Prolem Set #5 Solutions 4.4 Pulic Economics DUE: Dec 3, 200 Tax Distortions This question estalishes some asic mathematical ways for thinking aout taxation and its relationship to the marginal rate of sustitution etween goods. Consider individuals that have preferences u (c ; c 2 ; L) over two goods, c ; c 2, and leisure, L. Let p ; e the (efore-tax) prices of goods c and c 2. Let w e the (efore-tax) wage. The agent has an endowment of non-laor wealth of m. As usual, we assume that u is continuously di erentiale in (c ; c 2 ; L). Let u (c ; c 2 ; L) denote the marginal utility of c, u 2 (c ; c 2 ; L) denote the marginal utility of c 2, and u L (c ; c 2 ; L) denote the marginal utility of leisure.. Write the agent s udget constraint assuming no taxes and that the individual is endowed with unit of leisure so that l + L =, where l is the amount of time spent working for a wage. The udget constraint is given y any of the following expressions: p c + c 2 = m + wl p c + c 2 = m + w ( L) p c + c 2 + wl = m + w where m + w is the agent s "full income". 2. Derive the rst-order conditions for the agents maximization prolem y placing a lagrange multiplier,, on the udget constraint. Derive two equations that comined with the udget constraint characterize the solution to the maximization prolem, (c ; c 2; L). One condition should e for the marginal rate of sustitution etween c and c 2 and another for the MRS etween c and L. Explain the intuition of oth equations. [Note: you do not (and cannot in
general) solve for the solution explicitly; however, you should notice that you have 3 equations and 3 unknowns, so that given any well-speci ed function u one could solve the system of equations. Also, note that one could also write a condition for the MRS etween c 2 and L; however this would e redundant given the other two equations]. The lagrangian is given y u (c ; c 2 ; L) + [m + w p c c 2 wl] so that the FOCs are [c ] : u (c ; c 2 ; L) = p [c 2 ] : u 2 (c ; c 2 ; L) = [L] : u L (c ; c 2 ; L) = w so that the MRSs are given y and u (c ; c 2 ; L) u 2 (c ; c 2 ; L) = p u (c ; c 2 ; L) u L (c ; c 2 ; L) = p w The rst expression shows that the MRS etween c and c 2 is equated to the price ratio etween p and. Intuitively, the agent must e indi erent etween trading c 2 for c at price p. The second equation shows that the MRS etween c and L is equated to the price ratio etween p and the wage w. Intuitively, the agent must e indi erent etween consuming more leisure for one more unit of c at price p w. Now, suppose that the government levies a lump-sum tax of on all individuals, so that their net non-laor wealth is now m things for which individuals have no utility). (Assume for simplicity that this tax is used to nance 3. Set up the new udget constraint (assuming the agents never receive the money paid to the government and the money is used to nance things for which individuals have no utility). Derive the rst order conditions and provide the analogous two conditions for the marginal rates of sustitutions (for c vs c 2 and for c vs L). Explain the intuition of oth equations. Is the MRS distorted? (i.e. are they di erent than they would e in the asence of taxation?) Why or why not? The new udget constraint is now simply p c + c 2 + wl = m + w and the FOCs are the same as in part 2. The MRS is not distorted (always equals the price 2
ratio). Lump sum taxes provide no marginal incentive to consume one particular good over another particular good - it is simply a wealth e ect. Now, suppose that instead of the lump-sum tax, the government institutes a tax on c of, so that individuals must now pay p + per unit of c. 4. Set up the new udget constraint (assuming the agents never receive the money paid to the government and the money is used to nance things for which individuals have no utility). Derive the rst order conditions and provide the analogous two conditions for the marginal rates of sustitutions (for c vs c 2 and for c vs L). Explain the intuition of oth equations. Is the MRS distorted? (i.e. are they di erent than they would e in the asence of taxation?) Why or why not? The new udget constraint is given y (p + ) c + c 2 + wl = m + w and the FOCs are so that the MRSs are and [c ] : u (c ; c 2 ; L) = (p + ) [c 2 ] : u 2 (c ; c 2 ; L) = [L] : u L (c ; c 2 ; L) = w u (c ; c 2 ; L) u 2 (c ; c 2 ; L) = p + u (c ; c 2 ; L) u L (c ; c 2 ; L) = p + w The tax on c causes the MRS etween c and c 2 and etween c and w to e higher than it would otherwise e. Now, the agent needs to e willing to trade c 2 for c at relative price p +, instead of just p. Similarly, the agent needs to e willing to trade leisure L for c at relative price p, instead of just p. Thus the MRS is distorted - the agent adjusts her w w allocation so that her willingness to pay for c (MRS in terms of other goods) rises to equate to the after-tax price, p +. Now, suppose that, in addition to the tax on c, the government institutes a tax on c 2 of 2 and on laor earnings, w (so that the after-tax wage is w w ). 5. Set up the new udget constraint assuming the agents never receive the money paid to the government. Derive the rst order conditions and provide the analogous two conditions for the marginal rates of sustitutions (for c vs c 2 and for c vs L). Explain the intuition of oth equations. Is the MRS distorted? Why or why not? 3
The new udget constraint is given y (p + ) c + ( + 2 ) c 2 = m + (w w ) l (p + ) c + ( + 2 ) c 2 + (w w ) L = m + w w Note that a tax on laor is equivalent to a susidy on leisure. The FOCs are given y so that the MRSs are given y and [c ] : u (c ; c 2 ; L) = (p + ) [c 2 ] : u 2 (c ; c 2 ; L) = ( + 2 ) [L] : u L (c ; c 2 ; L) = (w w ) u (c ; c 2 ; L) u 2 (c ; c 2 ; L) = p + + 2 u (c ; c 2 ; L) u L (c ; c 2 ; L) = p + w Now, the tax on c, c 2, and l causes the MRS to e equated to the after-tax price ratios. The MRS etween c and c 2 is not distorted if and only if =p = 2 =. Likewise, the MRS etween c and L is not distorted if and only if =p = w! =w. Intuitively, if the tax rates across the goods are not the same, the taxes will induce a distortion in the MRS (i.e. the willingness to trade one good for the other). Now, consider a conceptually di erent, ut mathematically similar economy. Suppose there are two time periods, and 2. In the rst time period, agents can consume and work. In the second time period, agents can only consume. Agents are endowed with non-laor wealth of m in the rst period. Agents can save and/or orrow at a gross interest rate of R so that savings of s in period yield Rs in period 2. Assume that the price of consumption in oth periods is (in terms of money within the period). Also, assume the wage is equal to, w =. Agents utility functions are, as efore, given y u (c ; c 2 ; L). 6. Denote the agents net savings/orrowing position in the rst period y s. Write the agents two udget constraints, one for each time period. Then, comine these udget constraints into a single udget constraint over c, c 2, and L. Show how this economy relates to the economy descried in parts and 2 y providing prices for p, ; and w for the general economy (parts & 2) that make this economy mathematically equivalent (you should normalize p = ). 4
The udget constraint in period is given y c + s = m + l c + s = m + ( L) c + s + L = m + and in period 2 is given y c 2 = Rs so that the comined udget constraint is given y c + c 2 R + L = m + so that if we had p =, =, and w =, we are equivalent to the general economy R descried earlier. 7. Suppose the government institutes a tax on savings of (and no other taxes). What is the agent s marginal rates of sustitution etween c and c 2 and etween L and c? Explain the intuition of oth equations. Is the MRS distorted? (i.e. are they di erent than they would e in the asence of taxation?) Why or why not? [Note: no derivations should e necessary - just apply the results from parts -4] With the tax on savings, we have translated to = ( are given y ) instead of so that the MRSs u (c ; c 2 ; L) u 2 (c ; c 2 ; L) u (c ; c 2 ; L) u L (c ; c 2 ; L) = p = R ( ) = p w = So that the MRS is distorted etween consumption in period and consumption in period 2. The tax on savings makes individuals less likely to save: their willingness to trade etween period and 2 is now R ( ) instead of R. But, the MRS is not distorted etween c and L. 8. Suppose the government institutes a tax on laor earnings of (and no other taxes). Solve for the agent s marginal rate of sustitution etween c and c 2, and etween L and c. Explain the intuition of oth equations. Is the MRS distorted? Why or why not? [Note: no derivations should e necessary - just apply the results from parts -4] 5
With the tax on laor earnings of, the MRSs are given y u (c ; c 2 ; L) u 2 (c ; c 2 ; L) u (c ; c 2 ; L) u L (c ; c 2 ; L) = p = R = p w = so that there is no distortion etween the consumption of c and c 2, ut there is a distortion etween c and L, since leisure L is eing "susidized" (ecause its perfect complement, laor supply, is eing taxed). 2 Firm Taxation Consider an economy populated y a set of individuals who each supply laor, l, and capital, k, to a competitive market of rms at efore-tax prices w and r. Each rm has an identical production function F (k; l) = k l. For the rst part of the prolem, assume that the capital stock is supplied inelastically, so that k = for any set of prices. Also, assume laor is supplied according to a supply function l (w) = w where > 0.. Write the rms maximization prolem and solve for the demand for l as a function of w and, l w;. Solve for the equilirium laor quantity, l, the equilirium wage w, and the competitive price of capital, r, that equates demand with the inelastic supply at. Firms maximize pro ts = k l wl rk where r is the gross cost of capital and w is the wage. The maximization prolem yields k [l] : ( ) = w l l [k] : = r k since capital is supplied inelastically, we have k =, so that l w; = ( ) (w) Now, in equilirium we will have laor demand equal to laor supply so that l w ; = w 6
or ( (w ) ) = w ( ) = w (w ) ( ) = (w ) ( ( ) ) ( )!! = w = w = w so now we can solve for the equilirium quantity of laor l = ( ) l = ( ) and we can solve for the interest rate, r, using the marginal product of capital equation, l = r! ( ) = r ( ) ( ) = r ( ) ( ) = r so that we have l = ( ) w = ( ) r = ( ) ( ) 2. Suppose now the government institutes a laor tax of which requires agents to pay a tax to the government so that their after-tax wage is ~w = w ( ). Discuss graphically and 7
mathematically what happens to the agent s laor supply function (as a function of the pretax wage) as a result of the tax. Agents now only receive w ( ) instead of w. The agent will therefore supply l = w ( ) < w for any pre-tax wage of w. Graphically, we have w Post tax Pre tax l 3. Solve for the new equilirium allocation of laor, l, the equilirium efore-tax wage w, the equilirium after-tax wage ~w, and the competitive price of capital r. How do these relate to your solution in part? Why? In equilirium, laor supply equals laor demand, so that is the efore-tax wage, and ( ) l w ; = w ( ) ( ) (w ) ( ) ( ) ( ) ~w = ( ) w = ( ) ( ) = ( ) ( ) = w ( ) = (w ) a = w ( ) is the after-tax wage. Notice that the efore tax wage is higher than part, while the after tax wage is lower than in part. The tax makes laor more expensive, so rms have to e more willing to pay for laor (thus the efore-tax wage goes up). But, the rise in productivity is less than one-for-one; on net, the after tax wage drops as a result of the tax. 8
The new allocation of laor is given y l = ~w = ( ) ( ) = ( ( )) ( ) which is less than efore (since laor is taxed). Finally, the competitive price of capital solves l r = r = ( ( )) ( )! ( ) ( ) r = so that r is lower than in part. The tax on laor reduces the amount of laor supplied in equilirium. This reduces the marginal product of capital (since capital and laor are complements in our production function), which therey lowers the return on capital. 4. What is the net change in gross laor earnings (i.e. pre-tax wage x laor)? What is the net change in total capital earnings (interest rate * capital)? If capital earnings changed, explain why. The net change in gross laor earnings is E = l taxw tax l w = ( ( )) ( ) ( ( )) ( ) ( ) i = ( ) ( ) so that gross laor earnings fall. h( ) + < 0 The net change in total capital earnings is ( ) ( ) C = ( ) = ( ) ( ) < 0 Capital earnings fall ecause the tax on laor reduces the marginal product of capital. Since capital supply is inelastic, this reduction in the marginal product of capital in these rms leads 9
to a lower payment to owners of capital. Intuitively, since capital is supplied inelastically, it ears some of the urden of the tax on laor. Now, suppose that instead of a tax on laor, the government institutes a tax on capital, so that individuals return on capital is given y ~r = r ( ); for every r that owners of capital receive from the rm, they must pay r to the government. 5. Solve for the equilirium l, the equilirium wage w, the equilirium after-tax return on capital r. Without solving for the deadweight loss, is the tax on capital more or less e cient than the tax on laor? [Note: you are not required to make any unnecessary/duplicate calculations if you don t need to]. The equilirium is the same as in part & 2. The only di erence is that now the gross interest rate is r, while the after-tax interest rate is ~r = r ( + ). Since the tax introduces no distortions, it is more e cient than the laor tax. Now, suppose that capital is no longer supplied inelastically. Rather, let s make the polar opposite assumption. Let s assume that rms have access to an in nite amount of capital at a world price of ^r. 6. Without doing any math, discuss the impact of imposing a capital tax. How much of the tax would e paid y capital owners? How much y laor owners? Since supply of capital is perfectly elastic, the owners of capital will always receive an after-tax return of ^r. Therefore, their earnings will not change in equilirium (although they will invest less in the rms, they will invest more in their outside option which provides a return of ^r). Laor owners will ear the full cost of the tax. 7. Again without doing any math, discuss the impact of imposing a laor tax. How much of the tax would e paid y capital owners? How much y laor owners? Why? Again, since the supply of capital is inelastic, the owners of capital will not pay any of the tax. The laor owners will pay all of the cost of the laor tax. 3 Empirical Evaluation Barack Oama is ack from his trip overseas and is considering a change in the tax code to help reduce the de cit. He liked your advice from prolem set 3 and decided to give you a call ack. He asks you a couple of questions aout what would happen under various changes to the tax code. For each question, he asks you to do two things: Discuss rie y what economic theory predicts and what existing studies may have shown on this question. How con dent are we in these predictions/results? 0
Discuss a potential empirical method that would allow you to answer my question if you had access to any reasonale amount of data that could potentially e required. Discuss the potential limitations of your approach. He asks you the following questions:. "I m considering increasing the tax on savings (interest income), ut am worried that this might decrease the amount of savings. How much would this tax increase reduce savings?" A tax on interest income theoretically reduces savings, however theory does not predict the precise magnitude of the reduction in savings. The ideal empirical approach would randomly suject individuals to di erential interest taxation and see if they have a reduction in savings. However, the downside of this approach is that it requires randomly varying tax rates (which violates horizontal equity!). An alternative approach would e if some states had di erent tax rates on income taxation interest rate taxes and one could do a di erence-in-di erence around the times at which the policy changed. The downside here is that the states would need to have parallel trends in savings. Also, a downside to many approaches is that it s not clear whether we ll pick up the short-term or long-term response to taxation. In the short run, individuals may nd it di cult to adjust their savings holdings which are suject to taxation; ut over the long run they might adjust their savings more than we see initially. 2. "I m considering reducing EITC credits to the poor, ut am worried this might decrease their laor supply. To what extent does changing the level of EITC ene ts a ect laor supply?" Existing studies suggest that the EITC has very large e ects on laor supply, and the results are quite roust. Also, theory predicts that providing a susidy to laor will increase laor supply. Therefore, theory and existing studies suggest that reducing these credits would reduce their laor supply. A potential empirical approach to estimate the magnitude of this e ect would e to do an analysis of the laor supply efore and after of individuals sujected to the EITC as compared to individuals not sujected ot the EITC (i.e. do a di erence-in-di erence around the time of introduction of EITC etween those who are poor enough to qualify relative to those who are just aove the quali cation threshold). A potential downside of this approach is that those who are just aove the quali cation threshold may not e on parallel trends relative to those who qualify. 3. "I m considering raising the highest tax racket y from 35% to 40% to help alance the udget. But, I m worried that if I raise the tax rate this will decrease the amount of income President Oama explicitly mentions that some of his advice he received last time failed to mention the potential limitations/quali cations of the results. So he reminds you to include a rief discussion of the potential limitations of your proposed approach.
that people report (either through illegal evasion, or ecause the rich choose to work less in the face of higher taxation). How much will this increase lead to a decline in taxale income amongst the rich?" There has een sustantial work on the so-called "taxale income elasticity", and in general the evidence suggests that there is some reduction in reported taxale income, ut there is no real consensus (FYI: Martin Feldstein argues that this magnitude is huge, while Austan Goolsee has argued that it s large initially ut not in the long-run, since the rich individuals just change the timing of when they accrue their capital gains earnings - which iases the estimates of Feldstein). An ideal empirical approach would randomly assign tax rates to individuals and analyze how their reported taxale income varies as a function of the tax rate. However, this is not feasile for a variety of ethical and legal reasons. A potentially more realistic empirical approach would e to analyze a di erence in di erence in reported taxale income around the time when the tax rackets are changed. A time series estimator could e used (this is what feldstein uses), ut this is sujected to the signi cant prolem that individuals may know when their tax rate will change. If someone knows that they have a 35% tax today and a 40% tax tomorrow, they might shift some of the pro tale activity from tomorrow to today. therefore, the estimated quantity may e more of a short-term magnitude than the true long-run response. To counter this, one could look at longer time di erences from the time of the tax change (e.g. compare 2 years efore vs 2 years after). Or, one could analyze unexpected tax changes (which is hypothetically a possiility, ut may not e actually feasile). 4. "I m considering raising the ene ts provided to poor single mothers with children. But, how much will this increase the numer of single mothers?" As Prof. Gruer mentioned in class, current work on this is somewhat inconclusive, ut largely suggests that there s not a huge e ect of welfare ene ts on the numer of single mothers. Theory predicts that raising ene ts to single mothers would at least somewhat increase the numer of single mothers, ut does not predict a magnitude for this e ect. One potential empirical approach, among many, is to analyze past increases in ene ts provided to poor single mothers around some income threshold. If some states increased ene ts, while others did not, we could do a di erence-in-di erence estimation with the states that don t change their ene ts as control groups. A potential downside to this approach is that single mothers may move to states with greater ene ts, which would make it look like there was a ig e ect of the policy (since single mothers would move from the control to treatment group). Augmenting the study with data on migration patterns of poor single women could help rule out this possiility. 2
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