Department of Eonomis Boston College Eonomis 2202 (Setion 05) Maroeonomi Theory Pratie Problem Set 3 Suggested Solutions Professor Sanjay Chugh Fall 2014 1. Interation of Consumption Tax and Wage Tax. One basi idea of President Bush s eonomi advisers throughout his administration had been to move the U.S. further away from a system of investment taxes (whih we will disuss later in the ourse) and more towards a system of onsumption taxes. Disussion of a federal onsumption tax, whih would essentially be a national sales tax, has again emerged in reent poliy disussions. Here, you will modify the basi onsumption-leisure framework to inlude both a proportional wage tax (whih we will now denote by t n, where, as before, 0 t n 1) as well as a proportional onsumption tax (whih we will denote by t, where 0 t 1). A proportional onsumption tax means that for every dollar on the prie tags of items the onsumer buys, the onsumer must pay (1 t ) dollars. Throughout the following, suppose that eonomi poliy has no effet on wages or pries (that is, the nominal wage W and the prie of onsumption P are onstant throughout). a. Construt the budget onstraint in this modified version of the onsumptionleisure model. Briefly explain eonomially how this budget onstraint differs from that in the standard onsumption-leisure model we have studied in lass. Solution: The representative agent s net inome from working is now given by Y (1 tn) W n, where t n is the labor tax rate and the other notation is the same as in Chapter 2. He spends all of this inome on onsumption, whih now osts P (1 t ) dollars per unit (inlusive of the onsumption tax). Using the fat that n 1 l in the weekly model, equating the representative agent s labor inome with his expenditures on onsumption gives us P (1 t ) (1 t ) W (1 l). If we multiply out the right-hand-side of this expression and then move the term involving the labor tax rate to the left-hand-side we obtain P (1 t ) (1 t ) W l (1 t ) W. n n Then, solving this last expression for, we arrive at n
(1 tn) W (1 tn) W l. (1 t ) P (1 t ) P This last expression an now readily be graphed with onsumption on the vertial axis and leisure on the horizontal axis. As in the standard model, the horizontal interept is l 1. However, the slope is now (1 tn) W (1 t ) P. Clearly, however, if we set the onsumption tax rate to zero, we reover the budget onstraint in our standard onsumption-leisure model indeed, the model we studied in Chapter 2 is simply a speial ase of the model here. The reason the budget onstraint differs here from the standard model is simple: the onsumption tax is yet another tax for the onsumer to take aount of when making his hoies about onsumption and leisure. No matter the model under onsideration, the budget onstraint always desribes all the relevant tradeoffs between two alternative use of resoures, and the relevant tradeoffs involve all taxes. b. Suppose urrently the federal wage tax rate is 20 perent ( tn 0.20 ) while the federal onsumption tax rate is 0 perent ( t 0 ), and that the Bush eonomi team is onsidering proposing lowering the wage tax rate to 15 perent. However, they wish to leave the representative agent s optimal hoie of onsumption and leisure unaffeted. Can they simultaneously inrease the onsumption tax rate from its urrent zero perent to ahieve this goal? If so, ompute the new assoiated onsumption tax rate, and explain the eonomi intuition. If not, explain mathematially as well as eonomially why not. Solution: From the analysis in part a above, we see that the slope of the budget onstraint depends on the relative tax (1 tn) /(1 t) (in addition to the term W / P, but you are told to assume that W and P remain onstant). Under the urrent tax poliy of a 20 perent wage tax and zero onsumption tax, the relative tax is (1 0.20) /(1 0) 0.80. So the slope of the representative agent s budget onstraint is urrently 0.80 W / P, on whih he makes some optimal hoie of onsumption and leisure. Now the government wants to lower the labor tax rate to tn 0.15 but wants to leave the representative agent s optimal hoie of onsumption and leisure unhanged. This means that whatever the government does, it must make sure that the slope of his budget onstraint does not hange whih means that the relative tax must remain 0.80. We an then solve for the new onsumption tax rate that yields this relative tax: (1 0.15) /(1 t ) 0.80 means that the government must set a onsumption tax rate of t 0.0625. The eonomi reasoning is that the relative tax has two free variables in it, 2
the labor tax and the onsumption tax. There are an infinite number of ombinations that yield any partiular value of the relative tax. Think of the following simple example: if you have two numbers x and y, and you are asked to ome up with a ombination of the two variables suh that x/ y 0.80, there are obviously an infinite number of ombinations that work.. A tax poliy is defined as a partiular ombination of tax rates. For example a labor tax rate of 20 perent ombined with a onsumption tax rate of zero perent is one partiular tax poliy. A labor tax rate of five perent ombined with a onsumption tax rate of 10 perent is a different tax poliy. Based on what you found in parts a and b above, address the following statement: a government an use many different tax poliies to indue the same level of onsumption by individuals. Solution: The statement is true, and it follows from the disussion given in part b above. If the government believes that W and P are unaffeted by its tax poliies (whih is not true we will address this issue soon), then it has two tax rates it an alter to ahieve its goals, but it is only the relative tax that affets the representative agent s budget onstraint. d. Consider again the Bush proposal to lower the wage tax rate from 20 perent to 15 perent. This time, however, poliy disussion is foused on trying to boost overall onsumption. Is it possible for this goal to be ahieved if the onsumption tax rate is raised from its urrent zero perent? Solution: We saw in the standard onsumption-leisure model that as the budget line beame steeper, onsumption inreases. This is still true in this version of the onsumption-leisure model. The urrent tax poliy has tn 0.20 and t 0 so that the relative tax is (1 0.20) /(1 0) 0.80. Any new tax poliy whih features a larger value of (1 tn) /(1 t) (and hene a steeper budget onstraint) will thus ahieve the desired goal of higher overall onsumption. With a labor tax rate of tn 0.15, we thus need Solving this inequality for t, we have that (1 0.15) 0.80. (1 ) t t ahieves the desired goal. So any tax poliy with 0.15 n 0.0625 t and t 0.0625 ahieves the desired poliy role. So the onlusion is: yes, the onsumption tax rate an be raised and the desired goal still be ahieved. 3
e. Using a Lagrangian, derive the onsumer s onsumption-leisure optimality ondition (for an arbitrary utility funtion) as a funtion of the real wage and the onsumption and labor tax rates. Solution: The Lagrangian is Ll (,, ) ul (,) (1 tn) W(1 l) P(1 t) The FOCs with respet to onsumption and leisure are (we ll ignore the one with respet to the multiplier beause in order to generate the onsumption-leisure optimality ondition, we atually don t need it): u(,) l P(1 t) 0 ul(,) l W(1 tn) 0 To generate the onsumption-leisure optimality ondition, we must ombine these two expressions by eliminating between them. Doing so, and expressing one side of the resulting expressing as the MRS between onsumption and leisure, we have ul(,) l (1 tn) W. u(,) l (1 t) P The left-hand side is the representative onsumer s MRS between onsumption and leisure, and the right-hand-side is the real wage rate (W/P) adjusted by both the labor and onsumption taxes. 2. Non-Bakward-Bending Labor Supply Curve. Consider an eonomy populated by 100 individuals who have idential preferenes over onsumption and leisure. In this eonomy, the aggregate labor supply urve is upward-sloping. For simpliity, suppose throughout this question that the labor tax rate is zero. a. For suh a labor supply urve, how does the substitution effet ompare with the inome effet? Solution: The upward-sloping region of any individual s labor supply urve arises beause the substitution effet of a higher real wage dominates the inome effet of a higher real wage. (See the disussion in Chapter 2.) Thus, if the individual s labor supply urve is always upward-sloping, then it must be that for this individual the substitution effet always outweighs the inome effet. With 100 idential individuals in the eonomy, the aggregate labor supply urve is simple the sum of eah individual s labor supply urve, and thus inherits the properties of the individuals labor supply urves. Extended Note: the labor supply urve annot literally be always upwardsloping. The upper-limit on the labor axis is of ourse (for the weekly model) 168 hours. One that upper limit is reahed (i.e., a person is doing nothing but working), any further rise in the real wage annot inrease hours worked hene the labor supply urve beomes vertial. But this latter effet should probably strike you as uninteresting beause then the individual does not enjoy any leisure at all. Indeed, if we have a usual indifferene map over onsumption and leisure, we will never have that an indifferene urve is tangent to the budget line 4
on either axis, a neessary impliation of an optimal hoie that has zero leisure (try drawing this to onvine yourself). b. Using indifferene urves and budget onstraints, show how suh a labor supply urve arises. Solution: We must have that any rise in the real wage leads to a higher optimal hoie of onsumption and a lower optimal hoie of leisure (with of ourse a natural zero lower bound on leisure see the Extended Note above), irrespetive of the urrent real wage. 168(W/P) 4 slope = -(W/P) 4 D 168(W/P) 3 168(W/P) 2 168(W/P) 1 slope = -(W/P) 3 slope = -(W/P) 2 C B A slope = -(W/P) 1 168 leisure In the above diagram, as the real wage rises from ( W / P ) 1 to ( W / P ) 2 to ( W / P ) 3 to ( W / P ) 4, the optimal hoie moves from point A to B to C to D, respetively. Clearly, as the real wage rises, the quantity of leisure demanded (and hene the quantity of labor supplied) rises, onsistent with a labor-supply urve that does not bend bakwards. 3. A Bakward-Bending Aggregate Labor Supply Curve? Despite our use of the bakward-bending labor supply urve as arising from the representative agent s preferenes, there is ontroversy in maroeonomis about whether this is a good representation. Speifially, even though a bakward-bending labor supply urve may be a good desription of a given individual s deisions, it does not immediately 5
follow that the representative agent s preferenes should also feature a bakwardbending labor supply urve. In this exerise you will unover for yourself this problem. For simpliity, assume that the labor tax rate is t 0 throughout all that follows. a. Suppose the eonomy is made up of five individuals, person A, person B, person C, person D, and person E, eah of whom has the labor supply shedule given below. Using the indiated wage rates, graph eah individual s labor supply urve as well as the aggregate labor supply urve. Solution: In the table below, the aggregate (total) number of hours worked by all persons in the eonomy at eah wage rate is now shown (this was not given to you). Nominal Person A Person B Person C Person D Person E Aggregate Wage, W $10 20 hours 0 hours 0 hours 0 hours 0 hours 20 hours $15 25 15 0 0 0 40 $20 30 22 8 0 0 60 $25 33 27 15 5 0 80 $30 35 30 20 15 0 100 $35 37 32 25 20 6 120 $40 36 31 27 25 21 140 $45 35 30 26 28 30 149 $50 33 29 24 25 29 140 The aggregate labor supply urve simply plots the values in the last olumn in the table above against the wage rate (with, reall, the labor tax rate held onstant at tn 0 throughout for simpliity), as shown below. Clearly, most of the aggregate labor supply urve is upward-sloping, with only the very top portion bakward-bending. For brevity, the individuals labor supply urves are omitted they are of ourse simply eah individual s hours worked plotted against the wage, and it should be lear even from the table that eah individual in the eonomy has a bakward-bending labor supply urve. 6
Now suppose that in this eonomy, the usual range of the nominal wage is between $10 and $45. b. Restriting attention to this range, is the aggregate labor supply urve bakward-bending? Solution: If the usual range of the nominal wage is $10-$45 in the eonomy, then learly no (see the Figure above), the aggregate labor supply is not bakwardbending.. At a theoretial level, if we want to use the representative-agent paradigm and restrit attention to this usual range of the wage, does a bakward-bending labor supply urve make sense? Solution: The point of the representative-agent framework is to represent theoretially the average person in the eonomy in all aspets of his eonomi life (in so far as suh theoretial modeling is possible ), inluding of ourse his labor supply deisions. The average person in the eonomy does not earn the highest wages in the eonomy. 7
d. Explain qualitatively the relationship you find between the individuals labor supply urves and the aggregate labor supply urve over the range $10 $45. Espeially address the bakward-bending nature of the urves. Solution: Over the range $10 - $45, the labor supply urves of person A, person B, and person C are bakward-bending, while the labor supply urves of person D and person E are not (notie that the labor supply urves of person D and person E do not bend bakwards until the range $45 - $50). The aggregate labor supply urve is always upward-sloping in this range of the wage. The fundamental issue here is that people are different from eah other in suh a way that the average person, over the range $10-$45, looks like only 2 of the 5 people in this eonomy (person D and person E). We ould easily onstrut another example in whih the representative agent s labor supply looked like well less than 40% of the population over some usual range of inome. This illustrates that miroeonomi phenomena (in this ase the bakward-bending labor supply urve) when summed together do not neessarily give qualitatively the same phenomena at the maroeonomi level a autionary note in using the representative-agent approah to maroeonomis. 8