The McKinnon Portfolio Balance Model

K.7 (#689 in RFD Series)

INTERNATIONAL FINANCE DISCUSSION PAPERS

THE MCKINNON PORTFOLIO BALANCE MODEL

by

Lance W. Girton

Discussion Paper No. 16, August 22, 1971

Division of International Finance

Board of Governors of the Federal Reserve System

The analysis and conclusions of this paper represent the views of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or its staff. Discussion papers in many cases are circulated in preliminary form to stimulate discussion and

comment and are not to be cited or quoted without the permission of the author.

August 22, 1972

The McKinnon Portfolio Balance Model*

Introduction

In his closing comments at the Chicago Conference on Monetary Problems of the International Economy Harry Johnson expressed the opinion that McKinnon's contribution to the conference was of fundamental importance and would have large impact on future work. The insights provided by McKinnon are extremely valuable, but I argue there are fundamental problems in the analysis, In this paper I will set out a model that preserves the general tenor of McKinnon's model but that is designed to eliminate internal inconsistencies in McKinnon's treatment. Perhaps one reason why McKinnon's model has not been more widely used is because of the problems that one encounters in attempting to understand and use the model as it now stands. Also, the model will be extended to cover explicitly the effects of policy changes on the employment of labor and capital. It seems rather strange that McKinnon did not attempt to analyze the effects of policy changes on factor utilization. McKinnon's model can best be appreciated perhaps as an attempt to find out where the Keynesian short-run analysis will lead one over the long-run when asset demands are made consistent with portfolio balance considerations. *I have benefited from discussions with Dale Henderson, John Morton, and Don Roper on different points raised in this paper. The opinions expressed

in this paper are the author's and cannot be taken as representative of the views of anyone else in the Federal Reserve System.

1/ The model discussed here is set out and analyzed in Section I of Ronald McKinnon "Portfolio Balance and International Payments Adjustment", in Monetary Problems of the International Economy. Edited by R. Mundell and A.K. Swaboda. The University of Chicago Press, 1969. Only the basic closed economy model is discussed here,

2/ Monetary Problems of the International Economy, p. 399,

The Model McKinnon uses the following four equations in the closed 5, economy version of his model.

- 1.1 E (¥,M,B,K,i) - yY = 0 Commodity market flow condition

1.2 L @,i) -M=0 Money market stock condition 1.3 R (,i) - B20 Bond market stock condition 1.4 A (W,i) - Kk =0 Commodity market stock condition

List of symbols E - Total flow magnitude of expenditures Y - Income M - Money supply B - Government bonds K - Capital stock - 4 - Interest rate on government bonds - Stock demand for money

Stock demand for government securities

> wt a

~ Stock demand for goods (capital)

(The symbol A has been substituted for McKinnon's C)

The spirit of McKinnon's analysis is Keynesian in that wages and prices are taken as fixed, people hold stationary expectations regarding income and the interest rate, and the relevant income variable is assumed to be actual measured income. The focus of the analysis is, however, on long-run asset equilibrium, not the short-run asset

disequilibrium situation at which Keynesian analysis is usually directed.

nS

1/ McKinnon, portfolio Balance and International Payments Adjustment", pe 210.

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In Mckinnon' s equilibrium the capital stock is in equilibrium so no nev investment is occurring. Also, McKinnon makes the demand for securities explicit and consistent with portfolio palance theory. He wakes equilibrium asset demands depend only on measured {neome and the interest rate. There was apparently some disagreement at the conference oF whether the nodel is long-run or short-run jn nature. Un the one hand Johnson in his summary discussion said that the model is directed at Long-run problems in terms of &@ long-run stationary state analysis» while Kruger in her comments said that “Nas the model presently stands, it is manifestly short-run in qature." The divergent interpretations arise perhaps first, pecause McKinnon deals with a period of time long enough for asset markets (inc Luding the market for physical capital) to clear for asset demands that do not depend on qnitial wealth levels; put 4 period of time short enough for prices t© be treated 85 fixed. This period of time may not exist. Second, without proper explanation McKinnon puts together three long-run asset equilibrium equations with

one shorter-run adjustment equation for the goods market. The

; 1/ Moneta Problems of the International Economy, PP- 399 and 249.

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short-run adjustment equation does not play any role in the McKinnon’ 8 Long-run comparative statics satyste-n

McKinnon Lumps rogether fires and households and writes desand functions that jnclude poth sectors’ demands, which I think introduces some possibility of conrusivn. Here 4 distinction will be made between demands by ultimate wealth holders and demands by cirms. Firms own physical capital (K) and employ Labor (nN) to produce 4 single output (Q) . profit maximizing pehavior py firms is assumed.

Households consume goods and hold money a)» variable jnterest rate

government ponds ()> and claims om firms (v). To simplify, without

° comsumption function should be spelleé out. Lf ultimate wealth holders Long-run demands for money and securities are satisfied, for the existing

in the more familiar form of c= £Gre where = MrBtV. When sY. The asset equilibrium conditions

and the consumption income equality are interdependent. I drop the —

latter for the Long-run comparative statics analysis done here-

5

altering the qualitative comparative statics conclusions, it is assumed that the firms hold neither money nor government bonds, and households do not hold stocks of goods.

- tieKinnon uses only one interest rate explicitly in his model though he consiters twe eceparate assets that provide the owner with a yield. His justification for doing this is “The return on the most liquid of these owner ship claims... ownership claims on hese which bear interest would be a rate of interest equal to a Some of the policy shifts that McXinnon contemplates change supplies of government securities relative to ownership claims on firms, without presumably altering the relation between the rates of return, How can this be except if the market treats the assets as perfect substitutes, at. least for ranges of relative asset supplies considered? At the risk of possibly misinterpreting the ‘antent" of iicKinnon, I assume that rates of returns on government bonds and ownership claims on firms to be equal because households treat the assets as perfect substitutes. The demand for government bonds and claims on firms then is written as

the demand for the sum of non-monetary assets of households.

a Ye 1/ “innon, p. 208.

2/ Treating all securities as @ homogeneous bundle, I think, is @ useful and appropriate simplification at this level of abstraction, given the problems that MecKinnon's model was built to deal with.

1) F# F(Y,i), where

2) F=BtV. In equilibrium

3) B+V=F(,i).

Equation (3) replaces wicKinnon's equations

1.3 B= R(Y,4) and

1.4 K= ay, 4).

The money equilibrium condition is

4) M=2 L(,i), which is the same as McKinnon's condition.

The fixed output price implies that the typical firm treats the demand for its output as piven” It is assumed that firms are competitive in the market for financial capital. The firm maximizes profits under conditions of fixed wages, elastic supply of funds, and rigid output price. The firm will adjust its use of capital and labor so as to produce the fixed output that it can sell at the going price, for the least cost -~ Loe, maximize profits. The cost minimizing condition is for the firm to use capital and labor in amounts s0 that the ratio of the marginal products is equal to the ratio of the rental

2/

rates. The marginal product of capital here has only a time dimension

ee

1/ Mundell, hn Exposition of Some Subleties of the Keynesian System", Welwirtschaptliches Archis (Dec., 1964) p. 220.

2/ Only fortuitously will the maximum profit position of the firm under these conditions be at a point where the marginal product of

labor is equal to the wage rate and the marginal product of capital is equal to the interest rate. The fixed amount of output that the typical firm can sell, given the fixed price of its output, prevents the firm from moving to a position where the marginal products are equal to the rental rates.

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since capital is merely accumulated consumption goods. The opportunity

cost of holding capital is the interest rate ~~ the cost of borrowing

funds to finance the holding of capital per time period. Algebraically

the condition is

5) 5- me where w t3 the wage zate and MPN (K,N) and

MPK (K,N) are respectively the marginal products of labor and capital. If production takes place under conditions of constraint returns

to scale, and i will determine the capital/labor ratio. The output

demand together with equation 5, and the production function,

6) Y= N£(k), where kw = K/N, will determine K and N given W and 2 The financial claims on a firm will eara their holders 4

stream of income that will be the difference between the revenue

of the firm and the wage bill. Given the assumptions made here

this earnings stream will be expected to continue indefinitely at

the present level. ‘The aggregate value of claims on firms will be

We have then the following equations in the comparative

statics model.

UB e 1/ See Henderson and Quandt, Microeconomic Theory, New York: McGraw-Hill 1958, pe Sle

2/ Since government spending and taxes are not needed to make the

points I want to make here, I assume they are zero except for a lump

gum tax on the population to pay for the interest on the government

debt. ‘This lump sum tax will change, of course, whenever i or B changes. See McKinnon's article for & discussion of the model with positive government spending and taxes. Government debt is treated as outside debt. Private disposable income is equal to the value of output since taxes just match the interest payment on government debt.

1/ + MeL Wi).

1 BaF Wi) - ES,

tr y= HEdd)-

wow, fc kh 2/ i £! .

These conditions are sufficient to determine the comparative statics equilibrium values of the four variables y, i, & and N for given levels of mM and B.

Eouations q-1v differ from vinnon' 8 equations 1.2, 1.3 and 1.4. McKinnon seems to have misspecified the asset equilibrium sonditiones It is argued here that the jnelusion of only one jnterest rate means that the jemand for non-monetary assets iS 4 demand for their sun. On.:the demand side McKinnon's R(Y,1) and the part of A(MY,2) that represents claims on firms has been aggregated to F(Y;1)- Firms have 4 demand for physical capital given by the production f£ynetion (equation II1), the marginal condition (equation Iv), and the levels of Yi» and We In the usual Keynesian analysis the capital stock is taken a6 fixed at other than its equilibrium value °° Loess net investment ts taking place. Muadell hes argued that in the short-run meynesian model with fixed prices» the marginal condition does not hold because with fixed capital stock the only decision

1/ 1 am assuming that there are no pank OF non-bank financial inter mediaries in the system.

2/ For the production £ynction (111) ¥ * weds), the wen = £ - Kf! and the HPK = ¢', where ¢' denotes the partial derivative of £ with respect to k and a" <0.

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the firm faces is how much labor to hire in order to produce the output demanded -- assuming this can be done profitably.2/ However, when the firm can choose its capital stock, and get the capital it wants, then profit maximization will establish the marginal condition (equation IV). Equations (III) and (IV) replace the part of McKinnon's equation 1.4 that represents demand for physical capital by firms. Given that prices are

fixed, the value of claims on firms will only fortuitously be equal

Y-wN

L

to the value of the capital owned by firms. The level of V = replaces K as the supply of claims on firms available to ultimate wealth holders.

Using the equations I-IV the model can be analyzed graphically. Equation I is the same as McKinnon's money market equilibrium equation and is plotted as the MM curve in Figure 1. The MM curve has 4 slope

di. Wy 0!

dy

Figure 1

—_———— 1/ Mundell, "An Exposition of Some Subtleties of the Keynesian System."

2/ Subscripts Y and i denote partial derivatives with respect to these variables.

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The SS curve is the securities market equilibrium schedule, taking {nto account the supply of claims on firms in addition to the value of government ponds. pifferentiating (II); holding B and W “gixed the fol Lowi ng is obtained:

8) o2 Gy” wi) av + Fi * v/i) at x wht oe

pifferentieting the production function and the marginal conditions

eliminating ak and solving for dN as 4 function of dr and dY, then ' 1/

9) ane 1/t & . wil ai, where oe WE 2 > OF cee CE-kke")

Substituting this expression into equation § and solving for the slope

in, ¥ space, then

v 10) $f = i Y 20. Ww

Yo i a To the right of the iM curve the demand for money is greater than the fixed supply of money implicit in the cu uTve and to the left the demand for money is smaller than the supply of money The + and signs on the MM curve denote areas of excess demand for and supply of

money - The + and - sign on the ss curve denote areas of excess demand

1 Differentiating the marginal condition (rv) : = ie, holding w constant then dk * =a di. pifferentiating the production function

(III) ¥ * née) we get d¥ = net dk + fd. substituting jnto the expression for dy the expres gssion for dk obtained by assuming firms are

minimizing cost, then dY = se ai + fdN. Solving for aN one obtains

1 | an.= = a¥ - wie di.

~lilo-

for and supply of securities. The slope will be negative 4£ the partial effect of am income change 60 the demand for securities exceeds the value of the partial effect of an income change oT the

supply of claims on f£irmss and the slope will be positive 4£ the

reverse is the case. One cannot be certain of the sig® of the slope of the SS curve. tt appears rhat the definite negative sign that McKinnon gave to the slope of his pond curve was the result of misspecification of the asset equilibrium conditions. t will assume

that if the SS curve is positively sloped, its slope is less than the 2/ raf slope of the YM curvee Along the ss curve the demand for securities

is equal to the fixed level of government ponds plus the supply of claims on f£irmss where the supply of claims om firms is consistent with cost minimizing behavior.

1/ The denominator of the slope must be negative since each term is negative. The numerator is Fy ° 1 (£22 F°7 Ll NE_- Nw Y j¥ —F se Y

LL (|

i\ NE.

Lf the marginal effect of the level of {ncome on the demand for securities 4s not smaller than the ratio of security holdings t° income by more than the ratio of government debt to income then the ss curve must have & negative slope. A positive sloped SS curve would not seem to be an unlikely possibility. For Latter reference ane ghovld vote that if the geeurt dcomnd fonction is homogeneous of degree one in {neome (here real {ncome) an p20 then the ss curve will have & slope of zeTOe

2/ The curved arrows are put in to indicate graphically the range of the admitted glopes of the SS Curves

3/ One would like to set out same simple dynamic postulates and argue that the sS~curve having 4 slope greater than the WM curve would not be consist with stability. There ares however » gome not 80 obvious problems in sett: out dynamic adjustment equations that are consistent with this type of mo Ir would seem natural to make ay/dt depend on the flow excess demand for

when the capital stock is not at desired leveis the marginal condition 4 have written qt will not hold. Second, the proper expression for flow investment demand cax/dt) 18 aifficult to formulate.

~12°¢

eriments done bY McKinnon put with ntical

erforn the exP The 1m curve is ide

One can P a clearer ynderstamding of what ‘is happening ibrium curves put the SS curve: incorporates: the qirative comparative

to his money equil model. The qual

ns made here in y MeKinnos as

@eiter atio

ics conclusion ea negative slope.

stat ka question

ssumed to hav fying the mode Does the

e $s curve is a

clearly speci by McKinnon.

Long 45 th

By more 1 we can as

not answered, long-run level of move in the same direction

employment of Labo

n this podel? new long-run

ase in the

as income i nt ust incre

Labor emp Loyme . injection of

Clearly is an outs ide

equilibrium e new jong-run equilibrium

and employment of is an outst

4f the shock

capital must increase in th de qnjection of money = Licy om the level of output and the effects om 1

tn each case the

expansionary po abor .

ther in their

employment in the second casee

4nforce each ©

capital/ 1 employment in the first case and capital However, 20 outside gncrease in the money supply ™47 have mees on the level of employment of labor 4n the Long-TUn» nt bonds has contren

e in the ‘gupply of governme

s on the Long-TU2 emp Loymen

tside {ncreas t of capital.

and an ov

dictory effect

1/ The outside injectior of money oF gov a trans x payment.

as being made 25

ernment ponds can be viewed

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To facilitate the graphical analysis of the question of the effects of policy changes on the long-run utilization of capital and labor, two additional curves are drawn in Figure 2. The W curve represents combinations of r and Y that will keep the profit maximizing utilization of labor constant and. the K curve represents combinations of x and ¥ that will keep the profit maximizing demand for capital constant. The BW curve must have 4 negative slope and the X% curve must have 4 positive siope.? Policy changes that cause the stock of

capital to rise will shift the intersection of the ™ and SS curve

1 ~ 6. T7 tn footnote 1 page 9 dN = : dy - Nog di was derived by assuming the

production function and the mar ginal conditions hold. Solving for the ay and di that will make dN = 0, We get dil_ = cf" < 0. ‘The definitior . _ AYN NE

of k differentiated gives dK = Ndk + kdN. Substituting in the dN that satisfied the production conditions» and dk = den di derived

footnote 1 p. %then aK = N (£-k£") di + K GY. ce" £ »4 Solving for the ratio of di to dy that will keep the desired capital stock constant then di|_ = ~ Kcf£"_ > 0d Y-kf'

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into the area to the right of the K curve and policy changes that cause the employment of labor to rise will shift thesintersection of the MM and SS curves into the area to the right of the N curve. Areas to the left of the K curve and the WI curve are analogously defined. By use of the diagram one can see the effects of policy changes on f{org,-Ten capitel aad dabor market utilization, as well as on income and the interest rate. . Policy: Comparative Statics Analysis

The model can be solved algebraically for Y,i,K and N for given changes in the policy variables by differentiating equations I-IV and using Cramer's rule. Doing this and substituting for dk and dN in the

money and securities equations then in matrix form .

|e Ly) ey yee | lal.

ca]

I assume the determinant of the coefficient matrix is positive

(A>0). This is the same condition as slope MM - slope SS > 0.

By using dN = lay - Nai and ag = RAY + NE'_ 8 ai,

£ ef"f ¥ eff" i which are implicit in the solution for di and dY, dN and dK can be solved for explicitly.

An increase in the supply of money by the outside method will shift the iff curve to the right. In the new equilibrium income

must be higher than in the initial equilibrium position and interest

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rates will be lower if the SS curve is negatively sloped, 1€es Fy > ve If the SS curve is positively sloped, 1.€es the supply of securities is more responsive to income changes than is the demand for securities, then an outside increase in the money supply will increase jnterest rates. The effect of the outside increase in the money supply on the long-run employment of capital will be positive unless the SS curve is positively sloped, and hes a greater slope than the Kk curve. 1/

A necessary though not sufficient condition for ax/dM <0 is Fy < uk” While dK/dil < 0 cannot apparently in principle be ruled out it would seem to be an unlikely case.

An outside increase in the money supply will increase employment unless the H curve is flatter than the SS curve (and the $s curve is negatively sloped). A necessary though not sufficient

condition for -an increase in the money supply to reduce employment is

that Fy > 7 = where W* = % is totally implicit

i

1/. The effects on Y, i, and KX of a change in money by outside method are

F. oo ¥ witf ' L ” oc. > (0) ? : ¢" ay/dM = i__icf N

eY- 3)

depending on the slope of the SS curve, and

ak f, + Vy NE'w [Fy - We! yy aM Y i cff"i XY e

wealth. One would think it unlikely that the income responsiveness of the depand for financial claims only will exceed the yatio of implicit wealth to income.

| It was pointed out previously that if there are 7° government gecurities qn the system and the demand function for securities is nomogencovs in income then The Sg curve js flat.

In this case an outside injection of money results in an unchanged interest rate, increased income» and Bigher levels of utilization of poth labor and capital. This is 4 curious result. Outside injections of money do not affect the interest rate but do effect other real magnitudes in the economy e It can be shown that 2 flexible price version of this model will demonstrate full classical neutrality under the same sort of assumptions - The only aifference is the weaker assumption of homogeneity in prices» put not real {ncomes in the demand for securities if needed.

1 The expression for the effect of an outside increase in the money supply om the emplo oyment © labor is: 4

an/aM = =

. rs) substituting 4 = at - < - Wy = ¥ - ‘ into the above expressions

“ur, +he wet & - 2

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Another method of changing the money supply is by an open market operation. This involves both a change in M, shifting the MM curve, and a change in B, shifting the Ss curve. An open market operations can be viewed as 4 combination of an “outside” change in uM and an “outside” change in B. Before analyzing this policy combination it is ueeful to analyze the effect of an outside increase in government bond supplies in jsolation.

An outside increase in government bonds shifts the SS curve

up inereasing the equilibrium level of income and the interest rate. The employment of labor must rise since both income and interest rate changes work in the same positive direction. The increase in government securities drives UP interest rates, leading to a shift to more labor intensive production. The employment of capital in the new equilibrium may increase or decrease depending on the relative slopes of the MM and K curves. Lf dK/di is small then the increase in interest rates will have only 2 gmall substitution effect and the employment of capital will expand as the demand for output rises. A decline in the equilibrium capital stock would seem to be a possibility. Tit is assumed that the factor labor is raw labor with no embodied

human capital. The source of the labor service flow is not bought or sold and is fixed in total amount.

2/ The effects on Y, i, HN, and Zz of a change in government bonds supplied to the public by outside methods are

L. : ly t d¥ . i di ow aN | LN (0,486, ) Bh? BO > an 7" ¥ ( cf" *y) > 0 - et and A _ offi Y ax = Sif tL 0.

dB

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An expansionary open market operation will shift the 1M curve to the right and the SS curve down. Interest rates will be lower in the new equilibrium 4£ the SS curve is negatively sloped. Only if the ss curve is positively sloped and ly * + Fy - u <0 will interest rates rise. Income will expand if the demand for the total of financial wealth is a positive function of interest rates, which we assume to nota The demand for capital in the new equilibrium position will have increased except if Fy + ly - - <0, surely a remote possibility. The long run employment of labor will increase unless Y + Fy > i/ie The drop in interest rates has a more severe dampening effect on long-run labor employmerit when the money supply is increased by gngide methods than when money is increased by outside methods because the decline in the supply of government securities intensifies the interest rate decline.

er ne . 1/ Solving the equation system set out previously for dB = - dii we get v

tnt te wilt x » and Ly Fy =W; > 0, where J, is tne partial effect of the 2 interest rate on the demand for total financial assets. A180,

L+FL JV ai (*Y * %). di A - 2/ The expressions for dk and dN are . r& (L, +F cw - NES Ww (dy * + Fy @K aK = LY ty cet i ¥ and

aN N

—

aN BT Gy + BE + F, = 1/i)' dM NE, Gy * *y a

cf" a

—A e

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One cannot carelessly identify the affects of policy changes on income with the effects on long-run labor employment. Policy changes: may, in the long-run, alter capital to labor ratios. Inside changes in the supply of money especially may have very different

affects on the employment of labor ané the fewei of income.

Lance Girton Economist International Finance Division

1.

2.

Bibliography

Henderson, J. and Quandt, R.> Hicroeconomic Theory. A Mathematical Approach. McGraw-Hill Book Company, lew York, 1958.

McKinnon, R. “Portfolio Balance and International Payments © Adjustment", Monetary Problems of the International Economy. Edited by R. itundell and A.X. Swoboda. Chicago and London:

The University of Chicago Press, 1969.

Mundell, R. "An Exposition of Some Subtietries in the Keynesian System". WYeltwirtschaptliches Archis (Dece, 1964) pp. 301-14. Reprinted in Edward Shapiro, Macroeconomics: Selected Readings

_Harcoust, Brace & World, Ince, New York, 1970.