Division of Research Graduate School of Business Administration The University of Michigan August 1984 THE RELATIONSHIP BETWEEN TAX AND REAL INTEREST RATE Working Paper No. 385 Lu Wang FOR DISCUSSION PURPOSES ONLY None of this material is to be quoted or reproduced without the expressed permission of the Division of Research.

THE RELATIONSHIP BETWEEN TAX AND REAL INTEREST RATE Lu Wang Assistant Professor of Finance School of Management The University of Michigan-Dearborn

This paper seeks to serve as a simple footnote of an earlier article of this journal (Shapiro [2]), writing down an algebraic condition for an income tax cut to result in a lower real market interest rate and thus higher investment expenditure to contribute more capital stock and thus higher aggregate supply capacity to the economy. As pointed out by Shapiro, "whether a tax cut will raise or lower the growth rate of real income through its effects on the supply side is a question.... The answer to the question posed depends on a multitude of factors and their complex interrelationships." [2] This is indeed the case. Before making a simple footnote concerning the above-mentioned interrelationships, we will first cite a recent empirical study (Arak [1]) that "In conclusion, tax cuts appear to raise consumption even more than the standard Keynesian model would predict." This empirical finding suggests that the IS curve shifts out after a tax cut and gives an affirmative answer to the title question of this article by Arak "Are Tax Cut Stimulatory?" This answer contradicts the assertion of those who believe that an income tax cut, viewed from all potential capital suppliers, means a higher post-tax real interest yield, making an incentive to supply mbre capital to the market (or, equivalently, making it more expensive to hold back the capital supply by doing activities such as consumption). The following will take into account this interest-rate effect on savings and answer how large the elasticity f of real savings with respect to the real interest rate would have to be in order to make the IS curve shift inward (or downward) after an income tax cut. What we will do is to make a comparative static analysis of the goods - market equilibrium condition: investment +.government expenditure + export = savings + taxes + import. Since the derivation is somewhat mechanical, we will save it in the following footnote1. And here we will just write down the

- 2 - I conclusion. The conclusion is as follows: For an income tax cut to shift the IS curve inward (or downward), we need sr, > P, where sr, is the elasticity of real savings with respect to the after-tax real interest rate (i.e., sr, = percentage change in real savings/percentage change in the after-tax real interest rate), MPC - the marginal propensity to consume, and APS E the average propensity to save. Footnote 1. First, we will note the following general mathematical relationships, as they will be repeatedly used in the derivation. For any arbitrary differentiable function Q = f(X,Y,Z), we have dQ - aQ dX + 2Q dY + -Q dZ = X Q dX y Q dY Z Q dZ, thus aX x aY Y 3z z dQ (X Q)dX + Y Q) dY + Z ) dZ Q Q a X Q Y Y Q DZ z orn simpler notations, Q* + QXX* + OyY* + QzZ*, where a '*' over a variable means the percentage change in that variable and a symbol such as QX means the partial elasticity of Q with respect to X. aXY Next, in particular, if Q= aXY then Q* = X* + Y* - Z*, by logarithmic differentiation (where a is a constant). Also, if Q = aX + bY, where a and b are constants, then

/ - 3 -aX bYY 2 (aX + bY)]Y* * = [ aX+bY a (aX + bY)]X* + [ aX+bY )]Y* aX bY aX+bY aX+bY Now, we proceed to make the comparative static analysis: Let i - real investment (which is a function of the real market interest rate r), g E real government expenditure, X - real export, s real savings (which is a function of (i) the real after-tax interest rate r' - r(l-t), t being the tax rate, and (ii) the real after-tax income y' 'Y (1-t), y representing the pretax real income), ti. other real indirect taxation (so that ti + yt = total real taxation), and m - real import. So the goods-market equilibrium condition is: i(r)+ g + X = s (r',y') + yt + ti + m And this 'quantitative comparative static analysis' continues as follows. (i(r) + g + X)* = (s (r',y') + yt + ti + m)*; assuming o = g* = X* = ti* = m* and i s i+g+tX Ws E s+yt+ti+m and _-. yt Wy - s+yt+t+m representinq the three weights Wyt s+yt+ti+m We have: i*i = s*Ws + (yt)* Wyt; but * = ir r* (shere ir is the real-interest-rate elasticity of real investment demand), and s* = sr r'* + Syy'* = s[r(l-t)]* + sy[y(1-t)]* -t Yt Sr[r* +(l-)t*] + Sys[Y* +(~ )t*]; thus, i* Wi = s*Ws + (Yt)*Wyt implies Wiirr* = U s jr'[r* + ( )t*]+Sy[y* + (lt)t*1 r I- Y 1-t~~~~~~~~~~~~ + lyt (Y* + t*)

-4 - / = WlsSr r* + Wssr ( )t* + w5sy/* + ISSy, (-T)t* + WytY* + Wytt* and -t -t+ r*(Wiir - WsSr,) = t*[Wssr,(l) + WsSy'(1) + Wyt + y* [WsSy' + Wyt]; let D - Wiir - Wssr' then D < 0, since ir,<O transposing D to the right - hand side, we have r* = t* D-1 [Wssr'(-t) + Wssy'(4) + Wyt] +y*D-1 [Wssy, + Wyt] Now, for a tax cut, which means t*< 0, to result in a lower equilibrium market real interest rate which means r* < 0, we need -t -t ] [Wssr'(l- ) + Wssy'(t) + Wyt] < O; i.e., Ws(l-t) (sr' + Sy) + Wyt < 0, or, Ws(yt) (Sr' + Syl) - Wyt >, i.e., Ws(T- -) (Sr' + Sy') > Wyt or, Sr, + sy, >Wyt(tW) = ( ) (1t) = ) (i-) hence, sr, > (yt) (lt) - y, = () (l-t)- Sy, (Y) (l-t) - y' _ ' 'y 's = ( (1 as s 8y'- s s s - s y' MPC = AS (1-MPS) = AP APS APS Therefore, other things being equal, for t* < 0 to result in r* < 0, the economy MPC needs to have Sr' > PSy

- 5 References [1] Arak, M., "Are Tax Cut Stimulatory?" Review of Economics and Statistics, February 1982. [2] Shapiro, E., " "Supply-Side" Economics: A Diagrammatic Exposition," The Nebraska Journal of Economics and Business, Spring 1981.