# Berkeley ECON 202A - Lecture Outline 4 (15 pages)

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## Lecture Outline 4

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- School:
- University of California, Berkeley
- Course:
- Econ 202a - Macroeconomic Theory

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Economics 202A Lecture Outline 4 version 1 3 Maurice Obstfeld Government Debt and Taxes As a result of the events of September 2008 government actions to underwrite the U S nancial system coupled with a massive recession and a huge scal stimulus plan are sharply increasing the U S federal debt Leaving aside the fascinating questions raised by the nancial crisis itself how do macroeconomists think about government debt and its e ects Should government debt matter at all after all leaving aside the possibility of borrowing from foreigners we owe any public debt to ourselves Because one logical possibility is that government debt somehow a ects capital accumulation and growth it is natural to consider the question in the context of our growth models The leading breakthrough on the subject is Peter A Diamond s American Economic Review 1965 adaptation of Paul A Samuelson s overlapping generations model to incorporate capital growth and public debt Incidentally this paper was written when Diamond was on the faculty here in Berkeley We shall study the Diamond model soon but before doing so we take a look at the debt question within the Ramsey Cass Koopmans RCK dynastic family setup There the answers are less interesting and perhaps less intuitive yet they provide an essential benchmark case for understanding the Diamond model s very di erent predictions Within the RCK framework we now wish to distinguish between the private sector and the government two sectors that add up to be the total economy of course As we are now therefore dropping the idea that a government planner makes allocation decisions we need to observe following basic welfare economics that the RCK allocation can be decentralized if private agents face the time path of real interest rates corresponding to that optimal allocation rt f 0 kt and earn real wages per unit labor given by the marginal product of labor wt f kt 1 f 0 kt kt Following Diamond 1965 I assume that the depreciation rate of capital is 0 otherwise the real interest rate would be r f 0 k A key step in showing this is to contemplate the government and private sectors budget constraints separately With respect to the private sector household assets at the start of period t are the sum of capital Kt and debt issued by the government Dt If we rede ne these stocks in per capita terms as kt and dt and also assume that the household pays per capita lump sum taxes t to the government each period then we may write the private asset accumulation equation in terms of real per capital wealth a k d as at 1 1 1 rt at wt 1 n t ct Above rt is the interest paid during period t on assets accumulated over t 1 It is now easy to see that if consumers invest at the real interest rate rt 1 between dates t and t 1 then the relevant Euler equation of optimality would be u0 ct 1 rt 1 u0 ct 1 1 At the same time the government s debt evolves according to the equation dt 1 1 1 rt dt 1 n t gt where g is per capita consumption of goods by the government Since debt represents negative assets simply subtract the second of these from the rst to get 1 1 rt kt wt 1 n 1 f kt kt ct 1 n kt 1 ct gt gt the aggregate relationship from the RCK model where g 0 The private and public asset stock ow relationships above imply in nitehorizon intertemporal budget constraints for the two sectors For the private 2 sector for example we may write for t 0 w0 1 n 0 a1 1 r0 1 r0 1 n c1 w1 c0 w0 0 1 r0 1 r0 1 r1 t 1 X Y 1 n 1 ct wt 1 r0 t 0 s 1 1 rs t Y 1 n 1 n lim at 1 t 1 1 r s s 1 a0 c0 1 1 n 1 r0 1 n 1 r1 a2 t Consider the reasonableness of imposing on households the condition that lim t 1 t Y s 1 1 n 1 rs at 1 0 In the Ramsey economy we can never have a negative capital stock But in the decentralized economy where households borrow subject to a real interest rate we can imagine someone borrowing to consume and then always borrowing more to repay the previous loans thereby never repaying at all The preceding inequality constraint rules out such a Ponzi scheme and thus is called the no Ponzi game constraint 1 Imposing it we obtain the intertemporal constraint t 1 X Y 1 n ct wt 2 1 r0 a0 t 1 rs t 0 s 1 This restrictions says that household initial assets along with their payout must cover any discounted excess of consumption over after tax wage income Can you see using 1 why the transversality condition will normally ensure that in equilibrium this condition holds as an equality The government faces an analogous constraint the excess of its tax receipts net of public spending discounted to the present must cover at least 1 If we do not impose such a constraint then anyone can consume in nite resources and there would be excess demand for output 3 its initial debt to the private sector Because government assets are equal to d we may write the public sector constraint as t 1 X Y 1 n 1 r0 d0 gt 3 t 1 r s t 0 s 1 In words this implies just multiply the last inequality through by 1 that the discounted present value of primary government surpluses g must be at least as big as the government s total initial debt obligations 1 r0 d0 Putting the last two inequality constraints together leads to t 1 X Y 1 n ct gt wt 4 1 r0 k0 1 rs t 0 s 1 for the economy as a whole 2 The proposition I now wish to explore is the neutrality of public debt in this economy with lump sum taxes and a single representative family The proposition is known as the Ricardian equivalence of debt and future taxes Suppose the government increases its own initial debt by showering a gift d of government bonds on people at the start of period 0 Think of the recent U S scal stimulus package To nance the payments on this debt the government raises taxes intertemporally perhaps far in the future by the amount t 1 X Y 1 n d t 1 rs t 0 s 1 recall that denotes per capita taxes in 3 Notice that this experiment changes the left hand and right hand sides of the household constraint 2 by equal amounts there is no change in intertemporal household consumption possibilities Accordingly private consumption behavior also is unchanged In other words the gift of government debt does not represent net wealth 2 In the RCK model with government consumption we would have kt 1 1 kt f kt ct gt 1 n …

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