1 With state-contingent debt

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1 STOCKHOLM DOCTORAL PROGRAM IN ECONOMICS Helshögskolan i Stockholm Stockholms universitet Paul Klein paul.klein@iies.su.se URL: Macroeconomics II Spring 2010 Lecture 2: Ramsey optimal taxation under uncertainty 1 With state-contingent debt Consider an environment without capital where a representative consumer maximizes [ ] E β t u(c t, l t ) where the resource constraint is t=0 c t + g t = h t = 1 l t where {g t } is a stochastic process living on the probability space (Ω, F, P ). Let {F t } be the filtration generated by {g t }. Suppose g 0 is deterministic so that F 0 = {, Ω}. Markets are complete so that all contingent claims are associated with a well-defined price. Moreover, there are no limits on the quantities of contingent claims held in any particular period. Let {Q t : F R + } be a sequence of price measures with the following interpretation. The contingent claim that delivers 1 unit of consumption in period t if ω A has price Q t (A). We normalize so that Q 0 (Ω) = 1. Let W t be the (after-tax) leisure price measure. 1

2 Meanwhile, define the stochastic processes p t w t via dq t = p t dp on F t dw t = w t dp on F t. With these definitions, the consumer s budget constraint (conceived as an equality) can be written as [ ] E (p t c t w t h t ) = b 0. t=0 By the consumer s optimality conditions, p t = β t u c,t u c,0 w t = β t u h,t u c,0. Hence the implementability condition becomes [ ] E β t (u c,t c t u l,t h t ) = u c,0 b 0. t=0 The equivalent sequence-of-markets representation of this condition is to assert the existence of an {F t }-adapted stochastic process {b t } with initial value b 0 which satisfies βe[u c,t+1 b t+1 F t ] + u c,t c t u l,t h t = u c,t b t for all t = 0, 1, 2,... lim t βt E [u c,t b t ] = 0. To compute the Ramsey equilibrium, you again have to fix λ, solve the model check implementability. If satisfied, done. If not, adjust λ. How to check implementability? Simulate the solution thouss of times with different seeds for the rom number 2

3 generator. Take the average to compute an approximation of the expected value of the left h side. What can be said about optimal policy in this framework? Well, one thing is that statecontingent government debt will be used as a shock absorber. If g t is above its mean, the return on government debt will be low vice versa. For an analysis of optimal tax policy over the business cycle, see Zhu (1992) Chari et al. (1994). 2 Without debt State-contingent debt is an attractive idea but it is also perhaps a bit artificial. What if there is no government debt at all? The balanced budget constraint in this case becomes: u c,t k t = βe [u c,t+1 (c t+1 + k t+1 ) u l,t+1 h t+1 F t ]. Can you derive it? 3 With risk-free debt Aiyagari et al. (2002), in a paper based on Marcet et al. (1996), consider a world where debt is not state-contingent, i.e. a government bond must be a riskless asset whose return is determined one period in advance. This topic is also covered in Ljungqvist Sargent (2000). It is natural to impose these constraints on the sequence-of-markets representation rather than the present-value representation. We had βe[u c,t+1 b t+1 F t ] + u c,t c t u l,t h t = u c,t b t 3

4 we now impose We get E[b t+1 F t ] = b t+1. βe[u c,t+1 F t ]b t+1 + u c,t c t u l,t h t = u c,t b t. In order for this predictability of debt to have any bite, we also impose b b t+1 b for each t = 0, 1, 2,... though we will ignore this constraint, hoping that it doesn t bind in equilibrium. Incidentally, these bounds guarantee A particularly simple case is lim t βt E [u c,t b t ] = 0. u(c, l) = c 1 2 (1 l t) 2 With these preferences, the government s budget constraint is βb t+1 + c t h 2 t = b t. Imposing the resource constraint c t = h t g t, we have βb t+1 + h t (1 h t ) = b t + g t Using the consumer s foc, we get τ h t = 1 h t hence βb t+1 + τ h t h t = b t + g t a result we might have figured out without so much fuss. 4

5 Now define the Hamiltonian via H = β t [h t g t 1 2 h2 t ] + β 1 λ t+1 [b t h t + g t + h 2 t ]. The optimality conditions are β t [1 h t ] + β 1 E t [λ t+1 ]( 1 + 2h t ) Now define We get E t [λ t+1 ] = βλ t. µ t = β t λ t. 1 h t + E t [µ t+1 ]( 1 + 2h t ) = 0 E t [µ t+1 ] = µ t which means that µ t is a martingale. Meanwhile, we have Solving for h t, we get 1 h t + µ t (2h t 1) = 0. h t = µ t 1 2µ t 1. What about the labour income tax rate? Apparently so τ h t = 1 h t τ h t = µ t 2µ t 1 so that although the tax rate is not a martingale, it is a function of a martingale therefore quite persistent, certainly more persistent than the underlying shock. 5

6 References Aiyagari, S. R., A. Marcet, T. Sargent, J. Seppala (2002). Optimal taxation without state-contingent debt. Journal of Political Economy 110, Chari, V. V., L. J. Christiano, P. J. Kehoe (1994). Optimal fiscal policy in a business cycle model. Journal of Political Economy 102 (4), Ljungqvist, L. T. Sargent (2000). Recursive Macroeconomic Theory. MIT Press. Optimal taxation without state- Marcet, A., T. J. Sargent, J. Seppälä (1996). contingent debt. Manuscript. Zhu (1992). Optimal fiscal policy in a stochastic growth model. Journal of Economic Theory 58,