income tax: GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/14/13 18:20:20 Page 1 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m C o m p i l a t i o n 1 *Situation 3 government suddenly impose an incom tax 2 3 *Define the indexes for the problems 4 SET I GOODS /1,2/; 5 6 *Create names for parameters 7 PARAMETERS 8 ALPHA Shift parameter in utility 9 BETA(I) Share parameters in utility 10 Y Income (Budget Constraint) 11 P(I) Price of goods faced by consumer 12 U0 Initial utility level 13 C0(I) Initial Consumption level 14 R Tax revenue same as tax revenue in situation 2 ; *Assign values to tax rate 17 P(I)=1; 18 R=25; 19 Y=100-R; 20 U0=Y; 21 C0('1')=35; 22 C0('2')=40; 23 BETA(I)=1/2; 24 ALPHA=2.89; *Create names for variables 27 VARIABLES 28 U Utility level 29 C(I) Consumption levels; *Assign initial values to variables, and set lower bounds 33 U.L=U0; 34 C.L(I)=C0(I); 35 C.LO(I)=0; *Create names for equations 20

21 38 EQUATIONS 39 UTILITY Utility Function 40 BUDGET Budget Constraint; *Define equations 43 UTILITY..U=E=ALPHA*PROD(I,C(I)**BETA(I)); 44 BUDGET..Y=E=SUM(I,C(I)*P(I)); *Define the model and solve 47 MODEL UMAX2 /ALL/; 48 SOLVE UMAX2 USING NLP MAXIMIZING U; COMPILATION TIME = SECONDS 3 Mb WEX Feb 14,

22 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/14/13 18:20:20 Page 2 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Equation Listing SOLVE UMAX2 Using NLP From line UTILITY =E= Utility Function UTILITY.. U - ( )*C(1) - ( )*C(2) =E= 0 ; (LHS = , INFES = ****) ---- BUDGET =E= Budget Constraint BUDGET.. - C(1) - C(2) =E= -75 ; (LHS = -75) 22

23 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/14/13 18:20:20 Page 3 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Column Listing SOLVE UMAX2 Using NLP From line U Utility level U 1 UTILITY (.LO,.L,.UP,.M = -INF, 75, +INF, 0) ---- C Consumption levels C(1) ( ) UTILITY -1 BUDGET (.LO,.L,.UP,.M = 0, 35, +INF, 0) C(2) ( ) UTILITY -1 BUDGET (.LO,.L,.UP,.M = 0, 40, +INF, 0) 23

24 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/14/13 18:20:20 Page 4 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Model Statistics SOLVE UMAX2 Using NLP From line 48 MODEL STATISTICS BLOCKS OF EQUATIONS 2 SINGLE EQUATIONS 2 BLOCKS OF VARIABLES 2 SINGLE VARIABLES 3 NON ZERO ELEMENTS 5 NON LINEAR N-Z 2 DERIVATIVE POOL 10 CONSTANT POOL 17 CODE LENGTH 11 GENERATION TIME = SECONDS 4 Mb WEX Feb 14, 2013 EXECUTION TIME = SECONDS 4 Mb WEX Feb 14,

25 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/14/13 18:20:20 Page 5 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Solution Report SOLVE UMAX2 Using NLP From line 48 S O L V E S U M M A R Y MODEL UMAX2 OBJECTIVE U TYPE NLP DIRECTION MAXIMIZE SOLVER CONOPT FROM LINE 48 **** SOLVER STATUS 1 Normal Completion **** MODEL STATUS 2 Locally Optimal **** OBJECTIVE VALUE RESOURCE USAGE, LIMIT ITERATION COUNT, LIMIT EVALUATION ERRORS 0 0 CONOPT 3 Feb 14, WEX WEI x86_64/ms Windows C O N O P T 3 version 3.15I Copyright (C) ARKI Consulting and Development A/S Bagsvaerdvej 246 A DK-2880 Bagsvaerd, Denmark The model has 3 variables and 2 constraints with 5 Jacobian elements, 2 of which are nonlinear. The Hessian of the Lagrangian has 2 elements on the diagonal, 1 elements below the diagonal, and 2 nonlinear variables. ** Optimal solution. Reduced gradient less than tolerance. CONOPT time Total seconds of which: Function evaluations = 0.0% 1st Derivative evaluations = 0.0% LOWER LEVEL UPPER MARGINAL ---- EQU UTILITY EQU BUDGET

26 UTILITY BUDGET Utility Function Budget Constraint LOWER LEVEL UPPER MARGINAL ---- VAR U -INF INF. U Utility level ---- VAR C Consumption levels LOWER LEVEL UPPER MARGINAL INF INF EPS **** REPORT SUMMARY : 0 NONOPT 0 INFEASIBLE 0 UNBOUNDED 0 ERRORS EXECUTION TIME = SECONDS 2 Mb WEX Feb 14, 2013 USER: GAMS Development Corporation, Washington, DC G871201/0000CA-ANY Free Demo, , sales@gams.com, DC0000 **** FILE SUMMARY Input Output C:\Users\Wei Zhang\Documents\gamsdir\projdir\Untitled_3.gms C:\Users\Wei Zhang\Documents\gamsdir\projdir\Untitled_3.lst 26

27 Situation (3) Exercise tax on both good 1 and good 2: GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/16/13 00:47:03 Page 1 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m C o m p i l a t i o n 1 *Situation 3 government impose equal excise tax on two goods to achieve ta rgeted tax revenue 2 3 *Define the indexes for the problems 4 SET I GOODS /1,2/; 5 6 *Create names for parameters 7 PARAMETERS 8 ALPHA Shift parameter in utility 9 BETA Share parameters in utility 10 Y Income (Budget Constraint) 11 U0 Initial utility level 12 C0(I) Initial Consumption level 13 P0(I) Initial Price of goods faced by consumer 14 R Tax revenue same as tax revenue in situation 2 ; *Assign values to tax rate 18 R=25; 19 Y=100; 20 C0(I)=30; 21 P0(I)=1.33; 22 BETA=1/2; 23 ALPHA=2.89; *Create names for variables 27 VARIABLES 28 U Utility level 29 C(I) Consumption levels 30 t(i) tax rate 31 P(I) Price of goods faced by consumer; *Assign initial values to variables, and set lower bounds 34 C.LO(I)=0; 35 27

28 36 *Create names for equations 37 EQUATIONS 38 UTILITY Utility Function 39 BUDGET Budget Constraint 40 TAX Total Tax Revenue on two goods equals R 41 PRICE1 Price of good 1 42 PRICE2 Price of good 2 43 Price3 Price equals; *Define equations 46 UTILITY..U=E=ALPHA*PROD(I,C(I)**BETA); 47 PRICE1..P('1')=E=1/(1-t('1')); 48 PRICE2..P('2')=E=1/(1-t('2')); 49 PRICE3..P('1')=E=P('2'); 50 BUDGET..Y=E=SUM(I,C(I)*P(I)); 51 TAX..R=E=SUM(I,t(I)*P(I)*C(I)); *Define the model and solve 56 MODEL UMAX2 /ALL/; 57 SOLVE UMAX2 USING NLP MAXIMIZING U; COMPILATION TIME = SECONDS 3 Mb WEX Feb 14,

29 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/16/13 00:47:03 Page 2 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Equation Listing SOLVE UMAX2 Using NLP From line UTILITY =E= Utility Function UTILITY.. U + (0)*C(1) + (0)*C(2) =E= 0 ; (LHS = 0) ---- BUDGET =E= Budget Constraint BUDGET.. (0)*C(1) + (0)*C(2) + (0)*P(1) + (0)*P(2) =E= -100 ; (LHS = 0, INFES = 100 ****) ---- TAX =E= Total Tax Revenue on two goods equals R TAX.. (0)*C(1) + (0)*C(2) + (0)*t(1) + (0)*t(2) + (0)*P(1) + (0)*P(2) =E= -25 ; (LHS = 0, INFES = 25 ****) ---- PRICE1 =E= Price of good 1 PRICE1.. - (1)*t(1) + P(1) =E= 0 ; (LHS = -1, INFES = 1 ****) ---- PRICE2 =E= Price of good 2 PRICE2.. - (1)*t(2) + P(2) =E= 0 ; (LHS = -1, INFES = 1 ****) ---- Price3 =E= Price equals Price3.. P(1) - P(2) =E= 0 ; (LHS = 0) 29

30 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/16/13 00:47:03 Page 3 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Column Listing SOLVE UMAX2 Using NLP From line U Utility level U 1 UTILITY (.LO,.L,.UP,.M = -INF, 0, +INF, 0) ---- C Consumption levels C(1) (.LO,.L,.UP,.M = 0, 0, +INF, 0) (0) UTILITY (0) BUDGET (0) TAX C(2) (.LO,.L,.UP,.M = 0, 0, +INF, 0) (0) UTILITY (0) BUDGET (0) TAX ---- t tax rate t(1) (0) TAX (-1) PRICE1 (.LO,.L,.UP,.M = -INF, 0, +INF, 0) t(2) (0) TAX (-1) PRICE2 (.LO,.L,.UP,.M = -INF, 0, +INF, 0) ---- P Price of goods faced by consumer P(1) (.LO,.L,.UP,.M = -INF, 0, +INF, 0) 30

31 (0) BUDGET (0) TAX 1 PRICE1 1 Price3 P(2) (.LO,.L,.UP,.M = -INF, 0, +INF, 0) (0) BUDGET (0) TAX 1 PRICE2-1 Price3 31

32 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/16/13 00:47:03 Page 4 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Model Statistics SOLVE UMAX2 Using NLP From line 57 MODEL STATISTICS BLOCKS OF EQUATIONS 6 SINGLE EQUATIONS 6 BLOCKS OF VARIABLES 4 SINGLE VARIABLES 7 NON ZERO ELEMENTS 19 NON LINEAR N-Z 14 DERIVATIVE POOL 10 CONSTANT POOL 17 CODE LENGTH 43 GENERATION TIME = SECONDS 4 Mb WEX Feb 14, 2013 EXECUTION TIME = SECONDS 4 Mb WEX Feb 14,

33 GAMS Rev 240 WEX-WEI x86_64/ms Windows 04/16/13 00:47:03 Page 5 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m Solution Report SOLVE UMAX2 Using NLP From line 57 S O L V E S U M M A R Y MODEL UMAX2 OBJECTIVE U TYPE NLP DIRECTION MAXIMIZE SOLVER CONOPT FROM LINE 57 **** SOLVER STATUS 1 Normal Completion **** MODEL STATUS 2 Locally Optimal **** OBJECTIVE VALUE RESOURCE USAGE, LIMIT ITERATION COUNT, LIMIT EVALUATION ERRORS 0 0 CONOPT 3 Feb 14, WEX WEI x86_64/ms Windows C O N O P T 3 version 3.15I Copyright (C) ARKI Consulting and Development A/S Bagsvaerdvej 246 A DK-2880 Bagsvaerd, Denmark The model has 7 variables and 6 constraints with 19 Jacobian elements, 14 of which are nonlinear. The Hessian of the Lagrangian has 4 elements on the diagonal, 7 elements below the diagonal, and 6 nonlinear variables. ** Warning ** The number of nonlinear derivatives equal to zero in the initial point is large (= 63 percent). A better initial point will probably help the optimization. ** Optimal solution. Reduced gradient less than tolerance. CONOPT time Total seconds of which: Function evaluations = 0.0% 1st Derivative evaluations = 4.5% 33

34 LOWER LEVEL UPPER MARGINAL ---- EQU UTILITY EQU BUDGET EQU TAX EQU PRICE EQU PRICE EQU Price3... EPS UTILITY Utility Function BUDGET Budget Constraint TAX Total Tax Revenue on two goods equals R PRICE1 Price of good 1 PRICE2 Price of good 2 Price3 Price equals LOWER LEVEL UPPER MARGINAL ---- VAR U -INF INF. U Utility level ---- VAR C Consumption levels LOWER LEVEL UPPER MARGINAL INF EPS INF VAR t tax rate LOWER LEVEL UPPER MARGINAL 1 -INF INF. 2 -INF INF VAR P Price of goods faced by consumer LOWER LEVEL UPPER MARGINAL 1 -INF INF. 2 -INF INF. 34

35 **** REPORT SUMMARY : 0 NONOPT 0 INFEASIBLE 0 UNBOUNDED 0 ERRORS EXECUTION TIME = SECONDS 2 Mb WEX Feb 14, 2013 USER: GAMS Development Corporation, Washington, DC G871201/0000CA-ANY Free Demo, , sales@gams.com, DC0000 **** FILE SUMMARY Input Output C:\Users\Wei Zhang\Documents\gamsdir\projdir\Untitled_4.gms C:\Users\Wei Zhang\Documents\gamsdir\projdir\Untitled_4.lst 35