# On integrating the Ricardian equivalence theorem and the IS-LM framework.

ON INTEGRATING THE RICARDIAN EQUIVALENCE THEOREM AND THE IS-LM FRAMEWORK

I. INTRODUCTION

It is well established that government bond sales have no real macroeconomic consequences if the Ricardian Equivalence Theorem (RET) is true. Specifically, sales of government bonds have no independent effect on the rate of interest, and a switch from tax finance to bond finance (or vice versa) does not alter the level of aggregate demand. While the conditions under which the RET holds are quite stringent, (1) many economists believe it should be taken seriously, at least as a first-order approximation of reality, and discussion of the theory has, in recent years, crept into the leading undergraduate macroeconomic texts. (2) It is our impression that many teachers at the undergraduate level would like to incorporate the RET into their courses, but have refrained from doing so under the mistaken belief that the technical demands are too great. Quite the contrary, we show in this paper that the RET is easily integrated with the textbook IS-LM model that we teach our students. Moreover, and somewhat surpringsinly, it turns out that the IS-LM model already incorporates a healthy dose of the RET in that bond sales to finance an increase in government purchases of goods and services have no independent effect on the rate of interest. This result is shown to derive from an implicit restriction on the arguments of the money demand function.

II. THE STANDARD IS-LM MODEL

We begin with the following wholly-conventional IS-LM model: y = c([y.sup.d]) + i(r) + g (1)

M/P = M(y,r) (2) [y.sup.d] = y - t, (3) where lower case letters denote real variables and upper case letters nominal variables. The IS curve (Equation 1) requires no explanation. The LM curve (Equation 2) is also conventional; however, we show below that there is an important, but neglected, relationship between the standard money demand function and the validity of the RET. Equation 3 defines disposable income in the usual way as total income minus net taxes.

Two conditions must be satisfied if this model is to be consistent with the RET. First, the level of aggregate demand (hence output) must be independent of the method by which government expenditures are financed. A switch from tax finance to bond finance (holding g constant) should leave output unchanged, and an increase in g financed by bond sales should have the same effect on output as an increase in g financed by taxes. Essentially, this is because in the RET households view government bonds sales as being equivalent, in present value terms, to an increase in future taxes. Second, government bond sales to finance a deficit should have no independent effect on the interest rate. This counterintuitive result arises because households increase their current demand for bonds in order to afford the higher taxes they expect to be levied in the future to pay interest on bonds being sold today. The increased supply of government bods is, therefore, passively absorbed into private sector portfolios.

To show that the RET does not hold in the textbook IS-LM model, consider bond-financed and tax-financed increases in g of equal amounts (see Figure 1). The bond-financed rise in g shifts the IS curve horizontally by [(1/1-[derivative]c/[derivative]y)][DElta]g from [IS.sub.0] to [IS.sub.2], while the tax-financed rise in g shifts the IS horizontally by only [Delta]g to [IS.sub.1]. The first condition required by the RET is not satisfied since the effects on y are not the same. What about the second condition that the bond sales have no independent effect on the interest rate? Curiously, it is satisfied. When the increase in g is financed by bond sales, the rise in the interest rate from [r.sub.0] to [r.sub.2] is fully explained by the increase in transactions demand for money induced by the increase in output from [y.sub.0] to y.sub.2. (3) We return to this point below.

III. RENDERING THE IS-LM MODEL CONSISTENT WITH THE RET

In the RET, households view government bond sales as a substitution of future taxes for current taxes. When the government finances a given level of expenditures by selling bonds, households recognize that the government will levy higher taxes in the future to pay the interest on bonds being sold today. Of course, this assumes that households have perfect foresight and that the government budget is intertemporally balanced. But if these assumptions are satisfied, then the current-period tax burden as seen by households is not current net taxes, as in the standard IS-LM model, but rather current net taxes plus the present value of future taxes required to pay the interest on current bond sales.

For simplicity, assume that each government bond is a consol paying \$1/year in interest so that the market price can be written compactly as 1/r. If B is the number of bonds outstanding, then B/r is the nominal value of the debt, and B/rP is the corresponding real value. Let [Delta]B denote the number of bonds sold in the current period to finance the budget deficit. The nominal value of these bonds is [Delta]B/r, and the yearly interest payments are r([Delta]B/r) = [Delta]B. The real present value of implied future taxes is:

PV = (1/P) [[Delta]B/(1+r) + [Delta]B/(1+r).sup.2] + [[Delta]B/(1+r).sup.3] + ...] = [Delta]B/rp, (4)

and the current-period tax burden on households consistent with the RET is:

[sup.t.RET] = t + [Delta]B/rP. (5)

If we assume that the government does not resort to moeny creation, then the current-period government budget constraint requires that:

g = t + [Delta]B/rP. (6)

Equations 5 and 6 imply that, in the RET, the appropriate definition of real disposable income is not y-t but rather:

[y.sup.d.sub.RET] = y - t.sub.RET] = y - g. (7)

In effect, the current burden of government is better measured by the goods and services that government absorbs than by the current taxes it levies--an insight we do not claim to have discovered. It follows that the IS-LM framework can be made consistent with the RET if the IS curve is rewritten as:

y = c(y = g) + 1(r) + g, (8)

which is a very simple modification of the standard specification. (4)

Consider, now, the macroeconomic implications of Equation 8. First, since current net taxes are no longer an argument in the consumption function, a change in t holding g constant will have no effect on output. For example, a tax cut financed by bond sales does not raise consumption. The reason is that bond sales imply higher future taxes having a present value equal to the (absolute) value of the reduction in current taxes. Disposable income, as defined by Equation 7, does not change so the IS curve does not shift. n5 Second, when the IS is given by Equation 8 a change in g has the same effect regardless of how it is financed. With this specification, the horizontal shift in the IS curve in response to a bond or tax-financed rise in g is simply [Delta]g, (6) which is identical to the horizontal shift in the IS in the conventional model when there is a balanced-budget change in g. There are no longer separate multipliers for changes in g financed by bond sales and changes in g financed by taxes. Why? Because bond sales effectively impose the same burden on households as th current taxes they replace. We conclude that the IS-LM framework consisting of the conventional LM (Equation 2) and the modified IS (Equation 8) is fully consistent with the RET.

IV THE RET AND THE LM CURVE

Respecifying the IS curve in the manner suggested above is sufficient to render the IS-LM model consistent with the RET. But his respicification responds only to the first of the two conditions required to satisfy the RET--namely, that the level of aggregate demand must be independent of the mix between bond finance and tax finance. Why is it unnecessary to respond to the second condition requiring that government bond sales to finance a deficit have no independent effect on the interest rate? It turns out that this condition is implicitly, if inadvertently, but into the conventional LM curve so that no further modification of the basic IS-LM model is required.

To demonstrate this, suppose that the LM curve were to be respecified as:

M/P = m(y,r,TFW/P) (9)

where TFW/P = M/P + B/rP is total private sector holdings of real financial assets. The economic rationale for including TFW/P in the money demand function is that, at any moment in time, financial wealth must be held in some form in private sector portfolios. And since money and bonds are the only assets in the IS-LM model, there must be an implicit financial wealth constraint requiring the real demands for money and bonds to sum to total real financial wealth (TFW/P). It follows that:

M/P + B/rP = TFW/P=m( ) + b( ), (10)

where m( ) and b( ) are the real demands for money and bonds, respectively.

This constraint enforces internal consistency on the model's asset demands by imposing a set of adding-up conditions on the partial derivativatives of the two demand functions. n7 Specifically, the partial derivatives of m( ) and b( ) with respect to y and r must sum to zero since a ceteris paribus change in either variable will alter the composition, but not the size, of the portfolio. (8) On the other hand, the partial derivatives of (real) money demand and (real) bond demand with respect to TFW/P must sum to one since total financial wealth must be demanded to some form. (9) In short, this adding-up condition reflects the fact that an exogenous increase in real financial wealth must raise the total demand for money and bonds by the amount of the exogenous change. The manner in which this increase is split between money and bonds is determined by the relative magnitudes of the two partial derivatives. In the general case, 0 [is less than] @m/@(TFW/P) [is less than] 1 and 0 [is less than] @b/@(TFW/P) [is less than] 1 so that an increase in TFW/P raises the demand for both real money balances and real bond holdings.

Now compare a rise in g financed by taxes with a rise in g financed by bond sales. In both cases, the Ricardian IS curve shifts rightward by an amount equal to [Delta]g (see Figure 2). The effects on the LM (as specified in Equation 9), however, are not the same. In the tax-finance case, the LM remains stable so the economy moves from point A to point B in Figure 2. Because there is some crowding out, the total rise in y is less than [Delta]g. In the bond-finance case, however, the LM does not remain stable. The increased supply of government bonds raises TFW/P from [TFW.sub.0]/P to [TFW.sub.1]/P which, in turn, raises the demand for both money and bonds. The higher demand for money shifts the LM curve leftware (10) and causes the economy to end up at point C in Figure 2 rather than at point B as in the tax-finance case. Output is lower, and the interest rate is higher. Intuitively, the increase in bond demand stimulated by the rise in TFW/P is insufficient to absorb the increased supply of government bonds at the existing interest rate. Thus, an additional rise in the interest rate (from [r.sub.1] to [r.sub.2]) over and above the transaction-induced rise (from [r.sub.0] to [r.sub.1]) is required to maintain equilibrium in the bond market. The result is greater crowding out and a lower equilibrium level of output.

This pair of outcomes is inconsistent with the RET since the model's real variables are no longer invariant with respect to financing considerations. Evidently, the LM curve must remain stationary in the face of the bond-financed rise in g if the IS-LM framework is to be consistent with the RET. How can this be ensured? Suppose that TFW/P were to be deleted from Equation 9--in effect returning us to the conventional LM curve of Equation 2. Now, government bond sales to finance an increase in g do not shift the LM curve, and the economy moves from point A to point B in Figure 2, the same change as in the tax-finance case. What is going on behind the scenes to ensure this result? Since the adding-up conditions ensure that @m/@(TFW/P) + @b/@(TFW/P) = 1, omitting TFW/P from the money demand function is equivalent to setting @m/@(TFW/P) equal to zero and @b/2(TFW/P) equal to one, so that an increase in total real financial wealth raises bond demand by an equal amount. Consider, then the effects of a bond-financed increase in g using the conventional specification of the LM curve. Total financial wealth rises one-for-one as the bonds are sold, and this rise in wealth necessarily generates an equal increase in bond demand. The bond supply and bond demand curves both shift rightward by the same amount, so there is no independent effect on the interest rate. The rise in r between point A and point B is entirely explained by the rising transactions demand for money. This distinctly Ricardian behavior is subtly, if inadvertently, embedded in the textbook IS-LM model that we routinely teach.

V. CONCLUSION

Ricardian equivalence can be built into the IS-LM model with a minimum of effort. In fact, much of what is needed is already there--namely the absence of a portfolio-induced wealth effect on the demand for money. Equation 8, the modified IS curve, and Equation 2, the conventional LM curve, provide an easily explained framework that yields Ricardian predictions. Defining disposable income as y - g rather than y - t ensures that changes in the relative proportions of bond finance and tax finance for a given level of g do not shift the IS curve. Moreover, this definition is not difficult to motivate on economic grounds. The absence of a portfolio effect from the money demand function is more difficult to motivate pedagogically because of the need to derive and discuss the adding-up conditions. However, having done this, it is our opinion that the economic justification for the absence of a portfolio effect in the money demand function is readily provided by the RET. This contrasts sharply with the justification, or lack thereof, currently employed by the vast majority of macroeconomic textbooks.

(1) For a discussion of these conditions, see Barro [1978].

(2) See, for example, Dornbush & Fischer [1987], Gordon [1987], and Hall and Taylor [1988]. Barro [1987], of course, presents a detailed treatment of the RET within a framework that is much more sophisticated than the IS-LM model.

(3) With a constant money supply, an increase in y creates an excess demand for money. The private sector then attempts to increase its money holdings by selling bonds, which raises the interest rate and, ultimately, eliminates the excess demand for money. This is also the explanation for the rise in the interest rate from [r.sub.0] to [r.sub.1] in the tax-finance case.

(4) Notice that the MPC must now be interpreted as the marginal propensity to consume out of Ricardian disposable income.

(5) What does change, of course, is household saving. From the household budget constraint (i.e., y = c + s + t) and Equation 7 it follows that the savings function consistent with the RET is s = y - c(y - g) - t, which implies that a change in t holding g constant (which alters the mix between bond and tax finance) results in an equal, but opposite, change in household saving. This contrasts sharply with the conventional IS-LM model in which changes in s are a small percentage of a change in t. Put somewhat differently, the RET implies that a change in t does not affect aggregate withdrawals from the circular flow of income because households change their saving (also a withdrawal) by the same amount in the opposite direction. Compare this result with the conventional view that an increase in t raises aggregate withdrawals from the circular flow.

(6) This result follows because the differential of Equation 8 is dy = (@c/@y)dy - (@c/@y)dg + dg which implies that (1 - @c/@y)dy = (1 - @c/@y)dg and dy = dg.

(7) To our knowledge, Brainard and Tobin [1968] were the first to recognize explicitly the adding-up conditions implied by the financial wealth constraint.

(8) For simplicity, we are ignoring any interest-induced changes in the size of the asset portfolio. Also, notice how these adding-up conditions are consistent with the "stories" that we tell our students. An increase in r, for example, induces the private sector to want to substitute bonds for money in its asset portfolio with no change in the total demand for financial assets. Similarly, an increase y causes the private sector to desire increased money holdings at the expense of bonds. Once again, the desired composition of the portfolio changes, but not the size.

(9) This result, too, is consistent with the "stories" we tell. For example, an exogenous increase in the money supply via a "money rain" creates an excess supply which the private sector tries to get rid of by purchasing bonds. Thus, the entire amount of this exogenous increase in financial wealth takes the form of increased bond demand. The demand for money does not change at least not directly. Thus, the partial derivatives of money and bond demand with respect to the increase in real financial wealth sum to one.

(10) There will also be an interest-induced change in TFW as r changes, which, in turn, affects the quantity of money demanded. This effect, however, is captured in the slope of the LM and need not concern us here.

REFERENCES

Barro, Robert J. "Public Debt and Taxes," in Federal Tax Reform; Myths and Realities, edited by M. J. Boskin. San Francisco: Institute for Contemporary Studies, 1978.

Barro, Robert J. Macroeconomics. New York: John Wiley & Sons, Inc, 1988.

Brainard William C. and James Tobin. "Pitfalls in Financial Model Building." American Economic Review, May 1968, 99-122.

Dornbush, Rudiger and Stanley Sischer. Macroeconomics. New York: McGraw-Hill Book Company, 1987.

Gordon, Robert J. Macroeconomics. Boston: Little, Brown and Company, 1987.

Hall Robert E. and John B. Taylor. MAcroeconomics: Theory, Performance, and Policy. New York: W. W. Norton & Company, 1988.
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