Internet Appendix for: Social Risk, Fiscal Risk, and the Portfolio of Government Programs

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1 Internet Appendix for: Social Risk, Fiscal Risk, and the Portfolio of Government Programs Samuel G Hanson David S Scharfstein Adi Sunderam Harvard University June 018 Contents A Programs that impact the tax base B Optimal debt management problem 4 C Approximate model solution with no borrowing 6 1

2 A Programs that impact the tax base In this section, we show how the analysis should be modified if the program under consideration adds to the tax base Suppose that the program adds to the tax base so that tax revenue is given by T t = l t τ t Y t + W t q = 1 ητ t τ t Y t + W t q Thus, household consumption is given by C t l t = Y t + W t q l t 1 τ t l t 1 + η η + Net trade govt bonds t = Y t + W t q l t 1 τ t l t 1 + η η + T t G t X t q = Y t + W t q l t l t 1 + η η G t X t q since T t = l t τ t Y t + W t q ] = Y t + W t q 1 ητ t 1 ητ t 1 + η η G t X t q since l t = 1 ητ t ] = Y t + W t q 1 1 ητ t G t X t q simplifying] The optimal tax rate is Thus, we have so And we have so C t τ t τ t = τ t T t = 1 1 4η T t Y t+w tq τ t Y t + W t q, C t τ t = ητ t = h τ t τ t T t 1 τ t τ t W t q = 1 ητ t τ t 1 τ t Y t + W t q, τ t W t q = 1 ητ t τ t ητ t 1 τ t = 1 ητ t τ t h τ t The first-order condition for D 0 is the same in the baseline case where the program has no effect on the tax base Specifically, we have 0 = u C 0 C 0 τ 0 T 0 + βe u C 1 C ] 1 τ 1 T 1 τ 0 T 0 τ 1 T 1 D 1

3 Using the fact that Ct τ t τ t T t = h τ t, T 0 = 1, and T 1 R = R f + D f 0, this implies that u C 0 h τ 0 = R f R f + D 0 βe u C 1 h τ 1 ] 1 However, the fact that the program impacts the tax base will impact the first-order condition for q As in the baseline model, the first-order condition for q still takes the form 0 = u C0 C 0 + C 0 τ 0 + βe u C1 C 1 τ 0 + C ] 1 τ 1 τ 1 But we now have C 0 τ 0 τ 0 = C 0 τ 0 W 0 τ 0 W 0 + C 0 τ 0 T 0 τ 0 T 0 = 1 ητ 0 τ 0 h τ 0 W 0 q h τ 0 X 0 q and C 1 τ 1 τ 1 = C 1 τ 1 W 1 τ 1 W 1 + C 1 τ 1 T 1 τ 1 T 1 = 1 ητ 1 τ 1 h τ 1 W 1 q h τ 1 X 1 R f q + D 0 This is because the program impacts both government expenditures and the tax base As a result the program has two independent effects on tax rates and, thus, the distortionary costs of taxation Thus, can write the effect of changing q on household consumption at time 0 as: C 0 + C 0 τ 0 τ 0 = W 0 q 1 η τ 0 X 0 q + 1 ητ 0 τ 0 h τ 0 W 0 q h τ 0 X 0 q = W 0 q 1 + τ 0 h τ h τ 0 X 0 q which follows from 1 η τ + 1 ητ τh τ = Similarly, the effect on consumption at time 1 is C 1 + C 1 τ 1 τ 1 1 η τ + 1 ητ τ ητ 1 τ = τ ητ 1 τη = τh τ = W 1 q 1 η τ 1 X 1 q + 1 ητ 1 τ 1 h τ 1 W 1 q h τ 1 = W 1 q 1 + τ 1 h τ h τ 1 X 1 q h R f τ 1 D 0 X 1 R f q + D 0 3

4 Thus, the optimal scale of the program satisfies 0 = u C 0 W 0 q 1 + τ 0 h τ h τ 0 X 0 q +βe u C 1 W 1 q 1 + τ 1 h τ h τ 1 X 1 q h R f τ 1 D 0 ] This expression can be compared to the expression in the baseline model which is 0 = u C 0 W 0 q 1 + h τ 0 X 0 q 3 ] +βe u C 1 W 1 q 1 + h τ 1 X 1 q X 1 q h R f τ 1 D 0 Comparing the two expression, we see that there is an additional benefit of the form W t q τ t h τ t associated with programs that expand the tax base When programs can add to the tax base, the need to manage fiscal risk can reinforce the government s desire to manage social risk, partially alleviating the normal tension between fiscal and social risk management Specifically, programs that add to the tax base are valuable because they help keep tax rates and the associated deadweight losses low And, programs are especially valuable if they tend to lift the tax base in time 1 states when it would otherwise decline B Optimal debt management problem Our model embeds the core intuitions present in the state-contingent debt management problem studied in Bohn 1990, Aiyagari, Marcet, Sargent, and Seppala 00, and Bhandari, Evans, Golosov, and Sargent 016 In this problem, the government chooses riskless debt and its issuance q of a risky security with time 0 price P = X 0 and state-contingent time 1 payoffs X 1 to minimize the cost of distortionary taxes This problem can be studied in our framework by setting W t = X t, so that the net benefits of the project are zero In this case, the choice of q only impacts household consumption insofar as it impacts tax revenues Specifically, the first-order condition for q says that the government wants to hedge background fiscal risk by issuing risky securities that have low returns in states where other government spending G 1 is unexpectedly high, so that the tax rate τ 1 is unexpectedly high Letting P and q denote the price of the risky security and the quantity of the risky security that the government issues, the government s budget constraints are T 0 + qp + D 0 = G 0 + D T 1 = G 1 + qx 1 + R f D 0 Letting z denote the representative household s holding of the risky asset, the household budget constraints are C 0 = Y 0 1 η τ 0 + D B 0 zp T 0 C 1 = Y 1 1 η τ 1 + zx 1 + R f B 0 T 1 4

5 Atomistic households choose their holdings of the riskless bond B 0 and of the risky security z taking prices and taxes as given Thus, the Euler equation for B 0 is and Euler equation for z is u C 0 = R f βe u C 1 ] u C 0 P = βe u C 1 X 1 ] Imposing market clearing B 0 = D 0 and q = z and substituting in the government s budget constraints into the household budget constraints, we have C 0 = Y 0 1 η τ 0 G 0 C 1 = Y 1 1 η τ 1 G 1 Thus, the government s problem is { max u Y 0 1 η q,d 0 τ 0 G 0 + βe u Y 1 1 η τ 1 ]} G 1, and where R f is implicitly defined by u Y 0 1 η τ 0 G 0 = R f βe u Y 1 1 η τ 1 ] G 1 and P is implicitly defined by u Y 0 1 η τ 0 G 0 P = R f βe u Y 1 1 η τ 1 ] G 1 X 1 Finally, τ 0 and τ 1 are defined by The first order condition for D 0 is 1 + q P Y 0 τ 0 1 ητ 0 = D D 0 + G 0 P q Y 1 τ 1 1 ητ 1 = R f D 0 + G 1 + X 1 q, h τ 0 u C 0 = βe ] u C 1 h R f τ 1 R f + D 0 where, for a small increase in D 0, 1 + q P/ is the marginal decline in tax revenue at 0 and R f + D 0 R f / is marginal increase in tax revenue at 1 The first order condition for q is P + q P ] h τ 0 u C 0 = βe u C 1 h R f τ 1 X 1 + D 0 where, for a small increase in q, P + q P/ is the marginal decline in tax revenue at 0 and X 1 + D 0 R f / is marginal increase in tax revenue at 1 Using the definition of the stochastic discount factor, this implies that at an optimum we 5

6 have Marginal fiscal return on riskless bonds Fiscal SDF {}}{{}}{ 1 = E h R τ 1 M 1 h τ 0 R f + D f q P and Marginal fiscal return on risky security Fiscal SDF {}}{{}}{ h R τ 1 1 = E M 1 h τ 0 X 1 + D f 0 P + q P Thus, both optimality conditions can be written in the standard Euler equation form 1 = E SDF RET ] Here the relevant stochastic discount factor the fiscal SDF is the product of the private SDF, M 1, and the ratio of marginal tax distortions at time 1 and 0, h T 1 /h T 0 Similarly, the relevant return here is not the private return Instead, it is a fiscal return that accounts for the general equilibrium effects that arise when u < 0 Specifically, we have R f = R fβe u C 1 h τ 1 X 1 ] + u C 0 h τ 0 P βe u C 1 R f D 0 u C 1 h τ 1 ] R f = R f βe h τ 1 u C 1 ] + u C 1 h τ 0 βe u C 1 R f D 0 u C 1 h τ 1 ] 0 0 where the former assumes that X 1 0 We also have P = u C 0 h τ 0 P + βe u C 1 h τ 1 X1] u C 0 + u C 0 h T 0 qp P = u C 0 h τ 0 P + βr f E u C 1 h τ 1 X 1 ] u C 0 + u C 0 h τ 0 qp which are ambiguous There is a parallel here to the literature on intermediary-based asset pricing He and Krishnamurthy 013], Brunnermeier and Sanikov 014] In those models, households indirectly hold financial assets via intermediaries And, because there are frictions between the household sector and the financial sector, the stochastic discount factor that prices financial assets equals households stochastic discount factors with an adjustment that captures these intermediation frictions A similar result obtains in our model because tax distortions mean that the government perceives a financing wedge term between itself and the households it represents C Approximate model solution with no borrowing In this section, we explore the approximate solution to our model in the case where the government cannot borrow at t = 0 In this case, the optimal value of q satisfies a quadratic equation that can be solved in closed form However, much like in the case where γ > 0 and η = 0 many of the comparative statics are ambiguous due to competing income and substitution effects 6

7 Suppose that D = 0 and that the government cannot borrow at t = 0 so that D 0 = 0 Then the optimal level of q satisfies where 0 = 1 γ C 0 C W 0 X 0 h + η T Y +βe 1 γ C 1 C W 1 X 1 T0 T Y 0 Y h + η T Y X 0 T1 T Y 1 Y X 1 ] T t = G t + qx t η C t = Y t Ȳ τ + h Tt T ˆη Y Yt Y + W t X t q G t Thus, we obtain η 0 = 1 γ Y 0 C Ȳ τ + h G0 + qx 0 T ˆη Y Y0 Y + q W 0 X 0 G 0 W 0 X 0 h + η T G0 + qx 0 Y 0 X 0 η 1 γ Y 1 C Ȳ +βe τ + h G1 + qx 1 T ˆη Y Y1 Y + q W 1 X 1 G 1 h W 1 X 1 + η T G1 +qx 1 Y 1 X 1 { }}{ γ W 0 X h η T X0 = q Y T +βe γ W 1 X h ] η T X1 Y T A { }}{ 1 γ Y 0 C Ȳ η τ h G0 T G 0 + ˆη Y Y0 Y η T X0 Y T γ W +q 0 X h W 0 X h + η T G0 Y 0 βe 1 γ Y 1 C Ȳ η τ h G1 T G 1 + ˆη Y Y1 Y ] η T X1 Y T βe γ W 1 X h W 1 X h + η T G1 ] Y 1 { }}{ 1 γ Y 0 C Ȳ η τ h G0 T G 0 + ˆη Y Y0 Y W + 0 X h + η T G0 Y 0 1 γ Y 1 C +βe Ȳ η τ h G1 T G 1 + ˆη Y Y1 Y W 1 X h + η T G1 Y 1 C In other words, the optimal value of q satisfies the following quadratic equation in q : B 0 = A q + B q + C 7

8 The solutions to this first-order condition are: q + = B + B 4AC A and q = B B 4AC A The second order condition for an optimum is Aq + B < 0 Thus, assuming that the discriminant B 4AC > 0, the optimum us q = q = B B 4AC A Comparative statics on q for any exogenous parameter θ, then follow from θ = θ q + B θ q + C θ] A A θ q + B θ] A θ q + B θ q + C θ ] As in the case where γ > 0 and η = 0, many of these comparative statics will ambiguous due to competing income and substitution effects Special case where γ > 0 and η = 0 Whether or not the government can borrow, the solution is: q = 1 γ E Y 1 G 1 ] C W 0 X 0 + βe W 1 X 1 ] γ W 0 X 0 + β E W 1 X 1 ] + β V ar W 1 X 1 ] Y 0 G 0 E Y 1 G 1 ] W 0 X 0 W 0 X 0 + β E W 1 X 1 ] + β V ar W 1 X 1 ] βcov Y 1 G 1, W 1 X 1 ] W 0 X 0 + β E W 1 X 1 ] + β V ar W 1 X 1 ] In this Ricardian case, we obtain the same solution whether or not the firm can borrow This is because the way that expenditures are financed in this case is irrelevant for welfare Special case where γ = 0 and η > 0 taxes over time, the solution is: When government cannot borrow at t = 0 to smooth q = W 0 X 0 + βe W 1 X 1 ] η/y X 0 + β E X 1 ] + βv ar X 1 ] ] h + η T G0 h Y 0 X η T E G 1 ] Y 1 βe X 1 ] η/y X 0 + β E X 1 ] + βv ar X 1 ] ] ] βcov X 1, G 1 T Y Y 1 X 0 + β E X 1 ] + βv ar X 1 ] ] 8

9 By contrast, when the government can borrow, the solution is: q = 1 W 0 X 0 + βe W 1 X 1 ] η/y 1 + β 1 X 0 + βe X 1 ] + βv ar X 1 ] h / η/y β 1 D + G 0 + βe G 1 ] T Y Y 0 + βe Y 1 ] 1 + β 1 X 0 + βe X 1 ] X 0 + βe X 1 ] + βv ar X 1 ] ] βcov X 1, G 1 T Y Y β 1 X 0 + βe X 1 ] + βv ar X 1 ] The differences between these two solutions stem from the fact that, when η > 0, government chooses debt in an attempt to smooth tax burden over time And, the ability to smooth taxes and the corresponding tax distortions impacts the optimal choice of program scale Specifically, comparing the first and third terms of these two expression we see that since 1 + β 1 X 0 + βe X 1 ] < X0 + β E X 1 ] the government adjusts program scale more elastically in response to changes in the discount net benefits W 0 X 0 + βe W 1 X 1 ] or fiscal risk hedging considerations βcov X 1, G 1 T Y Y 1 ] when the government is allowed to simulatenously choose program scale and debt issuance This is an instance of Samuelson s 1947 Le Chatelier Principle which says that comparative statics are smaller in magnitude when an optimizing agents is not permitted to adjust certain control variables Furthermore, comparing the second terms of these expressions, we see that the inability to borrow to smooth expected tax distortions over time naturally changes the expected fiscal costs calculusus Specifically, since the government cannot borrow to smooth taxes, it will adjust program scale in an attempt to smooth expected tax distortions Obviously, household welfare must be lower when we require government to set D 0 = 0 as we do here And, one would generically expect that imposing the constraint that D 0 = 0 would lead the government to choose a smaller level of q 9