5. Government spending in the one period economy

Transcription

1 5. Government spending in the one period economy Index: 5. Government spending in the one period economy Introduction Perfect competition and lump-sum taxes...3 Main assumptions...3 Competitive equilibrium...4 Multipliers...5 Graphical illustration Distortionary taxes...8 Model description and equilibrium...8 Multipliers...9 Graphical illustration...9 Laffer curve The case with imperfect competition...13 Model setup...13 Graphical illustration...13 Multipliers Coordination failure Optimal provision...22 The planner optimum...23 Competitive equilibrium, lump-sum taxes...23 Distortionary tax...24 Imperfect competition...25 Further reading...26 Appendix 2 - Optimal provision of a congestible public input...27 Review questions and exercises...30 Review questions...30 Exercises

2 5.1. Introduction In this note, we add the government sector to the one-period economy. The main reason underlying the role of government in the economy is the existence of market failures. Market failures include public goods, externalities, information failures, imperfect competition, missing markets and so on. In what follows, we focus on the level of government intervention, without entering too much in details regarding why intervention takes place. Our economy consists in a large number of equal households and firms, plus a government that purchases the amount G of a public good from firms, and finances these purchases with a tax on households. There is no financial market. Hence, all agents in this economy are bound to spend their full disposable income. Since we are not allowing time to be divided into sub-periods, the model can only be used to analyse once-an-for-all changes in the exogenous parameters. In this note, we analyse the implications of government spending under alternative assumptions. In Section 5.2, we start out with the neoclassical benchmark, assuming perfect competition and lump-sum taxation. In that case, a fiscal expansion impacts positively on output because households optimally decide to work more. Section 5.3, we analyse the case in which taxes are proportional to income, giving rise to a distortion in the consumption-leisure decision. In this framework we take opportunity to discuss the relationship between the tax rate and the tax base, that underlies the fiscal Laffer curve. In Section 5.4, we address the case with imperfect competition. In that case, a fiscal expansion, gives rise to a multiplier effect, through which expenditure increase profits and profits increase expenditure. In Section 5.5 the model with imperfect competition is extended to account for the existence of fixed costs in production. The novelty in this case is that low demand may cause the economy to be stuck in a low income trap, while a superior equilibrium was also feasible. In this model, an increase in government spending may rescue the economy from the bad equilibrium where it is trapped. Finally, in Section 5.6 we investigate how the optimal provision of the public good may change, depending on the market structure and on the availability of lump-sum taxation. 2

3 5.2. Perfect competition and lump-sum taxes Main assumptions Consider an economy with a large number of equal firms and households. In this economy, the government buys G units of output from firms, and finances these purchases coercing citizens to pay a lump-sum tax T. Prices and wages are flexible. The preferences of the representative consumer are given by 1 lnc 1 lnl lng U (5.1) Where C denotes for Private consumption, l for leisure and G for a nonexcludable good. By a non-excludable good it is meant a good that, once made available to one agent, it will be impossible to preclude any other agent from consuming it. The implication is that no agent in the economy will be willing to pay for that good: all agents will attempt to free ride on any eventual provision, and no revenues would be voluntarily raised to make the provision profitable. The government can fix the problem, because it has the power to coerce taxes. For the moment, we abstract from the optimal provision of the public good. Hence, G is assumed exogenous. In Section 5.6 we turn to the issue of optimum provision. Households are endowed with h units of time. Time can be allocated to work, N, or to leisure, l: h N l (5.2) For simplicity, let s assume a Ricardian production function: Q zn (5.3) Because the economy is closed, equilibrium in the output market requires production to be equal to domestic demand. The later consists in private consumption, C, and government expenditures. Thus, Q C G (5.4) Using (5.2), (5.3), and (5.4), the production possibilities frontier of this economy becomes: 3

4 h l G C z (5.5) The government finances its purchases levying a lump-sum tax T on consumers. The government budget must balance, that is: T G (5.6) Firms hire working time from households at the wage rate w and sell ouput (numeraire). The representative firm real profits are: Q wn. (5.7) Figure 1: the flow income chart with the government sector The household income consists in labour revenues and profits. Given the lump-sum tax, T, the household budget constraint becomes: l T C w h, (5.8). Or C wl Y T, with Y T wh T (5.9) In (5.9), Y denotes for gross income: the notional amount corresponding to all potential income, given the time endowment. The flow income chart of this economy is as described in Figure 1. Competitive equilibrium The household maximizes utility taking T and w as given, subject to the budget constraint (5.8). From the first order condition, one obtains: c 1 w (5.10) l 4

5 This condition states that the MRS shall be equal to the wage rate. Replacing (5.10) in the budget constraint, (5.8), one obtains the demand functions: Y T wl 1 (5.11) C Y T (5.12) The implied labour supply is: N S T h l h 1 (5.13) w Firms take the wage rate as given and choose N so as maximize profits, (5.7). This leads to a demand for labour of the form: w z (5.14) Implying zero profits: 0 (5.15) Since profits are zero, the equilibrium level of disposable income is Y T zh T. (5.16) Conditions (5.14) and (5.10) imply that MRT=MRS. This is an important implication of assuming that lump-sum taxes are available: the equilibrium in this economy is Pareto efficient. is: Given the consumption function (5.12), the equilibrium in the output market hz T G Q (5.17) Using the balanced budget constraint, (5.6), this gives 1 : Q zh 1 G (5.18) Multipliers 1 We invite the reader to show that the same level of output is obtained starting with the labour market equilibrium condition. 5

6 From (5.18), one may assess the impact of a balanced budget increase in government spending: Q G dgdt 1 0 (5.19) That is, a fiscal expansion increases output, with a multiplier that is less than one. Why is this? First note that a fiscal expansion financed with an increase in taxes implies a reduction in households disposable income, (5.16) on a one-to-one basis: Y T G dgdt 1 (5.20) With less disposable income, the household decides to consume less leisure (eq. 5.11) by the amount: l G dgdt 1 z Since N h l, labour supply increases by exactly the quantity needed do match the change in output, (5.19). Given the fall in disposable income, private consumption declines by a (equation 5.12). Hence, there is a crowding out, whereby the increase in government expenditures reduces private consumption: C G dgdt 1 (5.21) Note however that this crowding out is less than one: the reason is that output increases, because households demand less leisure. Summing up, although an expansionary fiscal policy is successful in boosting output and employment, the household disposable income declines, implying a lower demand for leisure and for consumption. Whether the household is better off or worse off, it depends on how valuable the additional public provision is, as compared with the sacrifices in terms of consumption and leisure. The fact that an expansion in government expenditures comes along with higher output and employment suggests an avenue to explain business cycles. 6

7 However, the model also predicts that private consumption declines when government expenditures increase: since in reality private consumption tends to be pro-cyclical, this model does not offer a good argument for the idea that business cycles are mainly caused by fluctuations in government expenditures. As a long term model, however, the implication that higher government expenditures expand the economy and crowd out private consumption makes perfect sense. Graphical illustration Figure 2 compares the equilibrium in our economy when T=G=0 with a case in which G=T>0. First, consider the case with no government. In this case, the household budget constraint (5.8) is defined by the segment h0 in the figure (remember that profits are zero). Given the preferences and budget constraint, the household optimal plan corresponds to point 0 2. Now consider the case with G=T>0. Because of the lump-sum tax, the household budget constraint (5.8) shifts down vertically. The household optimal consumption plan moves to point 1. Again, the PPF (5.5) mimics the household budget constraint, with the difference that we shall replace T for G. Figure 2: Government expenditures and consumer choices 2 Remember that the household budget constraint (5.8) takes exactly the same form as the economy PPF, (5.5). Hence, a benevolent planner wishing to maximize the household utility would choose exactly point 0. This is another way of saying that the competitive equilibrium is efficient. 7

8 5.3. Distortionary taxes We now examine the case where taxes are proportional to market income. Since we are assuming perfect competition, profits will be zero. Hence, the only taxable income is labour income. Model description and equilibrium With the income tax, the government budget constraint becomes: G Q zn (5.6a) Because of the income tax, the household budget constraint becomes wh l C 1 (5.8a). The first order conditions of utility maximization subject to the budget constraint (5.8a) imply: c 1 w1 l (5.10a) The firm problem does not change relative to the earlier case: the wage rate shall be equal to productivity (5.14) and profits are zero (5.15). This, together with (5.8a) and (5.9a) deliver the following demand functions: l 1 h (5.11a) C hz 1 (5.12a) 8

9 The implied labour supply is N S h (5.13a) In contrast to the case with lump-sum taxes, the labour supply is now vertical (invariant with the wage rate or with the tax rate) 3. The output market equilibrium condition requires, as before, that supply equals aggregate demand: Q hz1 G (5.17a) Using the balanced budget constraint, (5.6a), the equilibrium level of output becomes: Q zn zh (5.18a) Since the employment level is invariant with the tax rate, total output is also invariant with the tax rate. The other side of the coin is that the increase in government expenditures fully crowds out private consumption. Multipliers Because the labour supply is inelastic, output and employment are invariant with government expenditures. That is: Q G dgdt 0 (5.19a) The fact that output and employment do not change with government expenditures implies that private consumption is fully crowded out: C G dgdt 1 (5.21a) Graphical illustration 3 Remember that, when preferences are of the form (5.1), in the absence of exogenous income, the substitution effect and the income effect exactly cancel out. 9

10 Comparing (5.8a) with (5.5), and taking into account (5.14) and (5.15), we see that the household budget constraint no longer corresponds to the economy production possibility frontier. This fact is illustrated in Figure 3. In the Figure, the PPF is described by the line crossing points 1 and 2. This constraint is determined by the time endowment, technology and the level of government expenditures, G. Since the PPF does not depend on how government expenditures are financed, it is the same as with lumpsum taxation or with a central planner. The household budget constraint is defined by the points h and 2. The slope (5.8a) reflects the fact that taxes are proportional to labour income. The optimal household choice with distortionary taxation is described by point 2. Note that this is the only feasible point: it has to satisfy simultaneously the household budget constraint and lie at the PPF. Comparing with point 1, we see that the household decided to work less and to consume less, because leisure is now cheaper. Because in point 2 the slope of the budget constraint (5.10a) is less than the slope of the PPF ( w z1 ), this point is not Pareto efficient. Figure 3: Lump-sum versus distortionary taxation Laffer curve 10

11 The key implication of income taxation is that it reduces the relative price of leisure, distorting the consumption-leisure decisions. In case the labour supply is positively sloped it may happen that an increase in the tax rate, by inducing a fall in wages and thereby in employment and in output, ends up delivering a lower tax revenue. This is the argument that underlies the Laffer curve. To see this in terms of our model, let s assume that the proceeds of income taxes, instead of spent in a public good, are transferred back to households, in the scope of a social protection scheme. That is, the government budget constraint becomes: I Q zn (5.6b) Where I refers to the income transfer. This income transfer is a fixed amount, not depending on household choices. In this case, the household budget constraint becomes: C h lw I 1 (5.8b). Utility maximization in this case, delivers a labour supply of the form: N S 1 h 1 I w (5.13b) Note that this labour supply is positively sloped. This equation also reveals that the labour supply decreases monotonically with the level of government transfers. The impact of government transfers on the labour supply is twofold: on one hand, by shifting the budget constraint (5.8b) upwards, government transfers induce households to demand more leisure through an income effect. On the other hand, because higher transfers come along with higher income taxes, there is a fall in the after-tax wage rate: the lower slope of the budget constraint gives rise to a substitution effect through which the household switches away from consumption to leisure. The two effects reinforce each other in increasing the demand for leisure. Given (5.3), (5.13b) and (5.14), the output supply function becomes: 1 Q zh I 1 Now, we can substitute this in (5.6b) and solve for I, obtaining: 11

12 I 2 1 zh 1 This equation reveals that the proceeds raised with taxes to finance a transfer program are a non-monotonic function of the tax rate. This relationship is labelled as the Laffer Curve, and is described in Figure 4. Figure 4 The Laffer Curve To understand the slope of the Laffer curve, note that the increase in the tax rate has two conflicting effects on fiscal revenues: on one hand, the increase in the tax rate expands the government revenue per unit of output; on the other hand, the increase in the tax rate decreases the level of output, shrinking the tax base. According to Figure 4, when the tax rate is low, the first effects dominates; as the tax rate gets too high, the distortinary effect of taxation dominates, shrinking the tax base by an amount that more than offsets the increase in the tax rate. Thus, if the economy is on the wrong side of the Laffer curve, a decrease in the tax rate will deliver a higher revenue. The model also implies that there is tax rate that maximizes the government revenues. Note however that there is no particular reason for any government to maximize the tax revenues. A limitation in this simple model is that taxes are paid by the same individual that is beneficiary of the transfer. The reason is that we are using a representative consumer. In real life, taxes are paid by some households and transfers are paid to 12

13 others. Since these programs intend to reduce inequality, and inequality is something that societies dislike, there is scope for a well designed social protection scheme to be welfare improving, even though there is an efficiency loss The case with imperfect competition Model setup To analyze this case, let s assume that the economy is populated by a large number of imperfect competitive firms, each one producing a differentiated product. We are still assuming flexible prices and a competitive labour market. Because the number of firms is large, each firm ignores its strategic interactions with others firms. It is assumed that the number of differentiated products is fixed: there is no free entry, so monopoly profits do not erode over time. The fact that each firm is price-maker gives rise to a wedge between prices and marginal costs. Taking the consumption good as numeraire (P=1), the mark-up will be as follows 4 : 1 w z 1 Solving for the wage rate, we get: w z 1 (5.14b) Replacing this in the expression for profits, (5.7), one obtains: N Q Q z 1 (5.15b) Together, equations (5.14b) and (5.15b) reveal that imperfect competition is equivalent to a transfer from wage earners to profit earners. Graphical illustration 4 To avoid complications, the mark-up is assumed exogenous. The underlying assumption is that there is as competitive fringe that does not allow the monopolist to fully explore its monopoly power. 13

14 The implication of a wage rate that is lower than in the case with perfect competition is that the household budget constraint is now less sloped. Figure 5 illustrates this, for the case with no taxes (T=0). Point A describes the optimal consumer basket in the case with perfect competition, as already described in Figure 2. Figure 5: Implication of imperfect competition on relative prices and consumer choices Point B describes the case with imperfect competition. In the figure, the implied income transfer from wages to profits that comes along with imperfect competition translates into a budget constraint that moves upwards vertically (higher profits) and then slopes down (lower wages). The new optimal basket, B, is simultaneously determined by the income expansion path corresponding to the new relative price and the production possibilities frontier (5.5): remember that, irrespectively of how the relative price shows up to consumers, the production plan has to be feasible. Because imperfect competition implies lower wages than in the well functioning case, the demand for leisure in B is higher than in A. Hence, people work less and consume less than in point A. In a word, the distortion in the output market comes along with a lower employment level. 14

15 Based on figure 3, it is easy to guess that lump-sum taxes, by shifting the budget constraint vertically downwards, impact more in the demand for leisure in the imperfect competitive case than in the case with perfect competition: visually, note that the income expansion path in the later case is stepper. This property is very important to understand why a given fiscal expansion impacts more on output and on employment under imperfect competition than in the well functioning case. Multipliers To quantify the impact of an increase in government expenditures, let s consider again the condition describing the equilibrium in the output market: Q C G That is: Y T G z h G T Q 1 (5.17b) The novelty with imperfect competition is that profits are not zero. Hence, whenever aggregate demand increases, profits increase, which in turn translates into more income and more demand. The multiplier effect is summarised by equations (5.15b) and (5.17b) and described in Figure 6. In the figure, a balanced budget fiscal expansion causes an upward shift in the expenditure line, giving rise to higher profits and by then higher private expenditure and so on, until the new equilibrium is reached. Solving together (5.15b) and (5.17b), one obtains the equilibrium output/expenditure in this model: 1 Q wh G T 1 (5.18b) Figure 6 - The income multiplier effect 15

16 Thus, when taxes and government expenditures increase by an equal amount, the impact on output will be: Q G dgdt (5.19b) Note that when there is no mark-up, 0, the multiplier is the one obtained in the model with perfect competition, (5.19). As the mark-up increases away from zero, the fiscal multiplier also increases, approaching the limit value of 1. The intuition is as follows: imperfect competition implies a lower relative price of leisure. Hence, any given fall in net income Y-T will produce a higher contraction in the demand for leisure (higher expansion in the labour supply) than in the case with perfect competition. Because the impact on employment is higher, the crowding out of private consumption is mitigated. To see the impact on consumption, let s first compute the equilibrium level of disposable income: Y T wh T 1 1 z1 h G T T Which simplifies to 1 Y T zh G 1 Hence, 16

17 Y T G dgdt (5.20b) When 0, the later equation simplifies to (5.20). With a positive mark-up, however, the fall in disposable income (and the crowding out effect on private consumption) is lower than in the well functioning case. The impact on consumption will be C G dgdt (5.21b) Again, whether people are better off or worse off after the increase in government spending, it depends on how people value government spending (utility function, 5.1). Summing up, even though in this model there is a market failure arising from the fact that prices are above marginal costs, increasing government expenditures will in general crowd out private consumption. The reason is that we are imposing a balanced budget constraint: if one could expand government expenditures without rising taxes, then the income multiplier would be much higher. The problem, however, is that in an economy without financial markets there is no way for the government to run a fiscal deficit Coordination failure In this section, we explore the case where imperfect competition comes along with a fixed cost. In the presence of fixed costs, firms need a minimum amount of revenue to break even. Thus, the possibility exists of aggregate demand being insufficient for firms to cope with the fixed costs. To model this case, assume that the profit of each individual firm takes the following form: Q F, (5.7c) where F stands for a fixed cost. When profits are positive, the firm will remain in operations; when negative, the firm will shut down. In the following, it is assumed 17

18 that when a monopolist shuts down, production in that sector shifts to an informal sector (a competitive fringe, with a Ricardian production function and no fixed costs) 5. A key assumption in this model is that the amount F paid by each firm is equally spent across all firms in the economy. This generates a pecuniary externality to firms that break even to other firms, helping them to break even too. The possibility of firms shutting down implies that the households income, Y, depends on the number of firms which survive. To capture this, let s denote for the share of firms that break even: when s=0, there are no firms in the economy (all production is informal). When s=1, all sectors are operating under imperfect competition. The household disposable income as a function of s is: Y T wh s T (5.9c) Substituting (5.9c) in the demand for consumption (5.12), and taking into account that the final demand for products includes the fixed costs, the equilibrium condition in the product market becomes: z h s T G sf Q C G sf 1 (5.17c) Replacing this expression in the profit function (5.7c), and solving for profits, one obtains: s s hz1 sf G T (5.15c) Substituting in the disposable income equation, (5.9c), one obtains Y T 1 1 s z1 h s1 sf sg T T (5.16c) To interpret this, consider first the case with s=1. In that case, a balanced budget fiscal expansion will imply: 5 Without loss of generality, one could assume that the technology in the competitive fringe was q=n, implying w=1 (see Exercise 5.9 below). 18

19 Y T G dgdt s (5.20c) This is exactly the same we obtained in (5.20b). This reveals that, in case all firms remain in operation, the presence of fixed costs does not alter the magnitude of the fiscal multiplier. However, as we will see in a minute, the impact of fiscal policy may be quite dramatic when s is less than one. The key relationship in the model is equation (5.15c). This equation says that profits are a positive function of the number of surviving firms, s. Thus, the larger the number of surviving firms, the higher the demand for each individual firm (through F) and hence the higher the profits of each individual firm. In contrast, when s is too small, the possibility exists of a firm not breaking even. This, in turn, will imply a lower s and a lower demand for the remaining firms. Since in this model all firms are equal, there are only two possibilities: (a) all firms are profitable and remain in operations; (b) all firms are loss-making and remain closed. Hence, only two cases are candidates for equilibrium, s=0 and s=1 6. Whether both possibilities materialize or not, it depends on the specific parameter values. requires: The case with s=1 will be an equilibrium if 1 0. From (5.15c), this hw 1 G T F (5.22) requires: The case with s=0 will be an equilibrium if 0 0. From (5.15c), this F hw G T (5.23) 6 The assumption that firms are equal is obviously a simplifying one, but the student can easily guess what would happen if firms differed regarding the magnitude of fixed costs. 19

20 In case both conditions (5.22) and (5.23) hold, there will be multiple equilibrium. To illustrate this, we refer to Figure 7. In the figure, the function (5.15c) is displayed as a positive function of s. Since in this example 0 0, the case with s=0 is not an equilibrium: even if the economy starts out with all industries in the informal sector, each single firm would break even if investing alone. Thus, the economy would naturally gravitate to the equilibrium s=1. Figure 7 The case with single equilibrium Figure 8 The case with multiple equilibria 20

21 The interest case occurs in Figure 8. In that figure, there is multiple equilibrium, because 0 0 and 1 0. Hence, if the economy starts out with s=1, it will be profitable for all firms to remain in operations and the economy will stay with s=1. If however the economy starts out with s=0, no firm will find it profitable to stay in business, so the economy will remain trapped in the informal equilibrium. This equilibrium will be a poverty trap, because there is another feasible one (s=1) delivering higher production and welfare. The case in which the economy is trapped in the bad equilibrium illustrates the pervasive role of strategic complementarities: a strategic complementarity arises when the willingness of one agent to engage in a given action depends on whether the other are taking a similar action. In the current example, the fact that the other firms are not investing induces a single firm not to invest too. And still, all firms benefited if all invested at the same time. There is a coordination failure, because the market mechanism is not capable of coordinating the actions of agents so that the economy moves away from the bad equilibrium where it is trapped. The government can solve this by expanding government expenditures. To illustrate the role of government, we refer to figure 8. In the figure, suppose the economy starts out with s=1. Since 1 0, initially all firms are profitable and the economy remain with s=1. Then, suppose that a temporary shock (say, a decline in alfa) caused the profits function to fall such that 1 0. Since staying in business is no longer profitable, the only equilibrium is now s=0. To make the example more realistic, suppose that the economy did not jump immediately to the new equilibrium: due to some sort of heterogeneity (such as the timing of the payment of the fixed cost), some firms shut down earlier than others. In terms of figure 9, s starts declining continuously from 1 to 0. Of course, if private demand recovered to the former position before the critical level s=s* was reached, the shock would have produced only temporary effects: once the profits curve returned to the previous level, firms would find profitable to invest, and the good equilibrium s=1 would be reached again. If however the recovery was too slow, the case could happen that s was already below s* at the time of the recovery. In that case, even if the aggregate demand returned to the 21

22 previous level, the economy would proceed on its way to the bad equilibrium: in that case, a temporary recession would have produced permanent effects. Of course, the government could prevent the economy from moving to the poverty trap by expanding government expenditures during the recession: from (5.15c), we see that a balanced budget fiscal expansion shifts the profits curve up. If the fiscal expansion was the enough to preserve 1 0, a temporary fiscal expansion would deliver permanent economic gains. Figure 9 Aggregate demand contraction and government intervention 5.6. Optimal provision By now, we have considered government expenditures, G, as given. In this section, we discuss how a benevolent decision-maker would choose the level of G so as to maximise the households preferences, (5.1). In doing so, we first consider the planned economy case. Then, we consider alternative specifications for the decentralized economy: with lump-sum taxes, income taxes, and imperfect competition, respectively. 22

23 The planner optimum Consider first the case of a benevolent planner, which aim is to maximize the household utility function (5.1), subject to the resources constraints and technology, as summarized by the PPF, (5.5). The central planner problem is therefore: MaxU G, l 1 lnzh lg 1 lnl lng The first order conditions of this problem lead to c 1 z l C 1 G 1 (5.10d) (5.24) Equation (5.10d) is the usual condition stating that the opportunity costs of consumption and leisure in production and in consumption shall be the same. Condition (5.24) is a second efficiency condition, establishing that the MRS between consumption and the public good shall be equal to the corresponding MRT, which in this case is equal to one (one unit of government spending implies the deviation of one unit of output away from private consumption). Combining (5.5), (5.10a), and (5.24), the following optimal allocation path is obtained: * G Y zh (5.25) l * 1 h 1 (5.26) C 1 hz (5.27) Where Y zh As expected, the optimal provision of the public good is such that its share in gross income is equal to the corresponding weight in the utility function. Beyond this level, there is no scope for further fiscal expansion. Competitive equilibrium, lump-sum taxes 23

24 Assume now that the allocation of private goods is determined by the market mechanism, while the government decides the provision of the public good. Further assume that lump sum taxes are available. From Section 5.2, we know that the demands for consumption and leisure are given by (5.12) and (5.11). Replacing these and the balanced budget condition (5.6) in the objective function, the government problem becomes: MaxU G G z 1 ln zh G 1 ln1 h lng As you may check, the solution of this problem is exactly the same as in the central planner case, (5.25), (5.26), (5.27). Hence, as long as the government has lump-sum taxes, it can trust the private economy to optimally provide the private good, while the government only needs to care with the provision of the public good. Distortionary tax Now consider the case in which the government can only finance its purchases with taxes on income. In that case, we know from Section 5.3 that the optimal demands for consumption and leisure are (5.12a) and (5.11a). Substituting this in the utility function, and using the balanced budget condition (5.6a), the government problem becomes: MaxU 1 ln 1 hz 1 ln1 h ln wh The solution to this problem is: 1 (5.28) G h 1 * z G (5.25a) That is, in case only income taxes are available, the government will provide less of the public good than in the first best case. The reason is that, at the margin, the government has to balance the benefits for consumers of a higher provision of the public good against the costs imposed by the distortion in the labour market. The 24

25 second best solution (5.25a) corresponds to the optimal balance, given the constraint that lump sum taxes are not available 7. In this second best solution, the household consumes less and works less than in the case in which lump-sum taxes are available. Imperfect competition Now, consider the case with imperfect competition and lump sum taxes 8. In this case, the optimal demand for consumption and leisure are given by (5.12) and (5.11). The household disposable income is given by (5.17b). The government problem can be described as follows MaxU T Y G z 1 ln Y T 1 ln1 lnt 1 1 Where Y T zh G The solution to this problem is: G * z G (5.25b) Implying 1 1 l (5.26b) 1 * h 1 hz C (5.27b) 1 Interesting enough, the optimal provision of the public good is the same as in the well functioning economy. The reason is that, in the absence of distortionary 7 The example above suggests that income taxes are always bad. This is not, however, a general proposition. Income taxes may be the the first best tool to address negative externalities that increase porportionally to income. In Appendix 1, we present an example. 8 In what follows, we ignore the role of fixed costs. The model can however be easilly extended to include F, with no qualitative differences. 25

26 taxes, the planner does not need to deviate from the optimal condition (5.24) when deciding the amount of public good. The fact that the market structure is of imperfect competition impacts, however, on the optimal level of consumption and of leisure, (5.26b) and (5.27b). Had the planer an instrument to eliminate the market distortion, setting 0, and the first best allocation would be immediately achieved. Further reading Andolfatto, D., Macroeconomic Theory and Policy, Chapter 6. Dixon, Huw, Reflections on New Keynesian Economics; the role of imperfect competition, in Surfing Economics, Chapter 4. Mankiw, N. Gregory, "Imperfect competition and the Keynesian cross," Economics Letters, Elsevier, vol. 26(1), pages Matsuyama, Complementarities and Cumulative Processes in Models of Monopolistic Competition, JEL XXXIII June, Chapters 2A and 2D. Williamson, S., Macroeconomics. Pearson, Chapter 5. 26

27 Appendix 1 - Optimal provision of a congestible public input In Section 5.6 we saw that the first best policy to finance a public good was a lump sum tax. In this appendix, we explore an alternative specification for the publicly provided good, which leads to a different conclusion. Consider an economy under perfect competition, just like in Section 5.2, with a unique modification: government expenditures, instead of entering in the utility function (5.1), impact positively on the productivity of labour, as follows: G z (a.5.1) Q You may interpret this effect as capturing the role of government policies that reduce the costs of doing business, such as the provision of a laws, contract enforcement, and public infrastructure 9. As before, it is assumed that the public input is non-excludable, so no-one will be able to pay for its provision. However, if no one produced G, productivity z would be zero and there would be no economy at all. In this economy, government provision is essential to production. Due to the macroeconomic externality, the aggregate production function does not correspond to the simple sum of individual production functions. The true production function in this economy that is, after accounting for the externality is obtained substituting (a5.1) in (5.3) and solving for Q, which gives: Q G N (a5.2) Hence, while individual firms consider (5.3) in their maximization problem, in the aggregate the relevant production function (taking into account the macroeconomic externality) is (a5.2). The problem is that G is non-excludable and hence no profit-making firm will deliberately engage in its provision. The government 9 Note that the specification above does not refer to a pure public good. A pure public good would be non-rival. By dividing G by output, the specification implies that government services in this model are somehow subject to congestion. 27

28 can solve this problem coercing citizens to pay taxes, and then using the proceeds to provide the essential input. For convenience, let s assume that a proportional tax is imposed on production 10 : tq WN 1, (a5.3) In this case, profit maximization implies the following labour-demand 11 : w z 1 t (a5.4) The government budget constraint in this case becomes: tq G (a5.5) Implying that (using a5.1): z t (a5.6) The question that naturally arises is how much should the government provide of the public input. To solve this question, first note that the demand for leisure is inelastic, l 1 h (5.11a). Hence, the only concern of the benevolent planner should be to maximize private consumption 12. Using (a5.5) in (5.12a), the consumption function becomes: t C h 1 t (a5.7) The tax rate that maximizes this expression is: 10 Specifying a tax that is proportional to income is more than mere convenience: because government services are subject to congestion, a proportional tax on income is the first best policy to address the external effect that each individual firm imposes on all others when it decides to expand its production. 11 Note that is irrelevant whether the tax is imposed on consumption or on labour income. 12 In alternative, you can maximize the households utility subject to the condition q=c+g, with the expression for q given by (a5.2). You will get exactly the same solution. 28

29 t G Q 1 (a5.8) That is, a judicious government intervention is the one that implies a provision of public input exactly corresponding to the elasticity of that input in the production function (a5.2). No more, no less. The relationship between private consumption and the size of the government in this model is illustrated in Figure A1: when the tax rate is very small, public provision is too small, so it would be socially beneficial to increase government intervention. Beyond the optimal level, expanding government expenditures will come at a welfare loss. This relationship is often known as the Barro Laffer curve 13. Figure a1: Consumption and provision of the public input 13 Barro, R., Government spending in a simple model of endogenous growth. Journal of Political Economy 98,

30 Review questions and exercises Review questions 5.1. Consider a well functioning economy, with full information, flexible prices and perfect competition. Explain how a fiscal expansion impacts on output and employment Consider the case with perfect competition and income taxes. Describe the equilibrium in the labour market in this model, and explain what happens when the tax rate increases Explain why with imperfect competition the crowding out effect of a fiscal expansion is lower than in the case with perfect competition Consider the model with the public input examined in the appendix. Explain why in this model a tax that is proportional on income is preferable to a lump sum tax. Exercises 5.5. (The well functioning economy) Consider an economy where the preferences of the representative consumer and the production function of the representative firm are given, respectively, by U lc U lc and Q zn. Also assume that in this economy, government spending is equal to G=200, h=100 and Z=10. a) (Social optimum) Obtain the central planer solution and explain [A: l=40; C=400]. b) (Competitive equilibrium) Describe the competitive equilibrium, assuming that government spending is financed with a lump sum tax. c) (Comparative static) Sticking with the assumption of a lump sum tax, explain what would happen to (i) production; (ii) consumption; (iiii) working time, if: c1) Government spending increased to 400 [A: l=30; Q=700; C=300]. c2) TFP increased to 20 [A: l=45; Q=1100; C=900]. d) (Distortiary taxation) Returning to the initial formulation, assume now that government spending was financed by a tax on labour income. Compare the welfare level with that found in (a). [A: l=50; Q=500; C=300]. 30

31 5.6. (Laffer curve) Consider an economy where the preferences of the representative consumer and the production function are given, respectively, by U 0.5lnC 0.5lnl and Q zn, with z=2. The household time endowment is equal to h=100. a) (Competitive equilibrium) Describe the competitive equilibrium, assuming that price and wages are flexible. In particular, find out: (b1) the labour demand function; (b2) The labour supply function; (b3) the equilibrium employment and output. (b4) Is this equilibrium Pareto efficient? b) (Labour supply) Now, assume that the government creates a transfer program financed with a tax on income, that is, I Q. Reformulate the household problem and find out the optimal supply of labour as a function of and I. c) (Laffer curve) Find out the expression relating the tax proceeds, I, to the tax rate and represent it in a graph. Find out the tax rate that maximizes the government revenue. d) Suppose now that the government intends to spend I=30 in this program. (d1) Show that there are two possible tax rates consistent with this level of revenue. For each of the two tax rates, describe in a graph: (d2) The equilibrium in the labour market; (d3) the equilibrium in the production possibilities frontier. (d4) Conclude Consider an economy where output consists in an infinite number of varieties with total mass equal to 1. The preferences of the representative consumer are given by U 0.75lnC 0.25lnl, with C denoting for consumption, and l for leisure. The time endowment is h=100 and leisure is numeraire (W=1). a) (Households): Find out the optimal demands for consumption and for leisure as functions of the disposable income, Y d h T. b) (Perfect competition): Assume that the production function of a representative firm was Q N. Find out: (b1) the equilibrium price, and profits under perfect competition; (b2) the implied levels of consumption and of leisure as functions of T; (b3) the equilibrium level of output, Q, in the particular case in which T=G=0. c) (Fiscal expansion) Examine the impact of a fiscal expansion from T=G=0 to T=G=40. In particular, find out the impact on: (c1) consumption; (c2) leisure; (c3) employment, (c4) welfare, and (c5) output. Explain, with the help of a graph. d) (Entrepreneur): Consider now the problem of an entrepreneur deciding whether to adopt technology Q 2N, which comes along with a fixed cost F=40. Assuming that P=1: (d1) Find out the expression for profits as a function of output, Q ; (d2) Would this entrepreneur find it profitable to adopt the new technology if the initial output was the one determined in (b3)? (d3) What if the initial situation was (c4)? 31

32 e) (Imperfect competition). Returning to the case with T=G=0, suppose that all firms were already using the increasing returns technology described in (d), with F=40 to being equally spent across all sectors in the economy. Compute: (e1) the equilibrium levels of output and profits; (e2) the implied values of consumption and leisure. (e3) Are households better off in this case than in case (b2)? Why? f) On the basis of your results, explain why, departing from (b), a temporary fiscal expansion like (c) may deliver permanent gains. Illustrate with a graph (Multiplier) Consider an economy where the representative consumer maximizes the following utility function: U 0.75lnC 0.25lnl, subject to PC l Y d, where C is consumption, l is the number of resting days per month (leisure=numeraire) and Y d Y T refers to disposable income. Further assume that the labour endowment is h=30. a) Find out the demands for consumption and for leisure, as functions of the disposable income. b) Suppose that output was produced by a monopolist under the following technology N Q. Further assume that a competitive fringe prevented this monopolist from charging a price above P=1. Find out the expression for profits as a function of Q. c) Taking into account that the total demand is Q C G, where G refers to government spending, find out the expression for Q as a function of profits, T and G. d) Find out the expression relating the economy s production level q to G and T. Why are the multipliers of T and G different? (compare to the textbook Keynesian model). e) Find out the expression relating the economy s income level Y to G and T. f) Assume for a moment that G=T=0. Find out the equilibrium values of Y, C, l, Q, and. Show that in this equilibrium the demand for labour is equal to labour supply. (R: Y=43.2; l=10.8; =13.2). g) Now consider the case in which G=T=9. Find out the equilibrium values of Y, C, l, Q, and. Show that in this equilibrium the demand for labour is equal to labour supply. (R: Y=45; l=9; =15). h) Compute the consumer utility level in (f) and in (g) (Coordination failure) Consider an economy where output consists in an infinite number of varieties with total mass equal to 1. In each industry, the potential monopolist, equipped with an increasing returns technology, decides to adopt the technology or to stay in the informal sector (competitive fringe), 32

33 where Q j N j. The entrepreneur production function is given by Q 2N, where N refers to the number of days worked. Operating with this technology also involves a fixed cost, F=12, that is equally spent across all sectors in the economy. Household demand for each variety and for leisure (numeraire) are given respectively by C 0. 75Y d and l 0. 25Y d, where Y d Y T h s T, h 30 refers to the time endowment, and 0 s 1 measures to the fraction of industries that are operated under increasing returns. a) Find out the expression for a potential monopolist profits,, as a function of output Q b) Using the equilibrium condition in the market for this variety, Q C G sf, and the consumption function, find out the expression for individual profits, as a function of s, G and T. c) First assume that G=T=0, and s=0. c1) Is s=0 an equilibrium? (A: =- 0.75). c2) How much is consumption, leisure and the demand for labour in this equilibrium? (C=22.5, l=7.5, N=22.5) d) Now consider the case in which G=T=10. d1) Is s=0 an equilibrium? (A: =+0.5). d2) Is s=1 an equilibrium? (A: =+10.4). d3) Compute the equilibrium levels of consumption, leisure and demand for labour in this case. (A: C=22.8; l=7.6; N=22.4). e) Finally, consider the case in which, departing from s=1, the government decides to set T=G=0. e1) Is s=1 an equilibrium? (A: =+8.4). e2) Compare the consumer utility levels in (c), (d) and (e) and conclude (Optimal provision) Consider an economy where the preferences of the representative consumer and the production function are given, respectively, by U lnc 0.5lnl 0.25lnG and q zn, with z=4. The household time endowment is equal to h=100. e) Find out the central planner solution [G=100]. f) Find the out the optimal provision in the competitive equilibrium, assuming that lump-sum taxation is available [T=100]. g) Finally, consider the case in which taxes are proportional to income. Find out the optimal provision in this case and compare with b. Explain [t=0.4] (Public input) Consider an economy where the preferences of the representative consumer and the production function of the representative firm are given, respectively, by U lc and Y zn. Finally, assume that h=2. a) Solve the firm maximization problem and find out the wage rate [A: W=z]. 33

34 b) In this economy, consumers pay a tax t that is proportional to labor income. Solve the household problem and find out the demand for leisure as well as the demand for consumption, as a function of t and Z [A: l=1; C=w(1-t)]. c) Using the labour market equilibrium condition, compare the consumer utility levels when: (c1) Z ant t=.25. (c2) Z and t=0,5; (c3) Z=3 and t=1/3. d) Now assume that Z 27G Y 0. 5, where G is a non-excludable input. (d1) Explain this formulation. (d2) Find out the expression of the actual production function in this economy. (d3) Display, in a graph, the household consumption as a function of the fax rate. Explain the shape of this curve. (d4) Find out the optimal tax rate [A: t=1/3] (Exogenous labour + money) Consider an economy with a fixed number of identical consumers, each of which supplies labour in a fixed quantity N=100, and maximizes the utility function U lnc lnm P subject to C M P W PN M 0 P T, where M 0 refers to initial money holdings. In this economy, the production function of the 1 2 representative firm is given by Q 2N, and the government deficit is financed by seigniorage revenues: T M M P. G 0 a) Assuming perfect competition, find out the optimal demand for labour, and the implied profit function. b) From the household optimization problem, find out the demand for money and the optimum consumption as a function of M 0, P,, T, W. Using the government budget constraint, find out the aggregate demand as a function of M, P and G. c) (Flexible prices) (c1) Describe the equilibrium with flexible prices, when M=160 and G=4 [A: P=10, W=1]. (c2) Examine the implications of an increase in government spending financed with taxes to G=10. Describe the change in the AD-AS diagram and in the Keynesian cross [A: P=16]. d) (Sticky prices): Assume now that the price level was stuck at P=20, with M=160 and G=4. (d1) find out the implied equilibrium in the AD-AS diagram and in the Keynesian cross [A: Q=12]. (d2) Describe the labour demand and supply curves in the (W/P, N) space. (d3) Could fiscal policy be used to achieve full employment in this case? [A: G=12]. e) (Sticky wages) Assume now that nominal wages were stuck at W=1.6, with M=160. (e1) Describe the implied equilibrium in the AD-AS diagram and in the labour market [A: P=13.02, Q=16.28]. (e2) Could a fiscal expansion drive the economy to full employment in this case? [A: G=10]. f) (Real wage rigidity). Assume that real wages were constant at W/P=0.2, with M=200 and G=0. (f1) Find out the equilibrium levels of output and employment [A: Q=10]. (f2) Describe, in the AA-DD diagram, the implications of expanding government expenditures to G=2 [A: P=25]. 34