Macroeconomic Policy Issues in General Equilibrium I. Money and monetary policy in nominal models with flexible prices: . II. Fiscal policy in general equilibrium Florin O. Bilbiie HEC Paris Business School
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Contents 1 Money and monetary policy with flexible prices 1.1 Money as an asset . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Money in the utility function. . . . . . . . . . . . . . . . . . . . . 1.3 Steady state stationary equilibrium . . . . . . . . . . . . . . . . . 1.4 Dynamic effects of monetary policy . . . . . . . . . . . . . . . . . 1.5 Modern monetary theory: monetary policy without control of a monetary aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Bottomline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 2 3 5 7 8 10
2 Fiscal Policy in General Equilibrium 11 2.1 Lump-sum taxes, no debt . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 The labor market . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Government debt and Ricardian equivalence . . . . . . . . . . . . 15 2.2.1 Debt sustainability . . . . . . . . . . . . . . . . . . . . . . 17 2.3 The Fiscal Theory of the Price Level . . . . . . . . . . . . . . . . 17 2.4 Distortionary taxation and Ricardian equivalence . . . . . . . . . 18 2.5 The effects of government spending shocks under distortionary taxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5.1 Debt dynamics and the ’fiscal rule’ . . . . . . . . . . . . . 21 2.5.2 Crowding-in of private consumption . . . . . . . . . . . . 22 2.6 Optimal fiscal policy: a primer . . . . . . . . . . . . . . . . . . . 24
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CONTENTS
Chapter 1
Money and monetary policy with flexible prices The ’real’ models studied previously are by definition silent about issues such as the price level, inflation, nominal variables and hence the role of monetary policy. The reason is simple: the price of goods is indeterminate since the consumption good is the numeraire (recall that in a general equilibrium model with n goods we can only determine n − 1 prices). We know the price of apples in banana units and the price of bananas in apple units, but not both the prices of bananas AND apples. This points to a role for fiat money as an unit of account, where by fiat money we mean money that has no intrinsic value (as opposed to commodity money). Deep(er) theories of money try to explain the existence of money by exploiting various trading frictions. The real models just studied rely on the presence of a Walrasian auctioneer (clearing house) who quotes prices to all agents and then runs a clearing house equating supply and demand. In the real economy, something called the ’double coincidence of wants problem’ occurs. Sometimes, trade may not be arranged between two agents holding two different goods, and money may occur in equilibrium simply because it is an accepted medium of exchange: money buys goods and goods buy money, but goods don’t (necessarily) buy goods. These discussions may seem abstract but they are not: barter was common practice in the post-communist period in Eastern European countries (’you can’t eat money’, an Eastern European student eloquently said); dollarization is still common practice in many Latin American countries. Once we introduce money can potentially have a role for monetary policy in influencing the real economy and we can study monetary policy issues. 1
2CHAPTER 1. MONEY AND MONETARY POLICY WITH FLEXIBLE PRICES
1.1
Money as an asset
Suppose that fiat money is just one asset that brings no benefit to the consumer other than being merely an unit of account, numeraire. All prices are denominated in this unit, including the price level for the consumption good and the NOMINAL bonds that the household can purchase. The household would face the budget constraint: (1 + It )−1 Bt+1 + Pt Ct + Mt = Bt + Mt−1 + Et , where Et is non-financial income of the household taken for now as exogenous (later it will include labor income and government transfers); Mt are end-ofperiod t money holdings and Bt are discount bonds that promise a unit of currency tomorrow and cost (1 + It )−1 today. Therefore, as we saw earlier in the course, the net return on bonds will be It . Note that this is the NOMINAL return on bonds, i.e. the return in currency units. What about the return on money? Money ’costs’ in terms of goods P1t today: this is the purchasing power of one unit of currency. Otherwise put, if one apple costs 2 units of currency, one unit of currency buys you half an apple, but choosing to hold it in currency costs you exactly this, half an apple. The payoff of a currency tomorrow is 1/Pt+1 (the purchasing power of a unit of currency tomorrow), so its gross return will be Pt /Pt+1 . Holding currency gives you as a return the rate of deflation. The household makes a portfolio decision, maximizing the present discounted value of utility of consumption by choosing bond and money holdings. We will solve this problem in nominal terms, assuming that the household chooses the amounts of B and M to hold, not the quantities (case in which we would divide by the price level and we would solve a ’real’ problem). The two methods should give the same solution. ∞ t Bt Mt−1 Et Mt −1 Bt+1 max β U + + − (1 + I ) − t t=0 Bt+1 , Mt Pt Pt Pt Pt Pt If we obliviously solved for the Euler equations, these would be: Bt+1 Mt
: (1 + It )−1 UC (Ct ) = β : UC (Ct ) = β
Pt UC (Ct+1 ) Pt+1
Pt UC (Ct+1 ) Pt+1
As long as the nominal interest rate is positive, It > 0 it is clear that both conditions cannot hold simultaneously. What happens is that the consumer optimally chooses to hold no money (hits a corner solution), since this is a dominated asset: bonds give always a higher return. The optimality condition t will be: Mt = 0, UC (Ct ) > β PPt+1 UC (Ct+1 ) = (1 + It )−1 UC (Ct ) (use your static optimization knowledge, Kuhn-Tucker conditions, etc. if you do not see this directly).
1.2. MONEY IN THE UTILITY FUNCTION.
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For further use, note that the Fisher parity condition holds: 1 + Rt = (1 + It )
Pt 1 + It = , Pt+1 1 + Πt+1
whereΠt+1 is net inflation and Rt is as before the REAL net return on a REAL bond, one that gives you the right to units of consumption. You would get this by no-arbitrage if you assumed that there were both nominal and real bonds in this economy. Hence, we need to give money additional roles, other than unit of account. You can think of many reasons why money has additional value: it is an accepted mean of exchange, it provides transaction services, holding currency avoids spending time trying to transform albeit liquid assets in the accepted means of payment, perhaps for many transactions we are required to hold cash balances in advance, etc. There are many theoretical papers on the deep reasons for the existence of money, and I recommend to those interested Kiyotaki and Wright and Kocherlakota. In these notes, we use a shortcut known as the money-in the utility function approach. Note that many other ways of modelling money can be cast in this form.
1.2
Money in the utility function.
We study a shortcut approach due to Sidrauski, which is an extension of the growth-RBC model by allowing money to enter the utility function directly. This provides for a reason to hold money. Suppose that the felicity function is: Mt U Ct , Lt , , UM > 0, UMM < 0 Pt Note that what enters the utility function are ’real money balances’, the real value of currency that the agent holds at the end of period t and will carry over to period t + 1. The budget constraint becomes: −1
(1 + It )
Bt+1 + Pt Ct + Mt = Bt + Mt−1 + Wt Lt + Pt St ,
where Mt are end-of-period t money holdings and Pt St are nominal transfers received from the government due to seigniorage revenues. Bt are discount bonds that promise a unit of currency tomorrow and cost (1 + It )−1 today. For now, we consider ’private’ bonds that agents will trade among themselves. We can write the constraint in real terms: (1 + It )−1
Bt+1 Mt Bt Mt−1 Wt + Ct + = + + Lt + St . Pt Pt Pt Pt Pt
Our problem is, after substituting out consumption using the budget constraint: ∞ t Bt Mt−1 Wt Mt Mt −1 Bt+1 max E β U + + L + S − (1 + I ) − , L , . 0 t t t t t=0 Bt+1 Mt Pt Pt Pt Pt Pt Pt , Pt
Pt
4CHAPTER 1. MONEY AND MONETARY POLICY WITH FLEXIBLE PRICES As before, we will have an intratemporal optimality condition governing the choice of hours worked: Wt UC (.) = −UL (.) , where UL < 0 Pt First order conditions: Bt+1 Pt Mt Pt
Pt UC (Ct+1 , .) : (1 + It )−1 = Et [Λt,t+1 ] , where Λt,t+1 = β , Pt+1 UC (Ct , .) Mt Mt Pt Mt+1 : UM ., − UC ., + βEt UC ., = 0, Pt Pt Pt+1 Pt+1
which after manipulation leads to a money demand equation: Mt Mt It UM Ct , Lt , UC Ct , Lt , = . Pt 1 + It Pt
(1.1)
This is like an LM curve from ISLM models. Money demand depends negatively on interest rates: if It goes up for a given UC , UM needs to go up, which happens only if real money holdings fall due to concavity of U (.). Money depends negatively on It because this is the opportunity cost of holding currency rather than bonds. Money demand and positively on consumption. Assume that utility is additively separable in money balances, i.e. UCM = 0. An increase in Ct for constant interest rates induces a fall in UC since UCC < 0. For equality to establish, UM ought to fall, which again by concavity means that money holdings must increase. This model features a version of the Quantity Theory of Money. To see this, note that 1.1 defines an implicit money demand function, say1 Mt = Ψ (It , Ct ) , where ΨR < 0, ΨC > 0. Pt Anticipating that in equilibrium all money will be consumed Yt = Ct , write this as: Ψ (It , Yt ) Mt = Yt , Pt Yt and defining the velocity of money Vt ≡ Ψ(IYtt,Yt ) , we have the quantity theory equation: Mt Vt = Pt Yt . One standard if old-fashioned view is that this is how the monetary authority determines the price level: for a given (and roughly constant) velocity of money and a given level of ’aggregate demand Y, picking a value of the monetary aggregate M fully determines the price level. Inflation will then be simply given
1 Example: 1 Rt
+ 1 Ct .
think of log utility in consumption and money, we would have:
Mt Pt
=
1.3. STEADY STATE STATIONARY EQUILIBRIUM
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by the rate of money growth. We now turn to our general equilibrium model in order to study these issues in a full-fledged model. Up to now we talked about households. On the firms’ side, we abstract from capital accumulation, for example assuming that the production function is linear in labour: Yt = At Lt (1.2) The real wage is given by profit maximization (this can be regarded as ’labor demand’ by firms): (1.3) Wt = At We have another agent in our economy, namely the government. The government in this model will simply rebate revenues from money creation to the household. The revenue from money creation is the change in the real value of t−1 money at given prices: Mt −M , and the government budget constraint will be: Pt St =
Mt − Mt−1 , Pt
where S is called seigniorage. The monetary authority (central bank) is in charge with choosing a path for the money supply Mt . We will assume that the money creation process takes the form: Mt = 1 + µt , Mt−1 where µt is the net rate of money growth. Equilibrium is obtained as usual when all agents optimize and markets clear. Substituting the government budget constraint, the production function and the expression for real wage, together with the goods market clearing condition Yt = Ct in the households’ budget constraint, we have that by Walras’ law the bond market also clears, namely: Bt = 0. This equation means that agents are indifferent between borrowing and lending and so bonds will be in zero net supply. There could be some trading in bonds among agents (some agents for whom bond holdings would be B > 0 would borrow from others, whose bond holdings would be B < 0) in case there was idiosyncratic uncertainty, but if you aggregated bond holdings across all agents you would get zero.
1.3
Steady state stationary equilibrium
We assume that money grows at a constant rate Mt+1 = (1 + µ) Mt and want to find the impact of the rate of money growth on all variables, where all real variables are constant. A constant real interest rate implies: 1+R =
1+I 1+Π
6CHAPTER 1. MONEY AND MONETARY POLICY WITH FLEXIBLE PRICES Since inflation will also be a constant, the nominal interest rate will be constant too. But this implies that real money balances will be constant, since: Mt I Mt UC C, L, UM C, L, = (1.4) Pt 1+I Pt Real money balances are constant, so the price level will increase at the same rate as money: there is constant steady-state inflation given by Π = µ. From the Euler equation for bonds: (1 + I)−1 1+I
1 , so: 1+Π 1+µ β
= β =
This effect of money growth on nominal interest rates is known as the Fisher effect. Since the real interest rate in steady state is 1 + R = β −1 , we have that I = (1 + µ) (1 + R) − 1 and using our usual approximation: I ≈ R + µ : money growth has an approximately one-to-one effect on nominal interest rates. Proposition 1 Neutrality of money: The level of money M does not influence real variables (note that only the ration M/P appears in all steady-state expressions). As the steady-state money stock M is increased, the only effect is on prices P, which increase by the same amount. When M increases, the marginal utility of money falls and people switch to holding goods, but this translates one-to-one in an increase in the price of goods. Note that this relies crucially on the prices being perfectly flexible - an assumption that many of us believe to be false and that we will relax in a later part of the course. Proposition 2 Superneutrality of money: The rate of growth of money does not affect real variables. The only effect of an increase in µ is on steadystate inflation and the nominal interest rate, but real variables such as C, Y, W and R are not affected. Note that both neutrality and superneutrality are robust to the introduction of physical capital: in that case, the real interest rate will not be a constant but will depend on the capital-labor share, as in the RBC model. However money is still neutral and superneutral (see Chapter 2 of Walsh for such a model). Proposition 3 Friedman rule: What is the rate of money growth (or inflation) that maximizes steady-state welfare? (we could label this ’the golden rule quantity of money’ by analogy with the capital stock)? Simply maximizing
1.4. DYNAMIC EFFECTS OF MONETARY POLICY
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steady-state utility with respect to µ and observing that by superneutrality C and L are not affected, we have the first-order condition: UM = 0 But we have seen that 1.4 money holds, so this boils down to: I = 0. This is intuitive: the economy is satiated with money (marginal utility of money is zero), since we have effectively eliminated the opportunity cost of holding money. People would be indifferent between holding money or other assets, since real returns on assets would also be now equal to the rate of deflation: 1 + R = (1 + Π)−1 , just as for money. Otherwise put, the central bank should deflate the economy at a rate equal to the inverse of the return on real assets. This is a result obtained in a very special economy. Most notably, Phelps has argued that since governments do not dispose of lump-sum instruments to rebate seigniorage revenues, but only of distortionary ways to do so, the conclusion would change as follows. The loss in government revenues occurring because of the reduction in the ’inflation tax’, will have to be made up through distortionary taxes. Optimality may then well require reliance to a certain extent on the (distortionary) inflation tax.
1.4
Dynamic effects of monetary policy
In the next homework, you will be asked to loglinearize this model and derive some dynamic implications pertaining to the effects of monetary policy. Assuming that technology does not change, At = A, and that the central bank picks a path for the money supply Mt , you will obtain that the price level obeys: pt =
1 ∞ 1 + α i=0
α 1+α
i
mt+i ,
(1.5)
where α is the elasticity of money demand to interest rates obtained when loglinearizing the money demand equation 1.1. Aside from the remarks below pertaining to whether modern central banks actually operate by changing the money supply, there are two problems with this model: 1. the effects of monetary policy on real activity in more complicated models (e.g. with physical capital) look nothing like those estimated in the data. 2. the price path 1.5 looks too much like an asset pricing equation; but goods prices are believed by many economists to be rigid, i.e. subject to frictions not present in asset markets.
8CHAPTER 1. MONEY AND MONETARY POLICY WITH FLEXIBLE PRICES Both of these remarks point to a need of modelling sticky prices, something we will turn to in due course. Note, however, that this model is useful in order to analyze the long-run effects of money growth on inflation (something that is present in the data, in contrast with the short-run effect that is largely absent). If we assume that money growth µt = mt+1 −mt is the instrument of the central bank, we obtain by taking first differences of 1.5: i 1 ∞ α pt+1 − pt = µt+i , 1 + α i=0 1 + α and a constant-money-growth rule µt = µ gives: i ∞ 1 1 α 1 pt+1 − pt = µ i=0 =µ α = µ. 1+α 1+α 1 + α 1 − 1+α Substituting this in the money demand equation we have: pt = mt + αµ. Therefore, this model predicts that an increase in the rate of growth of money leads to a jump in the price level. The relationship between money growth and inflation in the long run is not contradicted by the data, but in the short run it is. In the short-run data show a negative correlation between the rate of money growth and the nominal interest rate, contrary to the predictions of the model above. In U.S. data, an open market operation (an increase in the money supply) makes the nominal interest rate decrease, output increase after 6 to 9 quarters, and inflation increase after about 1 year.
1.5
Modern monetary theory: monetary policy without control of a monetary aggregate
Traditional monetary theory assumes that the central bank influences the economy and controls the price level through control of a monetary aggregate. By changing the money supply, the nominal interest rate will be determined by equilibrium in the money market M d (I, .) = M s , provided there exists a stable money demand equation. Modern monetary theory (and central bank practice) does the opposite: it assumes that the central bank controls the short term nominal interest rate and does not use the money supply at all. What about money then? The quantity of money is determined residually from the money demand equation, but this is not something that regards the central bank. At the provided interest rate, households demand that much money, the bank prints it, and that’s it (well, the story is actually much more complicated, depending on details of operating procedures of each central bank - see Woodford, Chapter 1, Section 3 for a useful discussion - but the main idea is this). Money plays no role in this framework except as a unit of account. This theory, developed recently by Woodford and
1.5. MODERN MONETARY THEORY: MONETARY POLICY WITHOUT CONTROL OF A MONETARY AGGR others, usually works within a cashless framework in which money is simply a unit of account - pretty much as in the highly stylized model with which we started. These ideas were outlined in Wicksell’s Interest and Prices more than a century ago! Under this ’Wicksellian’ view of monetary policy, the central bank chooses not the money supply, but picks the interest rate by adhering to an interest rate rule. For example, a rule that describes Wicksell’s view is: it = φp pt + εt ,
(1.6)
so the central bank responds to variations in the price level by moving nominal interest rates, where φp is the elasticity by which it does so and εt is a nonsystematic part of policy. Assuming that the real interest rate is exogenously determined by real factors such as technology, preferences, and other exogenous forces, substitution of 1.6 in the Fisher parity condition leads to: pt =
1 1 ∗ pt+1 + rt+1 − εt , 1 + φp 1 + φp
which solved forward leads to: pt =
1 ∞ 1 + φp i=0
1 1 + φp
i
∗ rt+1+i − εt+i
(1.7)
In contrast with the price path obtained under money supply control, 1.5, the existence of unique rational expectations equilibrium (i.e. equilibrium determinacy) is insured if and only if φp > 0, i.e. if monetary policy systematically and endogenously responds to the price level. Moreover, the price level is determined ∗ (apart from real factors influencing rt+1 ) by the future path of non-systematic policy decisions. What about the money supply? This is determined residually by the money demand equation. Given prices 1.7 and nominal interest rates 1.6, the quantity of money demanded is fully determined by the money demand equation. The central bank merely ’prints’ this money (or in fact pursues the open market operation that lead to this quantity). Practical, real-world policymaking is better described by what is known as a ’Taylor rule’ (from Taylor, 1993), whereby the central bank responds to variations in realized or expected inflation on the one hand and a measure of real activity (for example output), on the other. It is this type of rule that we will be using when building full-fledged models usable for policy analysis. For the moment, note that a simple version of such a rule can be expressed as: it = φπ π t + εt . Substituting this in the Fisher parity condition, we have: πt =
1 1 ∗ Et π t+1 + rt+1 − εt φπ φπ
Since π t is a forward-looking variable, we need to solve this equation forward; we are able to do this if and only if φπ > 1, i.e. if monetary policy is active. This
10CHAPTER 1. MONEY AND MONETARY POLICY WITH FLEXIBLE PRICES requirement is known as the Taylor Principle, and implies that the central bank will increase nominal interest more than proportionally in response to an increase in inflation. Under this policy, the path of inflation will be: πt =
1 ∞ φπ i=0
1 φπ
i
∗ rt+1+i − εt+i .
(1.8)
It is also immediately apparent why an interest rate peg (or an exogenous path for interest rates, a fully passive policy) is not a good policy recommendation: φπ = 0 would lead to an infinite path for inflation. We will study these rules in more detail when we deal with models with nominal rigidities.
1.6
Bottomline
We saw how to introduce money in a General Equilibrium model so that we can speak meaningfully about prices, inflation, nominal variables and monetary policy. We concluded, however, that the role of money is limited. In the simple (yet general enough) model we worked with, money was neutral and superneutral in the long run, and the short-run effects are in contrast with the data. Moreover, we have argued that price flexibility is perhaps not a realistic description of goods markets (in contrast with asset markets). Finally, we saw how monetary policy can influence the price level without controlling the money supply, but by setting nominal interest rates; this is not only a good description of central bank practice, but presents other advantages: it does not rely on a stable money demand equation, and ’works’ even in the completely cashless economy. In order to have a realistic model usable for policy analysis however, we need to introduce nominal rigidities, i.e. imperfect price adjustment. Otherwise, all changes in the monetary policy instrument are fully absorbed by the (fully flexible) prices.
Chapter 2
Fiscal Policy in General Equilibrium In this Chapter I want to familiarize you with the main ideas concerning the effects of government spending shocks in the RBC model, and with related fiscal policy issues. Given the difficulties of technology shocks to account for some of the data features, some authors have naturally looked at other shocks, for examples shocks to government spending. In particular, Christiano and Eichenbaum (AER, 1992) showed that adding this stochastic source of fluctuations helps in resolving an important puzzle, i.e. the discrepancy between the high procyclicality of the real wage implied by the baseline RBC model and the relative acyclicality observed in the data. However, government spending shocks have other undesirable properties: as will become clearer below, they generally imply a countercyclical consumption, in stark contrast with the data. Consumption falls in response to government spending shocks due to a negative wealth effect: government spending absorbs resources and makes the agent feel poorer by the present discounted value of taxes that are used to finance this spending. This makes the agent consume less and work more for a given real wage; the latter effect implies that output increases. Therefore, conditional on government spending shocks, consumption will be countercyclical. If you want to know more on these issues, I recommend reading Baxter and King (AER, 1993) and Christiano and Eichenbaum (AER, 1992). Some recent developments on these issues can be found in Gali, Lopez-Salido and Valles (2005) and in some of the references therein - in particular, you could check the ones whose author’s names start with B (Read these recent papers after you covered sticky prices and monetary policy issues). 11
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2.1
CHAPTER 2. FISCAL POLICY IN GENERAL EQUILIBRIUM
Lump-sum taxes, no debt
The main ideas pertaining to fiscal policy analysis in general equilibrium can be understood in the following highly stylized model. We abstract from capital accumulation, for example assuming that the production function is linear in labour: (2.1) Yt = At Lt The real wage is given by profit maximization (this can be regarded as ’labor demand’ by firms): Wt = At (2.2) We assume that there is an additional source of fluctuations: exogenous movements in government spending Gt . For simplicity, we assume that government spending is pure waste, i.e. it neither contributes to production, nor it affects utility directly: utility is still given by ??. Finally, government spending is financed via lump-sum taxes Tt on households. In the simplest case studies here, we assume that the budget is balanced each period, i.e. the government budget constraint reads: (2.3a) Gt = Tt . The budget constraint of the households becomes: Ct = Wt Lst − Tt
(2.4)
Note that there is no intertemporal dimension of the household’s maximization problem. The only optimality condition concerns the choice of consumption versus hours worked, which is the same as previously: vL (Lst ) =
1 Wt Ct
(2.5)
As before, this can be regarded as a ’labor supply’ schedule for a given level of consumption. Labor market clearing will again ensure that Lst = Ldt = Lt In the spirit of ’counting equations’ and ensuring we have as many equations as new variables, note that we have introduced 2 new variables (exogenous G and endogenous T ) and 2 new equations - the balanced-budget rule and an exogenous process for G. As before, by Walras’ law the economy resource constraint holds (convince yourself of this, by combining the equilibrium conditions above): Yt = Ct + Gt ,
(2.6)
stating that the good of this economy is used for either private or public consumption. Note that there are no endogenous dynamics: all dynamics are extrinsic, coming from dynamics in the exogenous processes for G. Nevertheless, I will
2.1. LUMP-SUM TAXES, NO DEBT
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apply the same techniques as in the model with capital (looking at the steadystate, loglinearizing, etc.), since in practice you may want to study fiscal policy in more complicated (and hopefully realistic) models. The loglinearized equations are very simple (we substituted the labor market clearing condition lts = ltd = lt already): yt wt gt ct ϕlt
= at + lt = at = tt WL T = (wt + lt ) − tt C C = wt − ct
The steady-state shares needed in the loglinearized budget constraint are easily found once we recognize that W L = Y and T = G, where G is given exogenously. We parameterize GY ≡ G/Y to average US government spending GY 1 T Y (about 0.2) and observe that WCL = YC = 1−G and C =G C = GY × C = 1−GY . Y The steady state is found by merely dropping time subscripts. Combining all the equations above, we have: vL (L) = L−
G A
1 W → WL − G
−1 = vL (L)
This equation implicitly defines steady-state hours worked as a function of exogenous forces. Note that due to the presence of government spending and taxation, hours depend on the level of technology. Hours are generally increasing in the level of government spending, since increased taxation introduces and extra income effect, on top of the two effects that, as we saw for this utility specification, cancel out: income and substitution effects due to wage variations. Nevertheless, hours could be constant (i.e. not growing) in steady state even if A grew if we assume that G grows at the same rate as A. Once we found L, we can easily find the steady-state values for all other variables.
2.1.1
The labor market
In most models of fiscal policy (and of business cycles in general), the labor market is crucial for understanding the propagation of shocks and the workings of the model. I try to illustrate this in our simple model. Labor demand in loglinearized form is horizontal in the (l, w) space: LD : wt = at .
(2.7)
Labor demand shifts here only because of exogenous shifts in technology: an increase in productivity a makes firm want to hire more labor at the prevailing real wage.
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CHAPTER 2. FISCAL POLICY IN GENERAL EQUILIBRIUM Labor supply, for a given level of consumption, is: LS : ϕlts = wt − ct .
This is an upward-sloping schedule in the (l, w) space that shifts due to endogenous movements in consumption. Recall first what happens if a positive technology shock hits in the baseline RBC model with no capital and no government spending (let G = 0 too) this is the same as the previous discussion where we now set α = 0. Labor demand shifts upward and the real wage increases. Because wage increases, the household feels richer and is willing to work less at a given real wage - so the LS schedule shifts leftward. In equilibrium hours do not move and all the increase in productivity translates into an increase in real wage. This can be seen analytically by (we have imposed labor market clearing and since there is no government spending at all we have ct = yt = at + lt ) ϕlt = wt − ct = wt − at − lt → wt − at = (1 + ϕ) lt Combining this with labor demand (wt = at ) we obtain lt = 0 and wt = at . This is an extreme version of one finding of RBC models that is at odds with the data, where we see the opposite: a lot of movement in hours and little movement in wages. Take an increase in government spending now, assuming that technology does not move, at = 0 - which also makes the real wage constant. Let’s first work out the effects analytically. First substitute the budget constraint and the balanced-budget rule in the labor supply equation: ϕlt −GY wt 1 − GY
= wt − ct → 1 GY = ϕ+ lt − gt . 1 − GY 1 − GY
Substituting the labor demand schedule we get: lt wt
GY gt ϕ (1 − GY ) + 1 = 0 =
Geometrically, the effects of a g shock are easily analyzed: the labor demand schedule is unaffected, and labor supply shifts rightward as the household feels poorer and is willing to work more at a given real wage (this is the negative wealth effect on hours). Moving along the labor demand, this implies that hours increase and wages are constant. If labor demand were downward sloping (as in the more general model with capital) as opposed to horizontal, we would get a fall in real wage and a (somewhat smaller) increase in hours worked and hence in output. This is the sense in which government spending shocks make the RBC model fare better in what concerns real wage fluctuations. Since the conditional response of the real wage to technology is highly procyclical, appending one more
2.2. GOVERNMENT DEBT AND RICARDIAN EQUIVALENCE
15
shock that implies a (conditionally upon the new shock) countercyclical real wage means that the unconditional (or more precisely, conditional upon both shocks) response of the real wage will be less procyclical, or even acyclical. However, a shock to g also has unattractive implications. While, as noted above, it implies an increase in hours worked (and in output, since without technology yt = lt ) it also generates a fall in private consumption: ct = −ϕ
GY gt . ϕ (1 − GY ) + 1
Intuitively, when the household is taxed it feels poorer and has two margins to adjust: labor and consumption. The relative size of the adjustment along these two margins is governed by the elasticity of labor supply. When labor is inelastic (ϕ very high), most adjustment takes place by reducing consumption. When labor is elastic (ϕ very low) the response of consumption becomes negligible and most adjustment takes place through increasing hours worked. This finding (the negative correlation between consumption on one hand and hours worked and output on the other) is at odds with the data.
2.2
Government debt and Ricardian equivalence
We now re-introduce intertemporal features in the fiscal policy model by assuming that the government can issue bonds that the household purchases for saving purposes. For simplicity, we assume that all bonds are riskless, one-period discount real bonds paying one unit of the consumption good at the end of the period. The budget constraint of the household becomes: d (1 + Rt )−1 Bt+1 + Ct + Tt = Btd + Wt Lt ,
(2.8)
d where Btd are the bonds the household enters period t with, Bt+1 the bonds that the household decides to purchase during period t at price (1 + Rt )−1 . Note that Bt is a state variable, whose initial value B0 is dictated by history.
Exercise 4 What is the transversality condition in this economy? (Hint: try to rule out 2 scenarios: one in which the household accumulates too many assets by the ’end of time’ and one in which the household ends up with debt -a negative Btd !- at the ’end of time’). The Euler equation is, as we found before: (1 + Rt )−1 where Λt,t+1
= Et [Λt,t+1 ] , Ct UC (Ct+1 ) = β =β , UC (Ct ) Ct+1
(2.9)
is the stochastic discount factor, that would be used to price any assets as discussed previously. For riskless bonds, this equation states that the price of a
16
CHAPTER 2. FISCAL POLICY IN GENERAL EQUILIBRIUM
discount bond (which is equal to the inverse of its gross return) is the discounted value of the ratio of marginal utilities at two successive points in time. The government budget constraint dictating debt dynamics is: (1 + Rt )
−1
s Bt+1 + Tt = Bts + Gt ,
(2.10)
i.e. total government revenues from taxation and issuance of new bonds equal total spending on goods and on repaying existing debt. Asset market clearing will imply d s Bt+1 = Bt+1 ≡ Bt+1 .
Note that we need one more equation dictating the dynamics of tax revenues, i.e. specifying how much of an increase in spending is covered by debt issuance and how much by increased taxes. We specify this arbitrarily for now, for reasons that will become clear: Tt = Φt (Xt ) , where X could contain any of the variables in the model. Substituting the asset market clearing condition, the government budget constraint (debt dynamics equation) and the equations from the firm’s side that are the same as in the case without debt into the household’s budget constraint we obtain the economy resource constraint: Yt = Ct + Gt .
(2.11)
Now think about it for a minute. We want to find the equilibrium of this economy, i.e. the decision rules for C, Y, L and prices, namely W and R. We can do this by solving the following 5-equation system: = Ct + Gt . = At Lt 1 vL (Lt ) = Wt Ct Wt = At Ct (1 + Rt )−1 = βEt Ct+1 Yt Yt
(2.12)
Proposition 5 ’Ricardian neutrality’. Debt (or equivalently, the timing of taxation) is irrelevant for the equilibrium allocation 2.12. This can be seen by direct inspection of 2.12. Note that the first four equations are identical to the ones used to determine equilibrium in the extreme case without government debt. The fifth equation merely defines the interest rate (price of bonds) of this economy that is consistent with optimal consumption choices. The timing of taxation is completely irrelevant, as is the number of bonds issued. If the government cuts taxes today, the household correctly anticipates that taxes will be increased in the future. It
2.3. THE FISCAL THEORY OF THE PRICE LEVEL
17
hence increases its bond holdings (’lends’) by buying the bonds that the government issues to cover the tax cut. (The opposite holds for a tax increase.). The equilibrium path of the interest rate insures consistency with these choices. Note that this is a very special result, which fails as soon as we depart from our special assumptions (the next exercise makes you think about this issue). The aspiring macroeconomists amongst you should really read Chapter 13 (and Chapter 10) of Ljungquist and Sargent for a more general exposition of Ricardian equivalence in an asset pricing context. Notably, they allow the government to issue any type of bonds, not just riskless one-period bonds as assumed here. Exercise 6 Enumerate three different assumptions under which Ricardian equivalence would fail. Explain why.
2.2.1
Debt sustainability
Here is where debt does make a difference though. Let us still work under the assumption that the path of taxation is determined somehow arbitrarily and see how far we can get. The budget constraint of the government iterated forward (we iterate forward because Rt > 0) gives: Bts Bts Bts
s = (1 + Rt )−1 Bt+1 + Tt − Gt → ∞ s = lim Et Λt,T BT + Et Λt,t+j (Tt+j − Gt+j ) → T →∞
= Et
∞
j=0
Λt,t+j (Tt+j − Gt+j )
(2.13)
j=0
This is the intertemporal government budget constraint; it states that the path of taxes cannot be chosen entirely arbitrarily, but rather to ensure that this constraint holds for any sequence for Gt and any prices. Note that the process for Tt+j is the only process that can deliver this adjustment, since Gt+j , Λt,t+j and Bts are given: the first one exogenously, the second one by equilibrium Λt,t+j = ji=0 (1 + Rt+i )−1 and the third one because it is a state predetermined variable. Therefore, the tax process Tt = Φt (Xt ) needs to be chosen so that to satisfy this intertemporal constraint.
2.3
The Fiscal Theory of the Price Level
The above assumed that all government debt is real, i.e. denominated in apples and oranges. Most government debt, however, is nominal, i.e. denominated in currency units. This is a key point, since (2.13) is a constraint only when bonds are real. Some really smart people believe in what is called: the fiscal
18
CHAPTER 2. FISCAL POLICY IN GENERAL EQUILIBRIUM
theory of the price level, FTPL. If bonds are nominal (2.13) reads: ∞ BtN = Et Λt,t+j (Tt+j − Gt+j ) , Pt j=0
(2.14)
where Pt is the price level of the economy (i.e. the price of the final good) and BtN is the nominal value of bonds (this is the relevant state variable now). Bluntly, FTPL says that if the path of budget surpluses Tt+j −Gt+j is exogenous, since Bts is given the only thing that can ensure adjustment is Pt , the price level of this economy! Therefore, the price level is determined by fiscal policy!!! In this case, the equation above is NOT a constraint but a valuation equation for the real value of government debt, which can end up being an equation that determines the price level. This case is labelled ’non-Ricardian’ (see Woodford 2003, e.g.) for very good reasons: the path of taxation ends up influencing the price level and hence all variables in the economy. If instead the surplus is determined to ensure that (2.14) holds for any price level Pt fiscal policy is labelled ’Ricardian’ and then the price level will not be determined unless we introduce a specific type of monetary policy. We have no time to get into details about this fascinating, counterintuitive and highly controversial topic - see e.g. John Cochrane’s paper ’Money as Stock’ (and the references therein) for a nice exposition of these ideas.
2.4
Distortionary taxation and Ricardian equivalence
Suppose that government spending is financed via a tax on labor income at rate τ˜t and there are no lump-sum taxes available. The production function and the real wage equation are unchanged, and so is the Euler equation for the choice of bond holdings. The government budget constraint reads (note that labor income is equal to total income since the production function is homogenous of degree one): s (1 + Rt )−1 Bt+1 + τ˜t Yt = Bts + Gt . The budget constraint of the households becomes: d (1 + Rt )−1 Bt+1 + Ct = Btd + (1 − τ˜t ) Wt Lt ,
(2.15)
The only optimality condition that modifies concerns the choice of consumption versus hours worked, which is now: vL (Lst ) = (1 − τ˜t ) Wt
1 Ct
(2.16)
Finally, we need one more equation: we have to specify a rule for the adjustment of tax rates τ˜t = Φτt (Xt ) . (2.17)
2.5. THE EFFECTS OF GOVERNMENT SPENDING SHOCKS UNDER DISTORTIONARY TAXATION19 To see whether Ricardian equivalence holds, we want to find out whether this temporal path of taxes has any effect on the equilibrium allocation. Note that by combining the household and government budget constraints with the bond market clearing equation and recalling that Yt = Wt Lt we find (or by Walras’ law): Yt = Ct + Gt . Let us list the equilibrium conditions as we did in the lump-sum taxes case in 2.12: Yt Yt
= Ct + Gt . = At Lt 1 − τ˜t vL (Lt ) = Wt Ct Wt = At Ct −1 (1 + Rt ) = βEt Ct+1
(2.18)
The path of taxation DOES matter now, and hence Ricardian equivalence fails, since the tax rate affects the intratemporal optimality condition governing the leisure-consumption trade-off. Therefore, we need the tax rule 2.17 in order to complete equilibrium description in 2.18. Moreover, we need to ensure that the government’s intertemporal budget constraint holds: Bts = Et
∞
Λt,t+j (˜ τ t Yt+j − Gt+j )
j=0
2.5
The effects of government spending shocks under distortionary taxation
We can understand how exactly does the intertemporal path of taxation matter for the equilibrium allocation by studying, as before, the loglinearized equilibrium. But before we do that, note that we can easily calculate the effects of shocks even without loglinearizing in the balanced-budget case, i.e. whereby τ˜t Yt = Gt . Substitute the consumer budget constraint Ct = (1 − τ˜t ) Wt Lt , into the intratemporal optimality condition to get vL (Lt ) =
1 − τ˜t 1 Wt = (1 − τ˜t ) Wt Lt Lt
Hours will now be constant (in levels, so both in steady state and outside!), as opposed to the lump-sum taxes case. For our specification for v, : 1
Lt = L = χ− 1+ϕ
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CHAPTER 2. FISCAL POLICY IN GENERAL EQUILIBRIUM
The intuition for this is: if government spending is fully financed via increased tax rates, these rates do nothing else than reduce the after-tax real wage; there is no separate income/wealth effect of taxation (as was the case under lumpsum taxes). But for the assumed log utility function in consumption, income and substitution effects of real wage variations cancel out, so the labor supply is constant. Since there is no intertemporal substitution (no capital and no government debt), hours will not vary at all over time. We will then have (in the absence of technology): Yt = Y = L, W = 1 The only variables that do move are consumption and tax rates. Tax rates increase and consumption falls: τ˜t Ct
Gt Y = (1 − τ˜t ) Y = Y − Gt =
Let us now consider the general case, whereby part of the increase in G is financed by issuing government debt. We look at the loglinearized equilibrium conditions in 2.18, together with the loglinearized versions of the tax rate rule and the government’s budget constraint. Let’s take first the equations in 2.18 (we use the economy resource constraint to replace household’s budget constraint and assume technology is constant): yt yt wt ϕlt ct
= (1 − GY ) ct + GY gt = lt = 0 1 = wt − τ t − ct 1−τ = Et ct+1 − rt
(2.19)
Importantly, I have defined τ t ≡ τ˜t − τ , where τ is the steady-state value of τ˜t ; this is similar to how we defined the interest rate’s deviation from steady-state, since the tax rate is also a rate (hence it is already in percentage points!) and we need not take logs of it. One general point about the distortionary-tax model, and about why the intertemporal path of tax rates influences the equilibrium allocation, should be noted. Substitute the intratemporal optimality condition in the Euler equation, just as we did before, to obtain (since wt = 0): ϕ (lt − Et lt+1 ) =
1 (Et τ t+1 − τ t ) + rt 1−τ
This equation describes intertemporal substitution in labor supply induced by tax rate variations. When the tax rate is expected to be higher tomorrow than it is today, the household optimally decides to work more today (because today
2.5. THE EFFECTS OF GOVERNMENT SPENDING SHOCKS UNDER DISTORTIONARY TAXATION21 she is taxed less). This is how taxation at a time-varying rate distorts the labor supply decision of the household. Moreover, note that, as before, GY is something we know (steady-state share of government spending in GDP). We now need to turn to the ’fiscal block’. First, we have one exogenous process dictating the dynamics of gt , for instance an AR(1) process just as for technology. The remaining two equations are studied in detail in the next subsection.
2.5.1
Debt dynamics and the ’fiscal rule’
Loglinearizing the government budget constraint yields (use the tricks, and let Bt −B B ˆ Y = BY , bt B ):
(2.20) βBY ˆbt+1 − rt + τ yt + τ t = BY ˆbt + GY gt I will make the simplifying assumption that budget is balanced in steady-state, i.e. BY = 0, τ = GY . Because B is zero, we can no longer use ˆbt BtB−B , so instead we will define bt ≡ BtY−B = BYt = BY ˆbt , i.e. debt as a share of steady-state GDP. replacing these elements in 2.20 and keeping only bt+1 on the left hand side we get: bt+1 = β −1 bt + β −1 τgt − β −1 (τ yt + τ t )
(2.21)
You now see clearly why we need a ’fiscal rule’. If we ignored the last term, debt would be given by a purely autoregressive process with root β −1 > 1, and hence exploding. But by making total tax revenues respond to government debt we will be able to avoid this problem. Therefore, let us assume that tax revenues are determined by the following: τ t + τ yt = φg τ gt + φb bt
(2.22)
This rule says that tax revenues respond to government spending and government debt. When φg = 1 and there is no debt initially, (bt = 0) , this is the balanced-budget rule we had before: tax revenues respond one-to-one to increased spending. When φg < 1, there is some deficit financing as not all of the increase in spending is accommodated by taxes. Finally, note that this rule assumes automatic stabilization: tax rates do not have to increase proportionally to the increase in spending, if the latter induces an expansion in the tax base, i.e. in yt . In fact, if this expansion is strong enough the tax rate may even fall - but since yt is itself an endogenous process we cannot study this separately. Finally, the response to the outstanding level of debt φb will ensure that debt itself will not explode. To see this, substitute 2.22 into 2.21 to get: bt+1 = β −1 (1 − φb ) bt + β −1 τ 1 − φg gt (2.23)
22
CHAPTER 2. FISCAL POLICY IN GENERAL EQUILIBRIUM
This equation is uncoupled with the rest and we can solve it backwards if and only if: β −1 (1 − φb ) < 1 → φb > 1 − β. Doing so (do it!!!), you obtain current debt as the present value of all past government spending.
2.5.2
Crowding-in of private consumption
Equilibrium is fully described by 2.19, together with 2.22 and 2.23. We are now ready to study the responses of all variables to a government spending shock. To begin with, since the real wage is constant by assumption, wt = 0, the after-tax real wage changes only because the tax rate changes: wt −
1 1 τt = − τ t. 1−τ 1−τ
The intratemporal optimality condition becomes: ϕlt = −
1 τ t − ct . 1−τ
Substitute the production function yt = lt and the resource constraint yt = (1 − τ ) ct + τgt where we used GY = τ to get: [1 + ϕ (1 − τ )] ct = −
1 τ t − ϕτgt 1−τ
(2.24)
You already see from this equation that consumption may indeed increase in response to government spending shocks if and only if the response of taxes is such that: τ t < −ϕτ (1 − τ ) gt . That is, the tax rate has to fall endogenously ’enough’, and the more so, the more inelastic is labor supply. We can derive an exact parametric condition for the crowding-in of consumption by substituting the tax rule into 2.24: [1 + ϕ (1 − τ)] ct = −
1 φ τ gt + φb bt − τ yt − ϕτ gt 1−τ g
and substituting the resource constraint for yt : [1 + ϕ (1 − τ )] ct = −
1 φg τ gt + φb bt − τ (1 − τ ) ct − τ 2 gt − ϕτ gt 1−τ
Rearranging: (1 + ϕ) (1 − τ ) ct = τ
τ − φg 1 − ϕ gt − φ bt 1−τ 1−τ b
(2.25)
2.5. THE EFFECTS OF GOVERNMENT SPENDING SHOCKS UNDER DISTORTIONARY TAXATION23 This equation gives the equilibrium response of private consumption to government spending shocks and last period’s debt. Assuming that we start from steady state, and therefore bt = 0 initially, the impact multiplier of government spending on consumption is positive if and only if: ϕ<
τ − φg 1−τ
There are three interrelated conditions for g to crowd in private consumption: labor has to be elastic enough (ϕ low), φg has to be low - enough deficit financing; and τ has to be high enough. These three features lead to a fall in the tax rate and an increase in consumption for the following intuitive reasons. A highly elastic labor supply ensures that labor reacts as much as possible to changes in tax rates, therefore expanding the ’tax base’; when this effect is strong enough, the tax base expands so much that the tax rate falls. A high degree of deficit financing ensures that the tax rate does not increase by much on impact (just look at the fiscal rule), while a high τ ensures that the automatic stabilization part is significant enough - i.e., for a given increase in yt tax revenues will increase automatically more when τ is higher, calling for a lower increase in tax rates. The effect on output and hours is easily found using the resource constraint: yt (= lt ) =
τ 1 − φg 1 φb gt − bt 1+ϕ 1−τ 1−τ 1+ϕ
As expected from the above discussion, the multiplier on output and hours is also higher when τ is higher, φg is lower, and labor is elastic (ϕ low). The equilibrium tax rates evolve according to: τ 1 − φg 1 + ϕ (1 − τ ) τ t = τ φg − gt + φb bt . 1+ϕ 1−τ (1 − τ) (1 + ϕ) From period t + 1 onwards, an expansion in consumption is harder to obtain since debt will have increased and needs to be paid back by an increase in taxes. Indeed, at any time t + j we have: τ − φg 1 (1 + ϕ) (1 − τ ) ct+j = τ − ϕ gt+j − φ bt+j (2.26) 1−τ 1−τ b where bt+j is given by 2.23 evaluated at time t + j as the present value of past government spending: bt+j = β −1 (1 − φb ) bt+j−1 + β −1 τ 1 − φg gt+j = (2.27) j = β −1 τ 1 − φg β −i (1 − φb )i gt+j−i .
(2.28)
i=0
This increase in debt will generally lead to a fall in consumption in later periods.
24
CHAPTER 2. FISCAL POLICY IN GENERAL EQUILIBRIUM
What is (not) satisfactory about this model? To answer this question we need to think of two interrelated issues: 1. are the parameter restrictions making this happen reasonable?; and 2. is the mechanism by which we obtained this result realistic? Two of the parametric requirements for a positive consumption multiplier have empirical support : much spending is indeed deficit-financed; and there is a lot of automatic stabilization, precisely because much of taxation is linked to the level of activity (is distortionary). But the remaining requirement, as we learned already, does not have enough support - the elasticity of labor supply is not very high. Moreover, the very mechanism at the heart of this model is likely to be unrealistic: empirical estimates reveal that ’tax base’ effects are not likely to be relevant; moreover, tax rates do not vary significantly at business cycle frequency. However, on the last point one can argue that the model’s τ t is not a particular tax rate, but the effective tax rate on labour (and estimates of effective tax rates are known to be time-varying).
2.6
Optimal fiscal policy: a primer
Let us look at our model in which government spending is financed by distortionary taxation: the government charges a proportional tax rate τ t for each unit of income received by the household. We want to ask ourselves a series of normative questions, namely how should the government pick the path of taxes that finance its spending; we will assume for simplicity that there is no government debt, i.e. Bt = 0. We study two environments: a first-best one, in which the government has unlimited access to lump-sum instruments; and a second-best environment, in which the government can only use distortionary instruments. In order to talk meaningfully about welfare, we need to compute the first-best, i.e. the optimal equilibrium that t he planner would choose, similarly to how we computed the planner equilibrium in the RBC model. For this purpose, we assume that the planner takes the path of spending Gt as given. The planner chooses quantities Ct and Lt to maximize utility ln Ct − v (Lt ) subject to the resource constraint Ct + Gt = Yt = At Lt , where Gt is given. As in the competitive equilibrium without government debt, there is no intertemporal dimension of the problem, so the only optimality condition concerns the intratemporal choice between consumption and leisure: 1 vL (Lt ) = At (2.29) Ct It is immediately apparent that the equilibrium occurring here is different from the competitive one studied above. In the competitive equilibrium, distortionary taxation induces a wedge 1 − τ˜t between the marginal rate of substitution between consumption and labor and the marginal rate at which labor is transformed into the consumption good. In the planner equilibrium, this wedge is absent and the two rates are equalized. Optimal policy in a first-best environment.
2.6. OPTIMAL FISCAL POLICY: A PRIMER
25
We assume that the planner has access to unlimited lump-sum instruments to finance a given level of spending (this is why we call this a ’first-best environment’). The two optimality conditions become identical iff the tax rate is zero at all times, τ ∗t = 0. Government spending would hence be financed via -fully available- lump-sum taxes Tt∗ = Gt . Solving the ’Ramsey Problem’ (optimal policy in a second-best environment)1 . Suppose that the planner works in a second-best environment, i.e. there are no lump-sum instruments to ensure implementation of the first-best found above. We can find the optimal value of the tax rate in two ways (remember, importantly, that we are still in the balanced-budget case!): Dual Approach. First, we assume that the planner ’sees’ the private decision rules of the household and firms, that are contingent upon the tax rate: Ct (τ t ) , Lt (τ t ) , Yt (τ t ) , etc. and chooses the path of tax rates τ t in order to maximize the utility of the representative household (this is called ’the dual approach’ to the Ramsey problem). The private decision rules for the balanced-budget case are (I have already substituted Wt = At ): 1 − τ˜t At Ct ≤ (1 − τ˜t ) At Lt ,
vL (Lt ) = Ct
(2.30)
which yield: 1 ¯ solution to vL (Lt ) Lt = 1 At → Lt = L (1 − τ˜t ) At Lt ¯ τ t ) = (1 − τ˜t ) At L Ct (˜
vL (Lt ) =
Solving the Ramsey problem directly implies solving: ¯ τ t ) , Lt (˜ τ t )) → max ln Ct (˜ τ t) − v L max U (Ct (˜ τ˜ t
τ˜t
s.t. Gt
≤
¯ τ˜t At L
The solution of this maximization problem after substituting the private sector decision rules which act as constraint on the government’s problem is simply: τ˜∗∗ t =
Gt ¯ At L
Substituting this in the budget constraint implies: ¯ − Gt Ct = At L Primal Approach 1 For
more on solving the Ramsey problem and on the primal/dual approaches see e.g. Ch 15 in Ljungquist and Sargent and references therein.
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CHAPTER 2. FISCAL POLICY IN GENERAL EQUILIBRIUM
Second, assume that the planner solves this problem ’indirectly’ (the primal approach), i.e. she chooses consumption and hours rather than the tax rate, taking as constraints the optimality conditions of the household, firms, etc. For this second approach we try to eliminate the tax rate, such that we have only private-sector variables in the relevant constraints (the resource and implementability constraints). To eliminate the tax rate from the constraints we multiply the intratemporal optimality condition 2.30 by Lt to get the ’implementability constraint’: vL (Lt ) Lt =
1 − τ˜t At Lt , Ct
and using the private budget constraint we have: ¯ same as for the direct approach. vL (Lt ) Lt = 1 → Lt = L, Substituting the government budget constraint into the private budget constraint we obtain the resource constraint: Ct + Gt ≤ At Lt These two equations are the constraints for government’s ’Ramsey allocation problem’ to: max U (Ct , Lt ) . Ct ,Lt
Since Lt is fixed, the solution of this problem is simply (as for the direct approach!): ¯ − Gt Ct = At L which implies (by ’inverse mapping’ back into policies) τ˜∗∗∗ = τ˜∗∗ t t =
Gt ¯ At L
Note that we get the same solution regardless of whether we solve the problem by the primal or dual approach. How does this solution (the tax rate) compare with the tax rate found in the first-best environment? The ’Ramsey’ tax rate is different from the first-best one precisely because we are in a secondbest environment: having no lump-sum instrument available, and facing a fixed labor supply by the households, the government simply chooses a tax rate that ensures the budget is balanced every period. This is an almost trivial result, but on the way to obtaining it you have learned the main ideas and concepts behind solving Ramsey problems2 .
2 For more on solving the Ramsey problem and on the primal/dual approaches see e.g. Ch 15 in Ljungquist and Sargent and references therein.
Bibliography [1] Baxter, M., and R. G. King (1993): “Fiscal Policy in General Equilibrium,” American Economic Review 83: 315-334. [2] Christiano, L., and M. Eichenbaum (1992): “Current Real-Business-Cycle Theories and Aggregate Labor-Market Fluctuations,” American Economic Review 82: 430-450 [3] Christiano, L., M. Eichenbaum and Evans (2005): ”Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy“, Journal of Political Economy, vol. 113, no. 1, 1-46 [4] Cochrane, J, 2004, ’Money as Stock’, Forthcoming in Journal of Monetary Economics [5] Galí, J., J. D. López-Salido, and J. Vallés (2002): “Understanding the Effects of Government Spending on Consumption,” mimeo, CREI. [6] Kiyotaki, N and Wright, R., 1989, On money as a medium of exchange, Journal of Political Economy [7] Kocherlakota, N. 1998 "Money is memory", Journal of Economic Theory [8] Ljungquist, L. and T. Sargent (2004), Recursive Macroeconomic Theory, 2nd Edition, MIT Press, Cambridge, MA. [9] Walsh, C. "Monetary Theory and Policy", MIT Press [10] Woodford, M., (2003), Interest and prices: foundations of a theory of monetary policy, Princeton University Press, Princeton, NJ.
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