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Thayer Watkins
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Imperfect Capital Markets

THE EFFECT OF IMPERFECT CAPITAL MARKETS
ON MACROECONOMIC CONSUMPTION:
A SYNTHESIS OF THE KEYNESIAN, PERMANENT INCOME,
AND RATIONAL-EXPECTATIONS APPROACHES

I. INTRODUCTION

Imperfections in the capital markets for consumers, such as debt limitations or higher interest rates for borrowing than for saving, may result in some households living from paycheck to paycheck, exactly consuming their current income. Other households may not be affected by the imperfections and allocate the expenditure of their income over an extended period of time. Aggregate consumption will depend upon what portion of income goes to the two types of households. The response of the two types to a change in income is quiet different. Including the two types of households in an macroeconomic model provides an interesting mutation considerably different from the model of the past. Macroeconomic models, however, have evolved considerably since 1936. In addition to elaborations on the basic Keynesian model there have been several distinct mutations. In contrast to Keynes assumption that consumer purchases depend upon current disposable income Friedman's Permanent Income Hypothesis makes consumption dependent on a long term expected income which is only to a slight degree influenced by current income. Friedman's model and its offspring, the Rational Expectations models, have implications for macroeconomic policy quite at variance with the Keynesian models. According to the Permanent Income/Rational Expectations approach a tempary tax cut will not stimulate economic activity significantly because consumers will adjust their expenditure plans over such a long time horizon that negligible effects will occur in the year of the tax cut. Yet perhaps this line of argument may claim to much. This theory may apply to many consumers, perhaps even most, but there are quite possibly other consumers who cannot act upon an anticipation of a tax cut because of cash and credit limitations, and who, for the same reasons, tend to spend the tax cut quickly. The relevant theoretical analysis for this problem is the effect of imperfections in the capital market on the time allocation of consumption, and a brief treatment is given in Appendix I. For further details see Watkins [5, 6 , 7, 8]. One major result of this analysis is that at any given moment there will generally be two groups of consumers; those who allocate the expenditure of their income over an extended time horizon and for whom the conventional theory applies, and those who live from paycheck to paycheck exactly consuming their current disposable income. The behavior of this latter group is optimal in that they are maximizing lifetime utility subject to the constraints they face. Their time preference in relation to the interest rate is such that at that time they want to borrow more against future earnings than the credit institutions will allow or they want to borrow at the interest rate for savings and lend at the interest rate for debt.

II. THE MODEL

The theory of the time allocation of consumption under uncertainty; see Watkins [7] and Hall [3]; indicates that in the absence of debt limitations or differential interest rates on borrowing and lending, the conditional expectation of consumption at any time is a function of the level of consumption at the time the conditional expectation was formed; i.e.,


E{Ct:cs} = f(cs).
 

Under special circumstances indicated in Appendix I the functional relationship is linear; i.e.


(l) E{Ct:ct-1} = a Ct-1 + b,
 

where a and b depend upon the difference between the subjective rate of time discount and the aftertax real rate of interest. The deviation of actual consumption from expected consumption depends upon the amount by which expected lifetime earnings changed as a result of the events which transpired since past consumption was determined. Watkins [8.] If the distributions of noncapital disposable incomes are independent over time, then the change in expected lifetime earnings is the difference between current noncapital disposable income wt and its conditional expectation based on the information set φt-1 available when ct-1 was chosen. This means that


(2) ct = a ct-1 + b + m(wt - E{wtt-1})
 

where the coefficient m is the marginal propensity to consume out of unanticipated income and it depends upon the time horizon, the after-tax real interest rate, and the subjective rate of time discount. In the model there is no random element in the determination of current consumption o ther than the uncertainty of income.

Relation (2) applies only to those consumers who are not credit constrained in their spending at the present time; i.e., those that have i nterest earning assets. The credit-constrained are those who have borrowed all that the credit institutions are willing to lend or that decline to borrow because of the higher interest rate on borrowing than lending. These consumers spend


(3) ct* = wt*.
 

In the model the aggregate expenditure and income of the credit-constrained consumers as a group are denoted by c* and w* and those of consumers who are not credit-constrained are denoted by c and w. The sums over the two groups are denoted by the upper case letters C and W. The crucial question is the shares of noncapital income going to each group. It is assumed that there are constant marginal shares


(4) wt = s Wt + q'
and
wt* = (l-s)Wt -q'.
 

Furthermore it is assumed that aggregate noncapital income Wt is linearly related to GNP, Yt, i.e., Wt = vYt + q" such that svYt + q and wt = (l-s)vYt - q where v is the marginal share of GNP going to noncapital disposable income and q is a simple function of the intercepts;i.e., q = q' + sq.
 

Aggregate consumption Ct is ct+ct* and thus ct = Ct -ct* = Ct- wt*. This relation along with (2) implies


(6) Ct = a(Ct-1 -wt-1) + b
+ m(wt - E{wtt-1} + wt.
 

This equation combined with (5) yields an aggregate consumption function of


(7) Ct= a(Ct-1-svYt-1)+b' + m(l-s)v(Yt - E{Ytt-1}) +svYt + u1t
 

where b'=b+(l-a)q and u1t is a random disturbance of zero expected value which is the net result of the random elements in the distribution of GNP to the noncapital incomes of the two groups. The model is completed with the usual conditions


(8) Yt = Ct + It + Gt + Xt-Mt
(9) Mt = g + hYt + u2t, E{u2t} = 0,
 

where I,G,X and M stand for, respectively, investment, government purchases, exports, and imports. In order to focus the analysis on the consumer sector of the model it is assumed that all components of aggregate demand other than consumption and imports are exogenous. This is not an unreasonable assumption in any case in as much as both the link between GNP and the long term aftertax real interest rate and the link between investment and the long term after tax real interest rate are weak. This assumption implies that only the sum of investment, government purchases and exports is relevant in the model. The sum I+G+X is denoted by A (for autonomous aggregate demand). The model; comprised of (7), (8), and (9); has the equilibrium solution


(10) Yt = K(At+aCt-1-asvYt-1+b' + m(l-s)vE{Ytt-1} + ut
 

where At=It+Gt+Xt, ut=u1t-u2t, and K=l/(l+h-sv-m(l-s)v). K is the short-term Keynesian multiplier for the model. In order to estimate the parameters of the model there must be some resolution of the matter of the unobserved conditional expectation E{Ytt-1}. By Muth's Principle of Rational Expectations [4] the value of Yt which is determined by the model should be the basis for the determination of the conditional expectation. Thus


(11) E{Ytt-1} = K(E{Att-1} + aCt-1 -asvYt-1+b' - m(l-s)vE{Yt t-1)
 

and therefore,


(12) Yt - E{Ytt-1} = K(E{Att-1} + ut)
 

A reasonable method to estimate At - E{Att-1} is by means of an autoregressive scheme and the coefficient of At-1 in the first order autoregressive scheme is close enough to unity that At-At-1 is used as an estimate of At-E{At t-1}. Note that Yt-E{Yt: φt-1} also depends on ut as well as (At-E{Att-1}) but this dependence will be ignored in order to keep the estimating procedure econometrically simple. When K(At-At-1+ut) is substituted for Yt-E{Ytt-1} in (7) and the model solved the result is:


(13) Yt = K'(At+aCt-1-asvYt-1 + b'+m(l-s)vK(At-At-1+ut) + ut,
 

where K' = l/(l+h-sv) and K= l/(l+h-sv-(1-s)v).

Both K and K' are short term multipliers. K gives the impact of an unanticipated increase in autonomous demand and K' gives the impact of an anticipated increase.

ESTIMATES OF THE MODEL PARAMETERS

In order to focus the analysis on the consumer sector of the model it is assumed that all components of aggregate demand other than consumption and imports (investment, government purchases and exports) are exogenous. This is not an unreasonable assumption in any case in as much as both the link between GNP and the long term aftertax real interest rate and the link between investment and the long term after tax real interest rate are weak. This assumption implies that personal dividend income plus rental income of persons with a capital consumption adjustment plus personal interest income less interest paid by consumers are endogenous. The estimating equation used is


(14) Yt = K'At+K'aCt-1-K'asvYt-1 +K'm(1-s)vK(At-At-1)+K'b'+ut.
 

The ordinary least squares estimate of (14) has significant autocorrelation of the residuals. The autocorrelation is eliminated when the quasi-differences zt-0.5zt-1 are used. The value of 0.5 was found to minimize the the autocorrelation of the residuals. The OLS estimate of the quasi-differenced form of (14) is


(15) Yt = 1.206 At+1.074 Ct-1-0.232 Yt-1+0.045 (At-At-1)+26.66
 

The coefficient of determination of the equation is 0.9966 and the Durbin-Watson statistic is 1.97. Because of the lagged endogenous variables in the equation the Durbin-Watson d test is not appropriate. Durbin's h statistic has the value of 0.3 which indicates no autocorrelation. Estimates of some structural parameters may be easily derived from the reduced form coefficients of (15); e.g. a=0.891; but the model is overidentified and so it is more convenient to use the two stage least squares estimates (2SLS). The 2SLS estimates of the import function and the relationship between noncapital disposable income and GNP are:


(16) Mt = -43.28 + 0.100 Yt
 


(17) Wt = -10.18 + 0.625 Yt.
 

Thus h=0.100 and v=0.625. The reduced form coefficients indicate that K'=1.206. Since 1/K'=1+h-sv this means that s=0.433. This is the estimate of the marginal share of noncapital income going to credit-constrained consumers; i.e., 43 percent. But there is an alternate estimate of s from the coefficient of Yt, which should be equal to K'asv. The value of this coefficient along with the previous estimates of K', a, and v imply a value of s of 0.346. In other words the model is overidentified. Later an estimate of s will be derived which from estimates in which the coefficients of At and Yt are constrained to be consistent with one value of s. But the unconstrained estimates of the coefficients are valuable in that one of the strongest tests of a model is to determine if, within the limits of statistical significance, the constraints implied by the model are satisfied by the unconstrained coefficients. The estimate of s from the cefficient of At implies that the coefficient of Yt-1 should be -0.291. The actual coeffient of Yt-1 is -0.232 and its standard deviation is 0.164, and therefore the difference between the actual and the computed is only 0.36 standard deviation units and not significantly different from zero at the 90 percent level of confidence.

The estimate of s from the coefficient of Yt-1 implies that the coefficient of At should be 1.131 rather than the actual value of 1.201. This difference is again only 0.36 standard deviation units. Thus the model passes this test superbly. Given the problems arising from the necessit;y of using proxy variables in the estimation this is a notable success. The coefficient of (At - At-1) should be equal to K'm(1-s)K. This implies values of m of 0.085 and 0.074, depending upon which estimate of s is used. Thus the marginal propensity of consumers who are not credit-constrained to consume out of unanticipated income is roughly 0.1.

IV. CONSTRAINED ESTIMATES OF THE MODEL PARAMETERS
 

If the model is corrrect the coefficients of the reuced form equation are not independent. If b1, b2, and b3 denote the coefficients of At, Ct-1, and Yt-1 in the reduced form equation, then consistency requires that


(18) (-b3/b2) + (1/b1 -(1+h)) = 0.
 

The vector of the coefficients B which minimizes the sum of the squared deviations subject to the constraint that g(B)=0 must satisfy the condition


(19) B = B0 + zM-1 (∂g/∂B).
 

where M is the moment matrix, B0 is the vector of the unconstrained least squares estimates, and z is the lagrangian multiplier. Equation (19) provides the basis for an iteration scheme for approximating the constrained least squares estimates of the coefficients. The constrained least squares estimate of the quasi-differenced reduced form equation is:


(20) Yt = 1.131 At +1.149 Ct-1 - 0.248 Yt-1 +0.075 (At-At-1)+28.1.
 

When the 2SLS estimates of h and v are used the other structural parameters have the values s=0.345, a=1.012, and m=0.134. The value of m is determined by the time horizon, the real aftertax interest rate, and the parameter a. The length of the time horizon consistent with the above estimates of a and m and a real aftertax of 0.02 is 7.5 years. The value of the short term Keynesian multiplier is 1.206, which coincidentally was the value of K' in the unconstrained estimate. This is a low value and probably lower than what would come out of a more complete version of the model. The general effect of the use of proxy variables in the model is to bias the estimates toward zero. Therefore the estimate of s, the share of noncapital disposable income going to credit-constrained consumers, is probably low. Therefore if the estimate of s is to rounded off to the nearest ten percent forty percent is a better choice than thirty. Forty percent as the share of noncapital income going to credit constrained consumers is consistent with cross-section studies and this figure is likely to survive any further refinement of the analysis.

V. The Dynamics of the Model

In the simple Keynesian model the entire impact of any shift in aggregate demand is concentrated in the period in which it occurs. In models involving Permanent Income or Rational Expectations the impact is distributed over an extended period. In the shift was anticipated then there is no impact from following through with the change. In the model presented in this paper the impact of a shift is may be relatively concentrated in the period in which it occurs, but there is also an impact over an extended period. The general nature of the response of the economy over time to an unanticipated change is shown in Table 1. The impact of an unanticipated unit increase in autonomous aggregate demand is K, which is equal to 1/(1+h-sv-m(1-s)v), and the first year impact on aggregate consumption is (sv+m(1-s)v times the first year's impact on output. The second year impact on output is the first year's impact times K'am(1-sv and the impact of all subsequent years is the parameter a times the previous year's impact.

VI. Conclusions

The theory of the time allocation of consumption under income uncertainty and credit rationing gives rise to a macroeconomic model which involves elements of both the Keynesian and the Permanent Income/Rational Expectations approaches. In the model there are two regimes of behavior for consumers. Some consumers choose to acquire interest-earning assets and allocate the expenditure of their income over an extended time horizon. The other consumers choose to live from paycheck to paycheck and thus not acquire interest-earning assets. For the consumers in this latter group consumption is equal to current disposable income.

This theory, supplemented by some simplified suppor squared deviations subject to the constraint that g(B)=0 must satisfy the condition


(19) B = B0 + zM-1 (∂g/∂B).
 

where M is the moment matrix, B0 is the vector of the unconstrained least squares estimates, and z is the lagrangian multiplier. Equation (19) provides the basis for an iteration scheme for approximating the constrained least squares estimates of the coefficients. model parameters raises some concern about the accuracy of the estimates but the results seem to indicate that roughly 40 percent of the noncapital income goes to consumers who are credit-constrained and live from paycheck to paycheck. The marginal propensity to consume of these consumers is 1.0. The marginal propensity to consume of those who are not credit-constrained is 0.1 and they have a time horizon of about eight years. Thus the aggregate marginal propensity to consume is (1.0)(0.4)+(0.1)(0.6) or 0.46. This along with a marginal propensity to import of 0.1 would result in a short term multiplier of 1.56. The short term multiplier effect is almost entirely due to the additional purchases of the credit-constrained consumers. The spending of those who are not credit-constrained serves to stabilize the economy since their spending constitutes a reservoir of demand that continues unabated despite temporary setbacks in the economy.

The model indicates the need to take into account both groups of consumers. Both the Keynesian and the Permanent Income approaches capture important facets of the economy. Because of this it is unlikely that either approach can be dismissed. Neither approach will give the complete answer to macroeconomic policy analysis and the model presented here shows that it is not necessary to eliminate either of them.

Although it is tempting to identify the credit-constrained consumers as the poor this is not necessarily correct. The credit constrained are the those whose subjective rate of time preference exceeds the real after-tax interest rate and expected rate of income growth. They are poor in terms of interest-earning assets but this is as a result of their choices and for them it is optimal. They are not necessarily poor in terms of income. On the other hand, the consumers who are not credit-constrained may not be rich but merely saving to compensate for an anticipated decrease in income. Rather than poor and rich a better characterization would be younger and older.

Table 1: The Impacts on Output of a Temporary Unexpected Unit Increase in Autonomous Aggregate Demand As a Function of the Marginal Share of Noncapital Disposable Income Going to Credit-Constrained Consumers
  The Marginal Share of Noncapital Disposable Income
Going to Credit-Constrained Consumers
Year  0.0 0.2 0.4 0.6 0.8 1.0
1 0.964 1.081 1.231 1.429 1.702 2.105
2 0.055 0.056 0.055 0.050 0.036 0.000
3 0.056 0.057 0.056 0.050 0.036 0.000
4 0.057 0.057 0.056 0.051 0.037 0.000
5 0.057 0.058 0.057 0.052 0.037 0.000
6 0.058 0.059 0.058 0.052 0.038 0.000
7 0.059 0.060 0.058 0.053 0.038 0.000
8 0.060 0.060 0.059 0.054 0.039 0.000

APPENDIX I.

Macroeconomics theory should be consistent with microeconomic theory. The relevant theory in this case is the time allocation of consumption. The decision on saving may be considered as an outcome of househol some simplified suppor squared deviations subject to the constraint that g(B)=0 must satisfy the condition


(19) B = B0 + zM-1 (∂g/∂B).
 

where M is the moment matrix, B0 is the vector of the unconstrained least squares estimates, and z is the lagrangian multiplier. Equation (19) provides the basis for an iteration scheme for approximating the constrained least squares estimaraint on a household's spending is that its financial assets, i.e. net worth, A(t) must not fall below what credit institutions will allow. If D(t) is the amount of debt that credit institutions will allow a household, then the constraint is that


(2) A(t) > -D(t).
 

Although credit do allow debt the amount of unsecured debt is relatively small, and therefore in order to keep the analysis relatively simple D(t) will be taken to be 0. Thus the constraint is that, for all t, A(t)>0. This is equivalent to the constraint


(3) ∫0tj(s)c(s)ds < ∫0t j(s)w(s)ds
 

where j(s) is the discount factor exp[∫0z - r(z)dz ] and w(s) is disposable noncapital income. In words the constraint is that cumulative discounted consumption expenditure at any age must be less than or equal to the cumulative discounted disposable noncapital income for that age. This, of course, treats unsecured consumer credit as being negligible. Graphically the constraint is of the form shown in Figure I. The Permanent Income Hypothesis takes into account only the constraint for the end of the lifespan; i.e., A(T)>0. For some consumers the only constraint that is restrictive is that A(T)>0, but for others, such as is illustrated in Figure I, the optimal consumption program involves their cumulative discounted consumption being on the constraint during some period of their lives. During such a period their consumption expenditure is equal to their current income, which necessarily entirely noncapital income. At other periods in the lives of these same consumers they may be off the constraint and have accumulated financial assets. During such periods the traditional analysis applies and an optimal consumption program requires that


(4) g(t) u'(c(t)) = v j(t),
 

where v is a constant, the Lagrangean multiplier. Consumers who are not credit-constrained pursue programs of consumption that take into account their income prospects over an extended period of time. The most pertinent bit of information for predicting the current consumption of these consumers is their past consumption. If there is no uncertainty (of income or interest rates, for example) then the necessary first order condition for an optimal consumption program is that


(5) (g(t)/j(t)) u'(c(t)) = (g(s)/j(z)) u'(c(s)).
 

If g(t) and j(t) are exponential in form; i.e.,


g(t) = exp(-kt) and j(t) = exp(-rt);
 

then the first order condition reduces to


(6) u'(c(t)) = exp[(r-k)(t-s)] u'(c(s)).
 

The parameter k is the subjective discount rate and r is the real rate of interest. If u'(c) is of the form c-m then the condition becomes


(7) c(t) = exp[(r-k)(t-s)/m] c(s).
 

This would mean that current consumption at t is just a scaled up or scaled down version of consumption in the past. More particularly, it means that consumption follows an exponential trend which depends upon the relative size of the subjective discount rate k compared to the real interest rate r. The trend would also depend upon m, the elasticity of the marginal utility function. If the marginal utility function is linear then the relationship between the consumption levels at different times is particularly simple. If u'(c)= q0-c and g(t)/j(t)= exp[(r-k)t] then


(8) c(t) = exp[(r-k)(t-s)]c(s) + (1-exp[(r-k)(t-s)]q0.
 

That is to say, consumption now is a weighted average of past consumption and the satiation level of consumption q0; i.e.,


(9) c(t) = ac(s) + (1-a)q0.
 

The weights of course depend upon the time period separating the two consumptions. Another way of expressing the relationship involved in (8) is that consumption now is a linear function of past consumption; i.e.,


(10) c(t) = a c(s) + b
 

In the above derivations c(t) represented the instanteous rate of consumption at age t. The same relationships could be expected to hold for the annual average consumptions. The analysis will now be in terms of annual consumptions, which will be denoted by subscripts. If s=t-1 then the linear marginal utility function leads to the difference equation


(11) q0 - ct = a(q0 - ct-1),
 

where a = exp(r-k). The solution of this difference equation can be represented as


(12) ct = q0 - at(q0 - c0),
 

even though the index on t starts at 1 rather 0. The parameter c0 constitutes a convenient way to express the Lagrangian multiplier for the optimization problem.

The dependency of ct on W is through its dependency on c0 which depends upon W.

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