Rates of return and the supply of government assets

JAMES P. DOW, JR. University

of California,

Riverside

Riuemide,

Cal$omia

Rates of Return and the Supply Government Assets*

of

An important issue in macroeconomics has been to determine the effect of an increase in government borrowing The “traditional” response, an increase in the interest rate, has been challenged by the notion of Fticardian equivalence. This paper argues that a positive correlation between debt and interest rates can exist in a model with optimizing rational agents through a portfolio substitution effect in response to changes in the supply of government bonds. A consumption/savings model with government borrowing is developed and the conditions for an increase in debt to increase interest rates are determined.

1.

Introduction

The effect of an increase in government borrowing on interest rates has long been debated. The traditional analysis of “crowding out” has been challenged by the principle of “Rica&m Equivalence” (Barr0 1974, 1989) which argued that people will anticipate future tax increases and so increase their saving proportionately, neutralizing any effect of additional debt. However, it is still felt that there is some relation between government debt and the level of interest rates. This has led to recent work trying to explain that relationship, while keeping the assumptions of rational expectations and flexible credit markets (for example, Barsky, Ma&w and Zeldes 1986). This paper offers one approach to that problem. Since risky and risk-free assets are imperfect substitutes in financial portfolios, under certain conditions, an increase in the supply of government bonds will make those bonds relatively less attractive and so require a higher return. Thus, increases in government borrowing can increase interest rates even in the absence of crowding out. The paper develops a tractable model that allows for multiple assets, each with limited supply, which can capture this relationship between debt and interest rates. The principle result of the model is that it identifies the sign of the correlation between tax payments and the return to stock as the key variable in determining the effect of this asset substitution. The structure of the model is based on an individual saving for the future with a variety of assets, including stocks, bonds, future income payments and tax obligations. Since each asset is an imperfect substitute for the other, and each has a limited supply, the price of an asset will be determined *I would

like to thank

an anonymous

referee

for helpful

comments

]mml of Macroeconomics, Spring 1994, Vol. 16, No. 2, pp. 281-293 Copyright 0 1994 by Louisiana State University Press 0164.0704/94/$1.50

281

lames P. Dow, Jr. by the supply and demand for that particular asset. Traditional models of security markets have modeled the demand side of financial markets using the portfolio selection problem of the consumer (for example, Lucas 1978) but have left the supply decision exogenous; however, supply effects may be a significant source of interest rate changes. In fact, many common govemment policies, including the two examined in this paper, a substitution of debt for taxes, and open market operations, involve a change in the quantity of Treasury securities. What makes these effects more significant is that the return on Treasury bills often stands for the interest rate, either as reported in government statistics, or in econometric studies. Changes in relative returns of assets will appear as changes in the average level of returns. Let us call the results of the change in asset mix the “portfolio effect” to distinguish it from other influences on interest rates. This portfolio effect makes changes between taxes and government debt non-neutral because the two assets are not perfect substitutes for each other. When individuals are forced to pay higher taxes in the future, they will respond by holding a variety of different assets, of which government securities are just one, so that the specific demand for treasury bills will not increase by as much as the supply. The amount of the increase will depend on the stochastic relationship of the tax obligations with other assets in the portfolio, specifically the covariances of the returns of the assets. This paper differs from other approaches to the question of govemment debt in that it does not rely on explicit liquidity constraints or a precautionary saving motive (for example, Chan 1983 and Barsky, Mankiw and Zeldes 1986). The model uses the principle developed in Kahn (1990) that moral hazard problems will prevent a market for complete income insurance, although it uses a different structure for the asset markets to allow for changes in the timing of taxes and changes in the supply of several assets. The paper proceeds as follows. A two-period model of the consumption/saving/portfolio decision is presented in Section 2. The approach is to use the marginal utility of consumption to price asset returns as was developed in Lucas (1978). The Euler equations from the consumption/savings optimization problem define the demand for assets. Given a supply of assets one can then determine the equilibrium interest rate. This is often done by allowing individuals to choose the amount of a security to hold, but in equilibrium, forcing the quantity of these claims to be zero. This paper expands on this approach by allowing multiple types of assets, all existing in positive quantities. This will allow the determination of the effect of a change in the quantity of assets supplied. The constraints on the supply of assets and government fiscal policy are then introduced, and the solution to the model calculated. In Section 3 the qualitative effects of the specific government 282

Rates of R&u-n and the Supply of Government Assets interventions will be examined and the necessary condition for changes in the timing of taxes to affect interest rates through portfolio adjustment will be determined. The paper concludes with Section 4. 2.

A Two-Period Model The model consists of a two-period consumption/savings problem, where the second period level of consumption is constructed as a portfolio with four types of assets. The first asset is the labor income of the consumer, which will be assumed to be stochastic and exogenous. It is assumed that there are moral hazard considerations which prevent income insurance across consumers as in Kahn (1990). The second asset is the tax obligations of the consumer. These will be lump sum, but may be correlated with the realization of income or dividends. The third asset is a risky security which represents ownership of equity. This will be interpreted as a portfolio of assets, where the individual can buy shares in the portfolio. The portfolios of all individuals will have the same expected return and variance, but different realizations of dividend payments. The justification for this is that individuals have idiosyncratic risk in other (unobserved) aspects of their lives, so that they will also construct distinct portfolios. Outside of the differences in dividend payments, individuals are identical, and face the same price for shares in their portfolios.’ The individual will be able to choose the number of shares of the portfolio to hold, but in equilibrium, they will be forced to hold the full supply of this security, the expected return on the security adjusting to equate quantity supplied and demanded. The final security will be government bonds. The model abstracts from concerns of inflation uncertainty, and so it is assumed that the real return on this security is known for certain. The division of financial assets into two categories is a significant simplification, given the continuity of riskiness available in modem asset markets, but it captures the spirit of this market, one group of risky assetsprovided by private firms, and one group of low risk assets, predominantly provided by the government. At the beginning of period 1, individuals lmow the prices of the securities and the first period income and taxation. They choose the amount to consume and the amount to save, and how to split their savings between stocks and bonds. The optimization problem is Max JW(Cd SJ

+ W(G))

‘An important assumption of the model is that individuals cannot perfectly diversify their financial holdings, otherwise, all individuals would have identically structured portfolios of wealth, including income and tax obligations. Empirically, this does not seem to be true. It is assumed that there is some imperfection in insurance and security markets to prevent this.

283

James P. Dow, jr. subject to C1=Y1-T1--9S--pB, Cz=Y2-tT2+dS+B, where, Ci = consumption in period i; Yi = income in period i; Ti = expected taxes in period i; S = holdings of stock; B = holdings of bonds; 9 = price of stock; p = price of bonds d = payout of stock, t = stochastic tax shock, B = subjective discount factor; E = expectations. Both bonds and stock are sold at a discount. The expected payout to both types of securities is equal to 1, with the payout to stock being stochastic. The tax obligation in period 2 equals the average tax obligation (T,} multiplied by a stochastic shock (t). The distribution of the three stochastic variables for the individual are E(YJ = yz;

E{fi)=y;+~$;

E(d) = 1;

E(d2} = 1+ c$j ;

E(t) = 1 ;

E{t2)=1+$;

E(dt) = 1 + ~QCQO~ = 1 + cov(d,t) ; E(dY2) = Ii + pdycr&y = y2 + cov(d,Y2) . The solution of the model is characterized by the two first-order conditions taken with respect to S and B: EN-9)U’G)

+

W”(C2))

EK-p)U’(CJ + VW,))

=0> =0.

Rearranging, the conditions can be expressed in terms of the returns to assets,

284

1= PE((U’(C2)/U’(Cl))(d/q))

>

1=

>

PEI(U’(C,)/U’(C,))(l/p))

Rates

uf Return

and the Supply of Governmmt

Assets

where (d/q) eq u al s one plus the return on the risky asset and (l/p) equals one plus the risk-free return.2 Total returns will be represented by R, and Rf respectively. Combining the equations we can solve for a CAPM type relation E(R,} = Rf[l - ~ov(R,,U’(C~)/U’(CI))]

.

The return to the risky security increases with its (negative) covariance with the return to the individual’s portfolio, the marginal rate of substitution of consumption. Financial models allow the “market portfolio” to proxy for the individual’s portfolio, effectively determining the supply of assets, leaving the risk-free rate as exogenous, and so using the CAPM relation to determine R,. This paper expands on this by determining both the risky and risk-free rates, and by including income and taxation in the supply of assets. This is essential since the largest part of most people’s portfolios are labor income and tax obligations, and these are the assets commonly affected by government intervention. Government spending per capita each period (GJ is given exogenously. Government spending in period 1 must equal taxes collected, plus revenue collected from issuing bonds. G, = T1 + Bp . Government spending in period 2 must equal expected tax revenue from period 2 minus bond payments. G, = T2 - B It is assumed that tax obligations are distributed independently across individuals, so that tax revenue collected will equal the average of the tax distribution each individual faces3 B is an exogenous variable which repre‘An infinite

horizon

version

of this model

1 = PE((u’(c,+,)/v’(c,))(y~+, 1 = pE((u(c,+,)/L”(C,))(pt+l

has equivalent

conditions,

+ 4+,VytN 3 + 4/pJl ,

where [u) is the payment on the risk-free asset. In the infinite horizon model, the return to the asset includes the expected future price of the asset, or equivalently, the discounted sum of expected future dividend payments. These two equations cannot be used to calculate the equilibrium returns, as will be done in the two period model, as the correlation of the marginal rate of substitution with the future price is unknown. Because of this, and the difficulties in incorporating tax changes, the two-period model is preferable to use in this case. %ince the individual returns to portfolios are correlated with individual tax obligations, this implies that the returns to portfolios are also distributed independently across individuals. An alternative to this assumption would be for the government to have a stock of assets (such as foreign currency) which it is allowed to buy and sell depending on the difference between government spending and actual tax receipts. Under this interpretation the government budget constraint implies that it chooses tax and debt policy so that the two-period budget is balanced in expectations.

285

James P. Dow, Jr. sents the specific fiscal policy. Given G,, G, and B both Tl and T2 can be determined (as functions of p). From the consumer’s problem we get the choice variables (S and B) as functions of the prices (p and 9) along with the distributions of the income and tax shocks. In equilibrium, B is determined by the government and S is given exogenously. Given the values of B and S we can equate the supply and demand for assets so that Sd(p,9) = s 7 Bd(p,9) = B > and so solve for the equilibrium prices. The utility function will be quadratic and take the form U(C) = uo + UlC + 1/2U&2 )

U&U1 > 0,

u2 < 0.

The quadratic function has three advantages. The most important is that it eliminates the effect of precautionary saving, and so isolates the portfolio effect. The second advantage is that it restricts the distribution of the stochastic processes to the first two moments. Finally, it allows for an analytical solution of the prices.4 Substituting the government budget constraints into the first-order condition, we can solve for 9 as 9 = (Q - Q2(& + 4&p

co~(t,d)B)“~ ,

$3 + 4&3cov(t,d)B

> 0 , $2 < 0 .

Terms not containing B have been incorporated into the constants $i, $s and Q3 (complete descriptions of the price functions are reported in the appendix). From the second first-order condition, we can write p as a function of 9. p=

$4@2Sq

- $5)-l 7

04<0>

05>0.

The expected rates of return of the risk-free and risky assets can be calculated as (1 -p)/p and (l-9)/9, respectively.

3.

Behavior of Interest Rates This section shows how the portfolio effect will produce interest rate changes under several government policies and establishes the necessary conditions for the changes to go in the hypothesized direction. The influences on interest rates can be classified into two types, those affecting tbe “average ‘?he equilibrium can easily be solved numerically for more complicated utility functions or for other variations to the model; however, the analytical solution provides a more transparent demonstration of the portfolio mechanism and so will be the approach used.

286

Rates of Return

and the Supply

of Government

Assets

return” of all assets, and those affecting the “relative return” of a given asset. The model of Section 2 includes both types of effects. Average returns will depend on the desire to save and the opportunities for saving. An increase in first period income will result in an increase in the desire to save in order to spread the extra income over time, and so with assets inelastically supplied, the returns to both assets will fall. An increase in the quantity of stock supplied will result in increases in both interest rates to encourage people to hold more assets. This change also has a secondary effect of increasing the overall riskiness of the portfolio (since stock is the risky asset), which will increase the value of the risk-free asset, and so reduce its return. On net, however, the return to both stocks and bonds will increase. The return on a given asset will also depend on the covariance of the asset with the entire portfolio, including income and tax obligations. For example, if the income process becomes more variable, the return on the risk free asset will decrease, and the return on assets highly correlated with income will increase. These effects are all standard results from asset pricing models. What will be shown in more detail is how changes in the supply of the risk free asset (government bonds) affect the returns to both assets. A Substitution of Debt for Taxes “Ricardian equivalence” refers to the proposition that a decrease in current taxes financed by increased debt will not affect consumption and interest rates, as consumers will save the additional income to pay future taxes. However, because of a limited supply of assets, changes in tax policy that affect the supply of govermnent bonds may affect the returns to these bonds. The nature of this portfolio effect is straightforward. Individuals will increase their saving in response to future taxation, but they will increase their saving by holding several different assets, of which government bonds are only one. The demand for government bonds will increase by less than the increase in supply of government bonds, and so the return to these bonds, “the interest rate,” will rise. This argument is subject to an important restriction. Individuals will wish to save with assets that have the same distribution as the tax obligations; that is, an asset that has an above average payment when the tax obligation is also above average. If taxes are certain, individuals will wish to hold a certain asset. In that case, individuals will choose to hold all of their additional savings in risk-free government bonds, and so Ricardian equivalence will hold. If taxes are uncertain then the change in interest rates will depend on the correlation of taxes and stock returns. The closer taxes are to stock returns, the more consumers will want to hold stock, and so the less the demand for bonds will increase. To measure the closeness between taxes and dividends the correlation of tax revenue and stock returns can be calculated (there are no capital gains

Jams P. Dow, Jr-. in the model since there is only one period of saving). This correlation should be positive. The reason for this relationship is that tax revenue is increasing with income, and dividends are part of income, so they should be directly correlated. In addition, dividends are also correlated with corporate profits which are in turn correlated with output. Since tax receipts are also positively correlated with output, this introduces an indirect correlation between dividends and taxes. This correlation was calculated using data on dividend income and tax payments. 5 A conservative estimate of the correlation was found to be 0.13 for the government data and 0.10 for the personal data. This number is small but agrees with the prior expectation of a positive value. The portfolio effect can be demonstrated by determining the effect of a change in the supply of bonds (B) on the prices of assets {p, CJ}.B indicates the fiscal stance of the government, an increase in B is associated with a decrease in first-period taxes and an expected increase in second-period taxes. From the price equations we can determine the effect of a change in B: dq/dB = -2($, + 4u~S~co~(t,d)B)-~‘~~~~S~cov(t,d) ?dilB = (-1)~4(~zSg - &J%&ky/~B>

< 0,

> 0, for cov(t,d) > 0 .

From these equations it is easy to see how the results depend fundamentally on the sign of cov(t,d). For a positive correlation an increase in debt yields an increase in the return to debt. Observationally, this will seem like a failure of Rica&an equivalence. Since the return to government bonds is the interest rate for most econometric studies, we find that an increase in bonds produces an increase in “interest rates.” In spite of this non-Ricardian result, savings are still increasing, and consumers are behaving in a “Ricardian” manner. Also, risky securities will require lower returns, encouraging capital investment by firms, the opposite effect from what is usually expected. Thus, the portfolio effect is able to produce the appearance of a violation of Rica&an equivalence by 5Data used were the following series from the Citibase database: personal dividend income, personal tax and non-tax payments, national income-dividend payments, federal government receipts-personal tax and non-tax payments. Income tax was also tried as a measure of tax payments and produced nearly identical results. Data were quarterly (at an annual rate) and seasonally adjusted. Data were converted to constant dollar/per capita and detrended (the trend, in part, reflected secular growth in dividends and taxes over the period, so detrending produces a downward bias in the estimated correlations. The correlations, only adjusting for inflation and population growth, were around 0.8). Because the data are quarterly there is a bias against any correlation since taxes on dividends are not commonly owed at the time of payment; however, longer time periods are likely to run into problems since there are significant positive trends in both series.

288

Rates of Return

and the Supply

of Government

Assets

inducing a correlation between the level of government debt and the interest rate without the corresponding negative welfare consequences. The portfolio effect is different from the effects reported by other approaches to the tax/debt problem. This result can be compared with the findings of Chan (1983) and Barsky, Mankiw and Zeldes (1986) who provide an alternative relationship between government debt and interest rates. These papers emphasize that, for “plausible” utility functions (those with expected marginal utility increasing with uncertainty), an increase in the future uncertainty of consumption will result in increased saving by the consumer. Chan (1983) argues that since future taxes are uncertain, a deferment of taxes will increase future uncertainty, and so increase saving. The demand for bonds will exceed the supply of bonds, and interest rates will fall. Barsky, Ma&w and Zeldes (1986) argue, on the other hand, that taxes are proportional to income, so that higher tax rates will mean less consumption uncertainty, lower saving and higher interest rates. In contrast, for the model presented in this paper, even if the policy increases the uncertainty of future income (which it will), there will be no effect as long as taxes are uncorrelated with stock returns. In this case, the tax payments can effectively be decomposed into two assets, a certain payment equal to the average tax payment, and security that is pure risk. The increase in average taxes is met by an increase in holdings of the equivalent assets, risk free government bonds. The change in risk has no effect because of the quadratic utility function; it is not the average risk that matters, but the relative risk across assets. The precautionary effect could be introduced in this paper by using a different utility function; however, the combination of the two effects would only serve to obscure the portfolio effect. Open Market

Operations

There are other government policies that can be evaluated with this model. In this section it will be shown that an increase in interest rates after Federal Reserve open market operations can be explained by the portfolio effect. Let open market operations mean the government printing of fiat money (money with no intrinsic value, but adopted as a means of transaction by convention or law) which is used to purchase government bonds, which are then eliminated. Assume that the existence of the additional money is neutral and all nominal prices adjust perfectly. This assumption is not important for the argument; in fact, the point is that open market operations will affect interest rates in spite of the neutrality of money. It is easiest to see the effects of open market operations by comparing it with the previous tax-for-debt experiment. In that example we observed an increase in debt and a shift in taxes to the future, which resulted in an increase in interest rates. Open market operations can equivalently be interpreted as 289

Jams P. Dow, Jr. TABLE 1. The Level of Government Debt and the Expected Return to Assets B

Return to Stock (RJ

Return to Bonds (%)

0.0 0.2

0.097 0.0854

0.0001 o.ooo9

a decrease in debt with a shift in taxes to the present. Since this is just the opposite experiment, the result is for a decrease in interest rates. To see why it is the opposite, open market operations can be broken down into its component parts. First, there is a decrease in the quantity of bonds through government purchase (by money creation). Second, the corresponding increase in the money supply reduces the value of real balances (the inflation tax), which increases taxes in the present. Finally, since the quantity of bonds is now smaller, the amount of taxes that must be collected in the future is less (fewer bonds to repay). And so, overall, we have a shift in taxes along with a reduction in bonds. The (constant expenditure) government budget constraint forces these kinds of connections; if the stock of bonds is reduced, that is, if government borrowing is reduced, there must be a corresponding increase in taxation in the present and a decrease in taxation in the future. In the case of open market operations the taxation takes the form of a “monetary” tax rather than a “fiscal” tax, but it is a tax just the same. Since this is just the opposite of the debt for taxes experiment we already know the results. Individuals will reduce their demand for bonds by less than the decrease in the supply of bonds, and so the return to bonds will fall. This produces the traditional result of an increase in the money supply producing a decrease in interest rates even though no restriction is placed on the ability of prices or investment to change. Magnitude of Interest Bate Effects The model is too simple to be able to effectively calibrate it in the style of Kydland and Prescott (1982); however, something can be learned about its qualitative behavior by numerically calculating interest rates for different values of parameters. Parameters were chosen so that there was a relatively low correlation between taxes and dividends and with the expected return to stocks significantly larger than the return to bonds (following Mehra and Prescott 1988).‘j The results of an increase in borrowing are reported on Table 1. The value of B increasing from 0.0 to 0.2 matches the recent ‘?heparametervalues usedinthe examplewereu 1 = 2.2,~~ = 0.85, Y, = 1.277,Ya = 0.88, G, = 0.3, G, = 0.3, p = 0.95, S = 0.2, pm = 0.2, pdy = 0.2, q, = 0.7, a, = 0.7 and o, = 0.7.

290

Rates of Return and the Supply of Government

Assets

increase in U.S. government debt relative to the net size of the equity market. The increase in B has the predicted effect of increasing the return to bonds and decreasing the return to stock. The change in the risk free return is relatively small in terms of percentage points, although larger in terms of the percentage increase. It was found that as the correlation between taxes and dividends increases the absolute change in the risk-free return increases, although the percentage change decreases. It was also found that for a wide variety of parameterizations the effect on the return to stock was much larger than the effect on the return to bonds. While the implication that the largest effect of an increase in government borrowing is on stock prices is surprising, it is not inconsistent with the experience of the 1980s when we saw dramatic increases in U.S. government borrowing and a significant rise in the stock market. Numerical results also suggest that the size of the effect of government borrowing on interest rates through the portfolio effect will depend on the relative amounts of the various kinds of assets, the size of the intervention, and the willingness of consumers to substitute between assets. The primary determinant of the latter is the consumers’ attitudes towards risk; the more risk averse, the smaller the change in return necessary for them to hold the additional amount of the risk-free asset. The parameterization of the model is a bit arbitrary since there are only two periods, and so the quantitative results are not very restrictive (there are too many “free” parameters); however, this exercise does provide an illustration of the kinds of results to expect and on how they might depend on the values of the parameters. The increased interdependence of economies should influence these results, and is a potential extension of the model. In an open economy framework, the demand for the domestic country’s assets will partly come from foreign investors. An increase in the supply of U.S. securities would then be held, in part, by people who do not pay U.S. taxes and so would not take their higher tax burden into account when deciding how many of the bonds to hold. Foreign investors would shift their asset holdings from stocks to bonds (since their desired portfolio hasn’t changed) which would dampen the change in relative returns, reducing the portfolio effect. Another possible extension of the paper is to include the endogenous supply of assets. Numerical results were calculated for a model where the supply of stock was assumed increasing in its price (cJ),and the rents earned by owners of the stock (price over physical cost of investment) were returned to consumers as “profits.” This model behaved in the same way as the model of Section 2; an increase in the supply of bonds caused a shift in returns, increasing the relative return to bonds. Again, this effect depended on a positive correlation between dividends and taxes. The size of the effect was somewhat smaller due to the increase in the supply of stock.

James P. Dow, Jr. 4.

Conclusion In a world with imperfectly substitutable assets, the supply and demand for a specific asset will determine its rate of return. Because of this, changes in the supply of government bonds should change the relative returns of assets. This paper has examined two examples of this, a change in the timing of taxes and open market operations. It is found that an increase in government spending associated with a shift in taxes will produce an increase in interest rates, even if consumers increase their saving to pay for future taxes. The necessary condition for this result is a positive correlation between future tax obligations and the return to other assets. If this condition is true, the increase in future tax obligations resulting from a decrease in current taxes will cause consumers to desire to hold more of both stocks and bonds. Since the supply of bonds has increased by more than the demand, the return to bonds will rise. On the other hand, the return to stocks will fall, as there is no increase in supply to accommodate the increase in demand. Since the return on government bonds is the standard measure of the interest rate, this will look like average interest rates are rising in response to the government issue of debt. This portfolio effect allows government debt policy to affect interest rates, even though consumers have rational expectations and capital markets are not directly constrained. Received: December 1992 Final version: September 1993

References Barro, Robert. “The Neoclassical Approach to Fiscal Policy.” In Modern Business Cycle Theo y, edited by Robert Barro. Cambridge, Mass.: Harvard University Press, 1989. “Are Government Bonds Net Wealth?” Journal of Political Economy’82 (1974): 109.!%1117. Barsky, Robert, N. Gregory Mankiw, and Stephen Zeldes. “Rica&m Consumers with Keynesian Propensities.” American Economic Review 76 (1986): 676-91. Chan, Louis. “Uncertainty and the Neutrality of Government Financing Policy.” Journal of Moneta y Economics 11 (1983): 351-72. Kahn, James. “Moral Hazard, Imperfect Risk-Sharing, and the Behavior of Asset Retums.“]ournul of Monetary Economics 26 (1990): 2744. Kydland, Finn, and Edward Prescott. “Time to Build and Aggregate Fluctuations.” Econometrica 50 (1982): 1345-70. 292

Rates of Return

and the Supply

of Government

Assets

Lucas, Robert. “Asset Prices in an Exchange Economy.” Econometrica 46 (1978): 142645. Mehra, Rajnish, and Edward Prescott. “The Equity Premium: A Puzzle.” Journal of Monetay Economics 15 (1988): 145332.

Appendix Derivation of the Price Functions The two first-order conditions u2Sq2 - (u2(Y1 - G1) + q)q

are (after

(A.l) is a quadratic appropriate interest

b2@1

- @2Bcov(t,d)

-

GJ

(assuming 4r2 = l/(2%$)

+uJ/(2~2S)

marginal

which

U(),Ul > 0 ;

cov(t,d)B)“2

= 0 ;

(A.l)

- G2 + S) = 0

equation in 9. Taking the root rates, and rearranging, we get

9 = Q1 - q2($s + 4&p h=

+ var(d)S))

- GI) + ul)p + @I + bz(Y2

b2@'1

out taxes)

+ p(ul + u2(Y2 - G2 + d2S - G2cov(t,d)

+ cov(d,y) u2Sp9 -

substituting

(A.2)

produces

the

u2 < 0 ;

< 0 >

utility

positive

for C1 = Y1 - G1) ;

< 0 ;

Q3 = (u2(Y1 - G,) + ZQ)~ - 4U$p(ul+ + cov(d,y)

u2(Ys - G2 + S - G2cov(t,d)

+ var(d)S))

‘& + 4&$ cov(t,d)B > 0’ is a necessary condition for red roots. A sufficient condition for this to be true is E {d(marginal utility of consumption)] z 0. To find the effect of a change in fiscal policy on stock prices we take the derivative of 9 with respect to B. 39/8B

= -2(+, Using

+ 4u;Sf3cov(t,d)B)-1’2+2u&kov(t,d) (A.2) to determine

p = Uu2S9 $4=

-P&I

~s=ul+u2(Y1

- 95) +u2W2

-1 -

p as a function

(cov(t,d)

> 0) .

of 9 we get

, G2+

S))

-G1)>O,

= -+4(u2S9

This term will take the opposite

(for positive

< 0 >

(for positive

To find the effect of fiscal policy derivative of p with respect to B. +/aB

> 0,

marginal

on the price

- $$2u2S@q/aB) sign from

the effect

marginal utility) of bonds

utility)

,

. we take the

~0 of fiscal policy

on 9.

293