An Optimizing Model of Household Behavior Under Credit Rationing

PETER MONT£ EL*

T HAS LONG BEEN RECOGNIZED that the transmission mecha nism linking monetary policy to the real sector depends on certainI structural and institutional characteristics that may be spe cific to a particular country at a given moment in time. 1 The proper formulation of monetary policy therefore requires analysis of how the transmission mechanism can be expected to operate under the set of circumstances that happens to be relevant to the particular economy under review.2 For a typical developing country, it may often be appropriate to assume that the private sector has no access to international capital markets, that the assets available to savers are limited to money (that is, the liabil ities of the banking system) and physical capital, and that interest rates on loans from the banking system are controlled at below market levels (that is, that credit is rationed). The analysis of the transmission mechanism for monetary policy in such a setting, however, is not well developed. The purpose of this paper is to take a first step in the analysis of the mechanism by which monetary policy is transmitted to aggregate demand under these circumstances by analyzing the

• Mr. Montiel, an economist in the Developing Country Studies Division of the Fund's Research Department when this paper was written, is now in the Macroeconomics Division of the Development Research Department, The World Bank. He is a graduate of Yale University and the Massachusetts Institute of Technology. 1 Such characteristics include the menu of available financial assets, the degree of substitutability among these assets, the extent to which various assets affect private net worth, the existence of various legal restraints such as interest rate ceilings or prohibitions on capital flows, and the exchange rate regime, among others. 2 Papers on the transmission mechanism for monetary policy under diverse circumstances include Modigliani (1963), Tobin (1978), Laidler (1978), Blanch ard (1980), and Modigliani and Papademos (1980}.

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behavior of a representative household under credit rationing in the general context set forth above. 3 The next section sets out a simple framework suitable for examining the issues involved. In Section II an optimizing model is developed for analyzing house hold behavior in the absence of credit rationing. The implications of imposing a constraint on the available supply of credit are examined in Section Ill. Section IV considers the effects of a change in the supply of credit and in the loan interest rate. A final section summarizes the main results.

I. Monetary Policy, Credit to the Private Sector, and Aggregate Demand

To investigate the issues, it is helpful to begin with a particularly simple framework. Consider a developing economy consisting of a government, a monetary authority, and a nonbank private sec tor. For simplicity, there is no commercial banking system. The government buys goods and services in the amount g, collects net revenues t, and covers any associated deficit in its budget by borrowing �da from the monetary authority and �fa from sources abroad. The government does not borrow directly from the do mestic private sector; its budget constraint4 is therefore

�d � g - (1) G + �G = I. The private sector accumulates assets either by investing in physical capital, i, or by hoarding, �m0. It finances this accumu

lation by saving, s, or borrowing from the monetary authority, �dp. The private sector's budget constraint is i + �m0=s + �dp. (2 ) FinaJiy, the monetary authority creates money, �m5, through the process of issuing credit to the public and private sectors, �da and �dp respectively, and acquiring foreign exchange reserves, �r. The behavior of the monetary authority is thus subject to the constraint (3)

3 Supply-side effects of credit policy are not considered here. For discussion of such effects in a developin� country setting, see Cavallo (1981), Taylor (1981), and van Wijnbergen (1983). 4 Interest payments by the government are not explicitly identified; for present purposes, they can be considered as already included in net revenues.

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Changes in the central bank's foreign exchange reserves arise from transactions with the private sector and with the govern ment, D.rpand D.rc respectively. If one assumes for simplicity that all government spending is on domestic goods, then �=�; � that is, the proceeds of net foreign borrowing by the government are exchanged for domestic currency at the central bank.5 The accounts of the public sector can be consolidated by substi tuting identity (1) into identity (3) and using equation (4) to produce

D.rp + [(g - t) + D.dp] = D.m5• (5) The expression in brackets is the exogenous component of the flow supply of money-that is, it is the portion of this flow that is directly controlled by the authorities under a fixed exchange rate regime. One can denote this exogenous component D.m5:6

D.m5= (g-t) + D.dp. (6) Identity (6) helps to illuminate the options faced by the authorities in this economy. They can choose to target the supply of money (through D.m5), the fiscal deficit, or the volume of credit to the private sector. However, targets for only two of these variables can be chosen independently. The third will be determined by identity (6). Whatever the combination of variables chosen as targets by the authorities, identity (6) makes clear that the supply of credit to the private sector is the key policy instrument that allows mon etary policy to be conducted independently of fiscal policy in this type of economy. For example, if the authorities set targets for those variables which dominate macroeconomic policy discussions in industrial countries-the supply of money and the fiscal defi cit-then the supply of credit to the private sector must be set as a residual to reconcile these policy objectives. Of course, the combination of instruments that should in prin ciple be targeted by the authorities depends on the relationships

5 Allowing for government spending on foreign goods, government receipts of income from abroad (for example, through transfers), or for both would merely clutter up equation (6) without adding anything of substance. 6 Identity (6) can also be written as 5 !:1m = 6.d + 6.fc; that is, the exogenous component of the flow supply of money differs from the flow of domestic credit by the amount of external government borrowing.

©International Monetary Fund. Not for Redistribution 586 PETER MONTIEL of these policy instruments to the ultimate goals of macroeco nomic policy. The supply of credit to the private sector has not, in fact, typically been relegated to a residual role but has become an important policy instrument in its own right. One role of fiscal subceilings under domestic credit ceilings in Fund-supported ad justment programs, for example, is to promote availability of credit to the private sector. This concern is based on the suppo sition that, in a context in which the supply of domestic bank credit is rationed and in which the private sector has no access to external funds, private aggregate demand will be constrained by the availability of bank credit (see Keller (1980)). More impor tant, a strategic component of private demand-private invest ment-is believed to be particularly sensitive to the supply of bank credit.7 According to this view, therefore, in this type of economy monetary policy works at least partially through a direct effect of the supply of credit to the private sector on private spending, specifically on private investment. 8 There are theoretical reasons to believe that, in a regime of credit rationing, the quantity of credit extended to the private sector would appear directly in private demand functions. Such a conclusion derives from the general principle discussed by Clower (1965) and Patinkin (1965), and extended by Barro and Grossman (1971), that under rationing the quantity constraints faced by agents will become arguments in their behavioral functions. Moreover, the argument can be made that private investment in particular can be expected to increase when the supply of credit expands: as long as the credit constraint is binding, the marginal product of capital exceeds the opportunity cost of funds, creating an incentive to expand the capital stock when the credit constraint is eased. In this setting, therefore, the transmission mechanism from monetary policy to aggregate demand would be relatively simple. The supply of bank credit to the private sector would itself appear as an argument in the private investment and consumption functions. But to say that the transmission mechanism from monetary policy to aggregate demand in the setting under scrutiny takes the form of the direct appearance of the supply of credit to the private sector in consumption and investment functions is obviously not

7 Empirical evidence of the sensitivity of private investment to the availability of cred1t has recently been provided by Blejer and Khan (1984). 8The effect is "direct" in that the effect on aggregate demand from supplying credit to the private sector is not transmitted simply through the money supply.

©International Monetary Fund. Not for Redistribution HOUSEHOLD BEHAVIOR UNDER CREDIT RATIONING 587 enough to satisfy the policymaker. One would also want to know what determines the total impact on aggregate demand of a given change in the supply of credit to the private sector as well as the components of this total effect-that is, the extent to which the effect on aggregate demand operates through changes in consump tion instead of changes in investment. These points can be illus trated by rewriting identity (2) as i(t::..dp, . .. )+t::..m0(t::..dp, ...)=s( t::..dp, ... )+t::..dp. (2a) In this form, equation (2a) illustrates the basic result of Clower (1965) that, under rationing, the supply of the rationed com modity enters private sector behavioral functions directly. Equa tion (2a) imposes a constraint on how changes in the supply of credit affect the behavior of the private sector: ai as at::..m0 - 1 (7) at::..dp at::..dp+ at::..dp = ; that is, changes in the flow of credit must be entirely consumed by changes in investment, dissaving (consumption), hoarding, or in a combination of these. Policymakers, however, are likely to be interested in the individual partial derivatives in identity (7). The direct effect on aggregate demand of a change in the supply of credit is given by ai/at::..dp -as /at::..dp. Thus the change in aggre gate demand will differ from the change in the flow of credit to the extent that private hoarding is affected (that is, am 0/at::..dp is non zero). Furthermore, if private investment is to be encouraged rather than private spending per se, the partial derivative ai/at::..dp will be of particular interest. There should be no presumption that an increased supply of credit will be devoted dollar for dollar to investment. As identity (7) makes clear, consumption and hoard ing represent competing uses of any funds that are made available. Although it is true that the marginal product of capital will exceed its opportunity cost under credit rationing, the same in general can be said of the marginal benefit derived by the household from consumption or from accumulating cash balances. Once this simi larity is recognized, it becomes apparent that it is not even neces sary in principle that the sign of ai/at::..dp be positive. That is, identity (7) does not constrain the sign of the partial derivative of private investment with respect to the supply of credit to the private sector. These issues are obviously crucial for the formulation of mon etary policy in the developing country framework. To derive the

©International Monetary Fund. Not for Redistribution 588 PETER MONTIEL shape of the functions in identity (2a), one needs to examine the determinants of a representative household's response to a change in the availability of bank credit. The next section, with its analysis of household consumption and portfolio behavior in the absence of credit rationing, begins that task.

11. An Optimizing Model of Household Consumption and Portfolio Decisions

The ultimate concern in this paper is analysis of private sector behavior under credit rationing. As a frame of reference, this section sets forth the basic model and analyzes private sector behavior in the absence of credit rationing. (A constraint on credit availability will be introduced in the next section.) In view of the absence of markets for equity shares in the economy being consid ered, households are assumed to hold physical capital directly. Thus the private sector collapses into simply a household sector, and the behavior of a representative household is analyzed. The household's objective is to maximize the discounted value of its own and its descendants' utility-that is, it seeks to maximize the discounted value of utility over an infinite horizon. Its instan taneous utility function takes the form u'>O, u"preference p, so that the objective functional becomes

V0 = f'u(c,)e-P' dt. (9)

At time t, the household's net worth is a,. As discussed, the household can hold assets in the form of physical capital k, and

money m,. Holdings of money and capital can exceed net worth because additional funds can be borrowed from the banking sys tem. With this borrowing denoted as d, the household's balance sheet at time t is (10) The household's motive for holding capital is Fisherian. Invest ment in physical capital yields a return, y, that is given by the production function: y, = y(k,); y ' >O, y"

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Holdings of capital can be adjusted both instantaneously and costlessly. Several alternatives are available to rationalize the holding of money. The most common are to include money in the utility function, in the production function, or in both (see, for example, Sidrauski (1967) and Levhari and Patinkin (1968)). The approach taken here is slightly different. Consumption is assumed to be a costly activity because of the existence of transaction costs, which are treated as explicit outlays and called H,. The household can reduce these costs by holding money. Transaction costs are thus given by H,= H(m, c,)

H2>0, H•• >O, H12 < 0. (12) We assume that His homogeneous in the first degree-that is, the transaction technology is such that costs per unit of consumption depend on the ratio of cash balances to consumption. Thus H can be written in intensive form: H,=h (m,lc,)c,; h'O =h(v,)c, (13) where v, =m, le, and the signs of the partial derivatives of h follow from those of H. 9 The household's problem is to choose time paths for c,, k,, and m, so as to maximize V. It must do so subject to both a budget constraint and a balance-sheet constraint. Its budget constraint governs the evolution of its net worth, a,, over time; that is,

a,= y, - rd, - (1 + h,)c, (14) where r is the interest rate charged on bank credit, and a dot over a variable indicates a time derivative. Equation (14) states that total saving a, is equal to total income minus interest payments minus transaction costs minus consumption. The second con straint on household behavior is that the present value of the. household's lifetime consumption cannot exceed the present value of its lifetime resources. To impose this condition, the

9 A similar transaction technology is used by McCallum (1982) to introduce a medium-of-exchange motive for holding money. The approach taken here dif fers from McCallum's only in that transaction costs are incurred in the form of goods rather than of leisure.

©International Monetary Fund. Not for Redistribution 590 PETER MONTIEL present value of the household's net worth is required to converge to a nonnegative number:10

lim e-n a, ;;::: 0. (15) The problem can now be summarized as follows:

max V= (u(c,)e- P' dt, (c,. kto m,) Jo subject to a0 given a,= y, - rd, - (1 + h,)c,

lim e-rt a, ;;:::0

c,, k,, m, ;;::: 0 for all t, where y, d, and h, are given by identity (10) and equations (11) and (13), respectively. The household's problem is to find paths for consumption, capital, and cash balances to maximize its dis counted utility over an infinite horizon. Applying the Maximum Principle, one has the current-value Hamiltonian: ,= u(c,) + A., a,- pA.,a,

where A, is the eo-state variable associated with a,. The first-order conditions are A.,[y '(k,)- r] = 0 (16) A,[-h'(m,/c,)-r)=O (17)

u '(c,)- A,{(1 + h(m, le,)]- h '(m, le,) m, le,} = 0 (18)

j,_, = (p- r)A,, (19) and, after the appropriate substitutions, a,= y(k,)- r(k, +m,-a,)- [1 + h(m,lc, )]c,. (14a) These conditions can readily be given economic interpretations. First, note that equation (14a) merely restates the budget con straint. It can be shown that, as long as u'> 0, A, will be nonzero

10 Sachs (1982) demonstrates that inequality (15) is equivalent to the condition that the present value of household consumption does not exceed the present value of household resources.

©International Monetary Fund. Not for Redistribution HOUSEHOLD BEHAVIOR UNDER CREDIT RATIONING 591 for allt in an optimal solution. 11 Equation (16) can thus be written as y'(k,) = r. This expression simply states the familiar result that the stock of capital should be chosen to equate the marginal product of capital and the interest rate. As long as r is constant (as has been as sumed), the optimal stock of capital-call it k* -will also be constant over time. The household's demand for capital therefore becomes k* = k(r), k'= lly"< O. (16a) Similarly, equation (17) becomes -h'(m, le,) = r, which imposes the restriction that the marginal gain in income (recall that h' < 0) derived from the reduction in transaction costs associated with an increase in cash balances should equal the opportunity cost of money. Again, with r constant this requires that the ratio m, le, be constant over time at its optimal level, denoted v*. Thus, the household's demand for money satisfies

m: = v*(r)c,, v' = - 11h " < O . (17a) Equation (18) states that the marginal utility of current con sumption should equal the marginal utility of saving. To see this, note from footnote 11 that A., can be interpreted as the marginal utility of wealth. But a one-unit decrease in consumption leads to more than a one-unit increase in wealth, since abstaining from consumption enables the household to avoid the associated trans action costs. These are given by h, - h,' v,. Thus the total increase in wealth from a one-unit reduction in current consumption is 1 + h, - h,' v, and therefore the term A.,(1 + h,- h,' v, ) represents the marginal utility of saving. According to equation (19), the marginal utility of wealth will be increasing along the optimal path if the rate of time preference exceeds the rate of interest. The rationale for this relationship is that, when the rate of time preference is "high," future consump-

" Let V� be the maximum of V when the problem is solved at time 1. v,• will be a function of the household"s initial net worth at timet; that is, V� = V(a, ). It can be shown (see lntrilisator (1971, Chapter 14)) that h, = V'(a, ). As long as the household is not satiated (u' > 0), an increase in a, that permits higher = consumption at some point must increase V*, so that V'(a,) h, > 0.

©International Monetary Fund. Not for Redistribution 592 PETER MONTIEL tion will be heavily discounted. Thus the optimal path will concen trate more consumption in the present, reducing the amount of wealth carried over to future periods; given diminishing (flow) marginal utility (u" < 0), the marginal utility of wealth will there fore rise over time. To analyze the properties of the solution, use equations (16a) and (17a) in equations (18), (19), and (14a). Differentiating the resultant version of equation (18) with respect to time and re arranging terms produces _}..1(1+h* -h '*) (18a) Cr- u"(c,) i..r = (p - r)A.r (19) (14a) where y* and h* are evaluated at k*(r) and v*(r). The qualitative properties of the solution will depend primarily on the sign of p- r. If p> r, i..rwill be positive for all t. Because (1 + h* -h '* )/ u"(c,) is negative, c, must also be negative. Thus, when the rate of time preference exceeds the rate of interest, the household will concentrate consumption in the early part of its horizon, and consumption will follow a declining profile over time. The initial level of consumption will have to be set such that the path for household net worth implied by equation (14a) does not violate the transversality condition (15). Of course if p

To see this necessity, note that, with a constant c (say, C), equation (14) becomes the differential equation: li1 =y(k*)- r(k* + v*c- a,) - [1 +h(v*)]c,

©International Monetary Fund. Not for Redistribution HOUSEHOLD BEHAVIOR UNDER CREDIT RATIONING 593 with solution

a, = (a0 + lt)en - li, where

_ y(k*) - r(k* + v*c) - [1 + h(v*)]c a = . r Because r > 0, this equation is unstable. If c > c*, a, will diminish continuously, and the present value of household net worth will converge to t lim e-r a, = lim e-''[(a0 + lt)e'' -a] r-oe r-oo =ao+li, which is negative if c > c*. 12 This negative sign violates the inter temporal budget constraint (15), so that paths with this property are infeasible. Paths with c < c* are feasible but cannot be opti mal, since for every such path it is always possible to find an

alternative path CI where CI = ec* + (1 - e)c, with 0 < e < 1 > which is also feasible but is preferred to the path c because it provides higher consumption at every instant. One may therefore conclude that, along the optimal path, c = c* and a, = a0. It can be shown that c* exceeds initial consumption for the case p < r and is less than initial consumption when p > r. The properties of c* can therefore be derived by solving equa tion (20) for c* : y(k*) -rk* +rao c* = 1 + h* + rv* = c(r, ao). (2 1) Differentiation of equation (21) with respect to a0 immediately yields the result that an increase in initial wealth increases house hold consumption uniformly at every instant-that is, c2> 0. The

effect on c* of an increase in r is more problematic. Note first that an increase in r is also an increase in p, since r is assumed to equal p. It can be shown that an increase in r will increase c*, leave it

12 Notice that ao +a= ,-1(ra + ra0) = y(k*)- r(k* + v•c- a0)- [1 + h (v*)]c. This can be written as y(k*)-r(k* +v*c* -ao)-(l+h (v*)]c* -[l+h (v*)+v*](c- c*). From equation (20), this in turn becomes -[1 + h(v*) + v'"](c- c*), which is negative when c > c*.

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unchanged, or decrease it according to whether r initially exceeds, equals, or falls short of a threshold value, F. This threshold itself depends on a0. 13 Equations (16a), (17a), and (21) describe the household's no tional demands. That is, these are its demands for capital, money, and consumption in the absence of constraints on its ability to borrow. Its notional demand for credit can be derived by substi tuting these equations in identity (10), yielding: d* = k* (r) + v*(r)c(r, a0) - ao

= d(r, a0). (lOa) The household's demand for credit thus depends on the interest rate and on the household's net worth. With consumption held constant, an increase in r would reduce the demands for both money and capital, thus reducing the demand for credit. Allowing for the effect of r on consumption, however, makes the overall effect ambiguous. If r 5 r, consumption will not increase, and the demand for credit will fall. If r > r, the increase in consumption might cause an increase in the demand for money that would be large enough to increase the total demand for credit. In contrast, an increase in household wealth will unambiguously reduce the demand for credit. The reason for this outcome is that, with increased own resources to finance its asset holdings, the house hold's demand for bank credit is diminished. Although consump tion will increase, the resultant increase in the demand for money is less than the increase in household resources. 14 To summarize, in the absence of credit rationing the household instantly adjusts its holdings of money and capital so that the marginal return from holding each is equal to the interest rate. With a constant interest rate, these asset holdings remain constant over time, so hoarding and investment are both zero. If the rate of time preference and the market rate of interest are equal, a flat time profile of consumption is optimal. The optimal level of this

13 From equation (21), dcldr = [(1 + h)(a0- k*)- vy]/(1 + h + r v)z. Let M = (1 + h)(a0- k*)- vy. Note that M is a continuous function of r and ao. The continuity of M and the properties of the roduction function imply the existence of an interest rate rsuch that M(r,ao) = 8. Since M.(r, ao) >0and M2( r, ao) > 0, the conclusions in the text follow. 14The properties of the function d can be established by substituting equa tion (21) in equation (lOa). This yields d(r, a0) = [ ( 1 + h)(k* -ao) + v•y]l (1 + h + rv*). Substituting from equations (16a) and (17a) and differentiating with respect to r and a0 y1elds the results discussed above.

©International Monetary Fund. Not for Redistribution HOUSEHOLD BEHAVIOR UNDER CREDIT RATIO ING 595 profile is that which sets consumption equal to permanent income. Thus there is no saving, and with constant net worth, the house hold exhibits a constant demand for bank credit equal to the difference between its asset holdings and its net worth.

Ill. Optimal Household Behavior Under Credit Rationing

In the previous section the household was assumed to face a perfectly elastic supply of bank credit at the interest rate r. Thus the demand for credit expressed in equation (lOa) was always satisfied. But what are the consequences for household behavior of restricting the supply of credit to the private sector while hold ing the rate of interest constant? Suppose the supply of credit is d, where d < d( r, a0). (22) The analytical consequence of inequality (22) is that the house hold is faced with an additional constraint: its effectivedemand for credit cannot exceed d. That is,

k, + m, - a, :5 d. (23) The household's problem may now be solved again on the as sumption that this constraint is binding. The constrained Hamil tonian is �= u(c,) + A.,a, - pA.,a,+ 7r,(d - k, - m, + a,), and the first-order conditions become A.,(y'(k,) -r] -7r, =O (24) A., [-h '(m,lc,) -r] -7r,=O (25) u '(c,) - A.,{( l + h(m,le,)] - h '(m,lc,)m, le,} = 0 (18) k, +m,=a, + d (23a)

X, = (p - r)A., - 'IT, (26)

a, = y ( k,) - rd - [ 1 + h (m, Ic,)] c , . (27) To interpret these conditions, note first that equation (23a) simply imposes the assumption that the credit constraint is binding by replacing inequality (23) with an equation. The significance of the constraint is that, since a, is fixed at any instant of time, the household can no longer adjust the total size of its portfolio in-

©International Monetary Fund. Not for Redistribution 596 PETER MONTIEL stantaneously, as was true in Section II. Its total asset holdings must equal a, +d. It can, however, change the composition of its portfolio between k, and m, instantaneously, subject to constraint (23a). This constraint is depicted as the portfolio opportunity locus AB in Figure 1. Its slope is -1. Equations (24) and (25) imply y'(k,) = -h'(m,/c,) . (28) Equation (28) states the rule for optimal portfolio composition that is, the marginal rate of return on holding capital should be equal to the marginal rate of return (the marginal reduction in transaction costs) on holding money, conditional on the level of consumption. In Figure 1, LL represents the locus of all com binations of k, and m, that satisfy equation (28), given c,. The slope of LL is dk, = _K>o · dm, LL J y"c, An increase in k, reduces the marginal product of capital. This decrease in turn requires a reduction in the marginal return from holding cash balances, which then requires an increase in m, le,.

Figure 1. Portfolio Choice with a Credit Constraint

a,+d A L

0 �------��------�----�B------m/ v•c, a,+d Ill,

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LL Given c,, m, must rise. Thus has a positive slope. Because both k and mic rise as one moves to the northeast along LL, the assumptions that Y11 <0 and h11 >0 guarantee that the (equalized) rates of return on both assets fall in this direction. The optimal composition of the portfolio is (m,E, k,E), given by the intersection of LL and AB at the point C, where the marginal revenues from holding money and capital are equal and the total amount of the two assets held equals total portfolio size. Algebraically, m,E and k,E solve equations (23a) and (28). The variables m,E and k,E represent effectivedemands for money and capital. It is convenient to express these as - m, E = m(c, a,+ d) -h;' m, lc,2 m1 = 11 hil >0 Yr - t le, ll y m2-- 11 hll >0 (29) Yr - t ICr and

k,E = k(c,, a, + (J) 2 h11 m lc k1 =-m�= 111 lz��/ <0. Yt- t e, h;' le, k2 = 1 - m2 = 11 h11 > 0. (30) Yt- t Ie,

Note that the interest rate r does not enter equations (23a) and (28) and therefore does not directly affect the effective demands m,E and k,E. The supply of credit d, however, affects each of these variables directly. The properties of the partial derivatives are LL readily established geometrically. An increase in c, causes to shift to the right in the same proportion (since h' depends on m, le,). Thus in Figure 1 the intersection C moves to the southeast along AB. The reduction in k, must be just offset by the increase in m, if the size of the portfolio is to be kept equal to a,+ a. An increase in a, + d, in contrast, causes AB to shift to the right by the same amount. Thus C moves to the northeast along LL, increasing both k, and m,. The total increase in k, and m, must equal the increase in a, + d. Figure 1 also illustrates the notional demands represented by equations (16a) and (17a). The former is a horizontal line from k,*, whereas the latter is a vertical line from v*c,. Because these

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notional demands satisfy y'(k*) = r = -h(v*), they also satisfy equation (28). Thus the horizontal and vertical lines must inter sect on LL. This point is labeled U in Figure 1. Because inequality (23) is satisfied by all points in the interior of the triangle OAB, the assumption that the credit constraint is binding means that the unconstrained optimum U must lie outside this triangle. Thus U is located to the northeast of the constrained optimum along LL: (31a) vf

y'(k,£)> y'(k*) = r (32a)

- ' v* = . -h'(v; )> h ( ) r (32b) Because U lies to the northeast of C, the constrained rates of return on both capital and money exceed the unco:,strained rates of return.

One can now interpret the ratio 'TT, /A., in equations (24) and (25). Equation (24), for example, can be written as

= n, y'(k,E) - r /A., .

The ratio 'TT, /A., is positive. It measures the amount by which the marginal return on assets in the household's portfolio under credit rationing exceeds the interest rate. Substituting from the preced ing expression in equation (26) yields

�� = [p - y'(k,E)]A.,. (26a) If p-y '(k,E) <0, it follows that along the optimal constrained path the marginal utility of wealth will be falling. In other words, the household can be expected to reduce current consumption in order to accumulate wealth over time when it is severely rationed in the market for credit. In effect, the rate of time preference p is less than the market rate of return, which under credit rationing is given by y '(k,E) = -h '(v;:). Yet if p-y '(k,£) > 0, the marginal utility of wealth will be rising, and the household will dissave along the optimal path. That such behavior indeed occurs along the optimal constrained

©International Monetary Fund. Not for Redistribution HOUSEHOLD BEHAVIOR UNDER CREDIT RATIONING 599 path can be verified by examining the first-order conditions (24)-(27), ( 18), and (23a). Equations (24), (25), and (23a) have already been shown to describe instantaneous portfolio behavior, summarized by the effective demands k,£ and m,£. To find the optimal path, turn now to the remaining conditions, equations (18), (27), and (26). Note first that equation (18), which governs the intertemporal allocation of consumption, is identical to its unconstrained version. It obviously has the same interpretation as made earlier. Budget constraint (27) differs from its uncon strained version (equation (14a)) only in that the credit supply constraint d replaces the notional demand for credit k, +m, -a,. The last condition, equation (26), was discussed immediately above. The implications of equations (18), (27), and (26) for the optimal path can also be examined by converting them into a set of two differential equations in c, and a, or in A.,and a,. The former option is chosen here. Differentiating equation (18) with respect to time and re arranging yields

= (�, /A.,)a,-�,a, c ' u" lu'-�,m, lc,' (33)

where m, a(c,, h,' > 0, a,= a,+ a)= 1 + h,- c, h:' m,m2 ��=�(c,a,+d)= >0. c,2 Substituting from equation (26) yields

. [p- y ' (k,E )]a, - �, a, c .o..:...._---;:-:--':-....:.....__,c.:---'---=-=----'- (33a) 1 = u"!u ' -�,m, le, Recall that k, and m, are both functions of c,and a, + d; equations (27) and (33a) therefore represent a pair of differential equations a,. in c, and To examine their behavior, consider first the case that p> r. Linearizing equations (27) and (33a) in the neighborhood of the equilibrium c, =a,= 0 produces �I]= [Qll Ql2] [de,] r = (34) la, a21 a22 da, '

©International Monetary Fund. Not for Redistribution 600 PETER MONTIEL where Y"k a-Qa au = = 0 u"lu�- !3�/c Y "ka2 -Qhl a12 t-' <0 =- u"lu 1 - j3m/c a21 =-a <0 a22 =-hI> 0. !3 !3 The definition of implies that = y"k�> so the numerator in the definition of a11 is zero. Given this implied definition of !3 and the !3, k , expressions for a, 1 and k2, the numerator in the definition of a12 can be written as

y"k2a(l k,h k2a y"k2 < - 11 ) = a l- 0. ( 1 +�h � ':�� J The determinant of the coefficient matrix is -a12 a2,� < 0. Thus the equilibrium defined by c, a, = 0 is a saddle point. Figure 2 illustrates the optimal= path for the credit-constrained household in the neighborhood of this equilibrium. The slope of the locus a, = 0 is given by

de, _ a22 . = > O da, a21 J a=O Because a11 = 0, the locus c, = 0, in contrast, is vertical in the neighborhood of the equilibrium defined by c, = a, = 0. Since a22 > 0, a, is positive when a, is to the right of the a, = 0 locus and negative when a, is to the left. In addition, the negative sign of a12 implies that c, is negative to the right of c, = 0 and positive to the left of it. This explains the directions of the arrows in Figure 2. From these arrows the stable path can be determined to lie in the first and third of the quadrants traced by c, = 0 and a, = 0--that is, the stable path must have a positive slope. 15 This stable path is denoted SS 1 in Figure 2. Given initial wealth a0, the household chooses a level of consumption c0 on this path, at point A. House-

15 The slope of the saddle path in the vicinity of B is

On _ a22 de A, > de > 0 = = ' da - da �-o ss· a21 a21 I root. I where At is the negative

©International Monetary Fund. Not for Redistribution HOUSEHOLD BEHAVIOR UNDER CREDIT RATIONING 601

Figure 2. Optimal Consumption and AssetAccumulation Under Credit Rationing, with p > r c, c,=o I / S' c' _j

_.L___ _ _

(}L..__L a, r 11, (/ hold wealth and consumption increase over time from (a0, c0 ) if a0 a' until the stable equilibrium values (a' , c ') are reached. To gain some intuition into this solution, consider the character istics of the stable equilibrium (a 1, c ') at point B. Setting c1 = a1 = 0 in equation (33), one finds that at point B,

y'(k' ) = p. Because p > r, k 1 < k* -that is, the equilibrium value of the cap ital stock under credit rationing is smaller than in the absence of credit rationing. Likewise, using the portfolio equilibrium condi tion (28), one has V1

©International Monetary Fund. Not for Redistribution 602 PETER MONTIEL

ciently small, asset returns will also exceed the rate of time pref 1 erence p. 6 This motivates the household to defer consumption until the future. The household depresses current consumption and engages in a positive level of saving. As wealth increases, the marginal utility of saving falls, so that the level of consumption rises over time, although saving remains positive. Thus con sumption and wealth rise together, as is shown along the stable path SS ' in Figure 2. Thjs pattern continues until household wealth increases sufficiently for asset returns to fall to the rate of time preference. Note that under credit rationing household sav ing decisions depend, at the margin, not on the controlled interest rate but on asset returns. Compared with the unconstrained case in Section 11, the pres ence of a credit constraint reduces aggregate demand by reducing consumption when a0

1t \� �tta\%ht\ot�-a"ld to e<::.t-ab\i<::.h th-at \"\oatding mu�t be po�iti-,c along SS '-that is, the household's effective demand for money at (a0 , c0), given by m;, falls short of its eventual value m', and cash balances are continually accumulated as m,E approaches m'. To see this, differentiate equation (20) with respect to time:

. . . £ mza,. m, = m1 c, + (35)

Because m1 and m2 are positive, and because c, and a, are also positive along SS', m,E must be positive. A similar procedure, however, yields ambiguous results in the case of investment. Differentiating equation (30) with respect to time yields (36)

Because k1 is negative and k2 is positive, the behavior of k, along paths on whlch c, and a, are both increasing depends on the rela tive magnitude of k1 and k2 as well as on the rate of increase of c,

6 1 Th is relationship can be shown by analyzing the solution path in (A, a) space. When ao p along the solution path.

©International Monetary Fund. Not for Redistribution HOUSEHOLD BEHAVIOR UNDER CREDIT RATIONING 603

Figure 3. Behavior of th e Cap ital Stock Along th e Optimal Trajectory

C n

() L------�------L------{/11 a ' a, relative to that of a,. Rearranging equation (36) allows one to write

. (c, k2) . k, = a,+ k1a,. k,

Thus k, > 0 if c, la, < -k2lk,. This condition can be given a geometnc mterpretation. The slope of the saddle path SS ' in Figure 3 is c, la,. Equation (30) generates a family of isocapital loci in (a, c,) space with slope -k2lk1 > 0. To each member of this family corresponds a particu lar value of k. Consider the locus that passes through the equilib

rium point (a', c '). The slope of this curve is steeper than that of 1 the a = 0 locus. 7 Since the curve passes through the equilibrium

17 From equation (30), one knows that -k2fk1 = c, lm1. The slope of a=0 is -h 1/(1 + h-h1mlc). Since

c h1 _ (1 + h)(c/m) + 0 m 1 + h -hIm le - 1 + h - hIm le > > the isocapital curve through (a', c ' ) is steeper than the a = 0 locus.

©International Monetary Fund. Not for Redistribution 604 PETER MONTIEL point (a', c') at point B, the set of points (a, c,) that constitute this locus satisfies k(c, a, + d) = k' = k(c', a'+ d).

In other words, along this curve k, = k '. To the left of this curve kt < k ', since a reduction in at at a given c, causes a reduction in k,. Likewise, to the right of this curve kt > k '. If the stable path SS ' approaches the equilibrium point from above this locus, there fore, it will cross members of the family of isocapital loci with successively higher values of k,, until k' is approached when SS'

intersects the locus k = k' at the equilib�ium point (a', c '). Thus k, would rise over time along SS', and k, would be positive. Be cause y '(k/;:)> p, this relationship must hold for the case at hand. With the initial capital stock below its equilibrium value, the point A must lie above the k = k' locus, and investment must be posi tive along the path SS ' up to point B. Thus, although initial consumption is lower in the rationed than in the unrationed case, investment and hoarding, which were zero and negative respec tively when p > r in the absence of credit rationing, become posi tive when a0 a', the household dissaves by reducing its stocks of both money and capital.

When the rate of time preference does no1 exceed the con trolled interest rate (p !5 r), the dynamics of household behavior are exactly as above while the credit constraint is binding. As long as 1ft > 0, the dynamic equations (27) and (33a) are unaffected. In this case, however, as assets are accumulated their rates of return will fall into equality with the interest rate r before reaching the rate of time preference p. When y'(k,) =-h'(m,lc,)=r, the credit constraint ceases to be operative (from equations (24) and

(25), 1T, becomes zero) because the household's notional asset demands are satisfied. Thus the credit rationing ceases to operate before the rationed equilibrium (point B in Figure 2) is reached.

The removal of this constraint will occur at the combination of c, and at that lies on the optimal trajectory and satisfies

k* + v*c, = a, + d, (37) where k* and v* are the unrationed values of k, and mt let from Section Il. Equation (37) is depicted as the upward-sloping locus emanating from point C in Figure 4. It intersects the saddle path at point B, with coordinates (a", c"). Everywhere on and to the

Figure 4. Optimal Consumption an d Asset Acc um ulation :5 Un der Credit Ration in g, with p r c,

li, =()

o �------�--��--�------11, 11, +k V right of this locus the credit constraint does not hold, essentially because (given the available supply of credit d) the household's own resources a, are sufficient to finance its notional demand for assets. The locus has a positive slope because the demand for money increases with c,, and this increased demand can only be financed with larger a,. From point B on, the household's behav ior is as described in the unrationed case of Section Il. 18 One may conclude that, faced with a credit constraint, the household is forced to restrict its total holdings of assets to less

18 When p = r, it can be shown that the equilibrium level of consumption under credit rationing excee ds the stationary nonrationed level c• of Section U. Al though this result may appear paradoxical, it reflects the fact that an equilibrium cannot be reached under rationing in this case until enough savings have been accumulated to finance the household's notional budget asset demands. Because thjs asset accumulation is achieved with less use of credit in the rationed case, steady-state interest payments are lower. and thus consumption can be higher. The accumulation of these additional savings, however, requires that con sumption be initially depressed below its unrationed stationary level. Since this rising time profile of consumption, although feasible, was suboptimal in the absence of rationing, it follows that higher long-run consumption is associated with lower lifetime utility in the ratjoned case.

©International Monetary Fund. Not for Redistribution 606 PETER MONTlEL than its notional demands. Under diminishing marginal returns, the resultant rates of return on assets differ from the rate of time preference. The household responds to this discrepancy by alter ing current consumption in order to save or dissave. Its total demand for output is not affected to the same extent as its con sumption demand, however, since it chooses to accumulate a portion of its savings in the form of physical capital. As its asset holdings change, the household alters its level of consumption over time. A stationary state is reached when the level of asset holdings reaches a level such that the rates of return on both capital and money equal the rate of time preference when p > r or,

if p :5r, when the household's wealth has grown sufficiently that it can finance its notional asset demands out of its own re sources-that is, when the credit constraint becomes inoperative.

IV. Monetary Policy

The analysis thus far has examined household behavior in the absence of credit rationing and in the presence of a binding credit constraint. The preceding section also compared the levels of consumption, investment, and hoarding in the constrained and unconstrained cases. The effects of monetary policy-in the form of changes in the supply of credit to the private sector and in administered interest rates-are now analyzed under credit ra tioning. This section is divided into two parts, in which these policy measures are discussed in turn. The discussion is restricted to the case in which p > r, so that the credit constraint is binding throughout.

Effects of Changes in Credit Supply

Consider an increase in the supply of credit to the private sector that leaves the credit constraint binding. Recalling that the house hold's long-run stationary equilibrium satisfies c, = a, = 0 in equa tion (33), one again has

y'(k,) = p. Thus, the long-run desired value of the capital stock remains k ', as in Section Ill. The household will move to the same long-run value for the capital stock, regardless of the amount of bank credit

©International Monetary Fund. Not for Redistribution HOUSEHOLD BEHAVIOR UNDER CREDIT RATIONING 607 that it is offered.19 Similarly, by the portfolio equilibrium equation (28), the ratio of money to consumption must remain v' in the long run. The long-run levels of consumption and household net worth, however, are affected by the supply of credit. Given k' and v', equation (38) implicitly determines the long-run level of con sumption under credit rationing as a function of the supply of credit a: 0. y(k') - ra - [1 + h(v')]c' = (38) This equation yields de ' = -r dd 1 + h(v') < O. Thus the long-run level of consumption falls when the supply of credit increases. From equation (38) one can verify by inspection

that, with lower CI and higher a> the long-run level of household wealth is also lower. These results are illustrated in Figure 5. An increase in d causes the locus a = 0 to shift to the left by

{ -1< - ( l+ .) 0. In contrast, since da -k2 1 = = _1' dd k;k" k2

it follows that the locus k = k' also shifts to the left, and this shift exceeds that of a = 0. Thus the new intersection of these loci, at point D, occurs at a lower level of wealth ii and consumption c than was the case with a lower supply of credit at point A.

'"This neutrality of the long-run capital stock with respect to the supply of credit is similar to that established by Sidrauski ( 1967) with respect to the rate of inflation. In both cases, the result comes about because the long-run stock of capital must equate the marginal product of capital and the rate of time prefer ence. The latter is exogenous in Sidrauski's model and in this paper.

Figure 5. Dynamics of Adjusrmem to an Increase in rhe Supply of Credir when p > r c,

li, =0 c.. /li,=O / / / I I..,""' ' c -;/"' "I / I A .., 1 1 "' I I I, ,... I I I I 1 I I 1 I I I I I I I 1 s I J I I I I I I I

() L------�------L------a, Ii a' These results are not as paradoxical as they may seem at first glance. The lower long-run levels of wealth and consumption sim ply reflect a change in the time profile of the household's con sumption toward more consumption in early years, exactly as intuition would lead one to expect when the credit constraint is loosened. Recall that it was the imposition of the credit constraint that depressed current consumption relative to its unconstrained profile. When the availability of credit increases, the household can instantaneously increase its holdings of money and capital. As it does so, the marginal product of its asset stock declines; as this marginal product moves closer to its rate of time perference, the household's incentive to save diminishes. This is the mechanism that produces the shift in the time profile of consumption. Thus a decrease in consumption in the distant future is more than compensated by an increase in consumption in the near term. These points can be illustrated with the aid of Figure 5. When the long-run equilibrium moves from A to D, the household finds itself with a nonequilibrium stock of wealth. It therefore moves immediately to the saddle path associated with the new long-run equilibrium, at B. Consumption thus increases immediately. Be-

©International Monetary Fund. Not for Redistribution HOUSEHOLD BEHAVIOR UNDER CREDIT RATIONING 609 cause B is to the right of the new k = k' locus, the household's capital stock is higher at B. With both consumption and credit higher at 8, the household's stock of money must also be higher there. The household therefore uses the additional credit in part to increase its stock of capital and in part to add to its cash bal ances. It moves gradually along the saddle path SS' to the new long-run equilibrium at D. Note that consumption exceeds its initial level over the early portion of this path (BC). Notice also that the household's demands for money and capital are only temporarily increased by the increase in the supply of credit. Although holdings of both money and capital are higher at B than at A, investment and hoarding are both negative along BD. As the household dissaves, it draws down its stocks of both assets. In the case of capital, this can be seen by observing that movement to the southwest along BD involves crossing members of the family of loci k(c,, ��+ d) = k corresponding to successively lower levels

of k, since k = k' is steeper than SS' in the vicinity of D (see Section Ill). In the case of money, dishoarding must be occurring along BD because both consumption and household wealth are falling. When point D is reached, the initial capital stock will have been restored, and cash balances will have fallen in proportion to consumption. To summarize, an increase in the supply of credit to the private sector causes the household to change its time profile of consump tion, causing an immediate increase in the level of consumption. The household will temporarily bold larger stocks of both money and physical capital, but these stocks will be drawn down over time because the higher levels of consumption are associated with dissaving in the form of both negative investment and dishoard ing. Both money and capital overshoot their long-run values. The long-run desired stock of capital, however, will not be affected by an increase in the supply of bank credit, and the long-run desired stock of money will actually fall.

Effects of Changes in Interest Rates

It is easy to see that, in the current context, changes in the administered loan rate that leave the inequality p > r in effect will leave the long-run capital stock unaffected. The administered loan rate r is irrelevant to the determination of the long-run desired capital stock because that capital stock depends only on the rate of time preference. Under credit rationing, capital is acquired by

©International Monetary Fund. Not for Redistribution 610 PETER MONT!EL an act of saving, not an act of borrowing, and thus the opportunity cost of a unit of capital is the cost of deferring a unit of consump tion, which is given by the rate of time preference. To complete the analysis of the household response to a change in the loan rate, consider once again equations (23a) and (27).

Imposing a, = 0, setting k = k I' V= v'' and differentiating these equations with respect to r produces de -Cl dr = 1 + h (v') < O

da' -v'd dr = 1 + h(v') < O. An increase in the loan rate reduces long-run consumption be cause it reduces the household's permanent income by increasing its interest payments to the banking system. Long-run wealth is reduced because the reduction in consumption leads to a decrease in the long-run demand for money. With a constant long-run capital stock, the household's diminished demand for assets can be financed with a smaller long-run stock of wealth. The path to this long-run stock of wealth is illustrated in Figure

= 0 6. An increase in r shifts the locus a to the right by da a dr = y'k2 -h'm2 I a•o

= -d/h' >0. The new long-run equilibrium is at point C. The stable path to C must lie below the original equilibrium at A. Consumption thus falls to B, accompanied by an instantaneous portfolio shift from money to capital. The qualitative characteristics of the path from B to C are the same as those of the path from B to D in Figure 5-that is, savings, investment, and hoarding are all negative. Again, the increase in the capital stock is only temporary. The household's demand for money, however, declines monoton ically. Note that under credit rationing the loan interest rate is relevant to the household's decisions only in the sense that it functions as a sort of lump-sum tax that affects the household's permanent income. Permitting interest to be paid on money is somewhat more complicated. Equations (25) and (27) of the first-order conditions for the credit-constrained problem would have to be modified to

'(m, = A.,(-h le, )+ r, - r] - 1T, 0 (25a)

a, = y(k,) - rd + rmmt - (1 + h(m, /c,)]c,. (27a) Thus the marginal return to holding money is the reduction in transaction costs plus the deposit rate. Portfolio equilibrium now requires (28a) in addition to equation (23a). LL The effect of introducing a nonzero r, is to shift the curve in Figure 1 to the right and thus shift portfolio composition in favor of money. The household's effective portfolio demands now satisfy m/:: ,+ d) = 1n (rm, c,, a (29a)

k,E = k(r,. , c,, a,+ d), (30a) k1 where m1 is positive and equals -m1• The analysis of the effects of a change in r,.now follows familiar lines. An increase in r, leaves the y'(k) = p locus undisturbed.

Figure 6. Dynamics of Adjustment to an Increase in the Loan Interest Rate when p > r

c - ti, =() - � ------.;:;" ------...... -:=------c - -

li 11

Although the long-run capital stock is once again unchanged, the long-run ratio of money to consumption v' increases because this ratio is now determined by equation (28a) rather than by equation

(28). The ci, = 0 locus shifts to the left . As a result, the two loci will intersect at a point to the northeast of the original long-run equi librium. Thus, long-run consumption and wealth both increase. This relationship can be verified by using equation (23a) and a version of equation (27) that has been modified to take account of interest receipts on deposits. Consumption increases because of an increase in permanent income as a result of higher interest receipts and lower transaction costs, and wealth increases to fi nance the additional demand for money attributable to the higher level of consumption and larger ratio of cash balances to consump tion. The transition to this long-run equilibrium involves an imme diate increase in consumption and an instantaneous portfolio shift from capital to money, along with a gradual increase in consump tion and accumulation of wealth in the form of both money and capital.

V. Conclusions

This paper goes part way in assessing how monetary policy affects household behavior under credit rationing. Although the model of household behavior analyzed here has several features that future models addressed to this issue might retain, certain assumptions could usefully be relaxed. The most important is the fixed rate of time preference, which ensures constancy of the long-run capital stock. Although a fixed rate of time preference is a common assumption in models such as that presented here, it is by no means innocuous. A second important assumption is that of costless and instantaneous portfolio adjustment. The description of household behavior in the paper would have gained added realism if allowance had been made for the existence of costs in adjusting the capital stock. Unfortunately, that complication ren ders the model much less tractable, and the primary effect is on the dynamic path to long-run equilibrium, not on the qualitative properties of that equilibrium. Nevertheless, the analysis of house hold behavior under credit rationing presented here points to several important conclusions that are likely to prove robust in the face of such complications. First, the transmission mechanism for monetary policy under

©International Monetary Fund. Not for Redistribution HOUSEHOLD BEHAVIOR UNDER CREDIT RATIONING 613 credit rationing is particularly simple and direct. The inclusion of the supply of credit to the private sector in private behavioral functions, far from representing an ad hoc procedure, is actually a consequence of optimizing behavior on the part of households. This fact has important consequences for the specification of sav ing, investment, and money demand functions in many developing countries.20 Second, because the supply of credit to the private sector there fore has a direct effect on private investment, a case can be made, on the basis of growth considerations, for setting ceilings on public sector credit in such a way as to avoid "crowding out" the private sector. It should not be presumed, however, that what would be crowded out in the presence of credit rationing would necessarily be private investment. It is true that under credit rationing the marginal product of capital exceeds the loan interest rate, so that an unsatisfied notional demand for capital exists. But unsatisfied notional demands for consumption and money also exist in this setting, so that expanding the supply of credit to the private sector may occasion additional hoarding and dissaving as well as addi tional investment. Indeed, whether additional investment is likely to be a result in a cumulative sense is itself problematic, since the long-run stock of private capital cannot be increased by expanding the credit supply if the household's rate of time preference is constant. Third, changes in the loan interest rate that leave households credit-constrained have allocative effects similar to a lump-sum tax. Only the rate of time preference and the rates of return on existing stocks of assets affect intertemporal allocative decisions in this context. In particular, an increase in the loan rate does not diminish the long-run capital stock. To the extent that it depresses current consumption by reducing permanent income, a higher loan rate increases the demand for capital in the short run. Finally, an increase in the deposit interest rate may increase household saving in the short run and wealth in the long run, but it will decrease household demand for capital in the short run and

20Wong (1977) presents a money demand function in which the supply of credjt appears as an independent variable. It is interesting that this direct de pendence of money demand on the supply of credit to the private sector is an additional reason that the "offset coefficient,"' which links the balance of pay ments to the supply of domestic credit, in the "fundamental equation'' of the monetary approach to the balance of payments should not be expected to be -1 (see Montiel (1985)).

©International Monetary Fund. Not for Redistribution 614 PETER MONTIEL leave such demand unaffected in the long run. Only the short-run effect can be overturned in this model, by allowing for the in creased saving to yield an increased supply of credit. In the long run, the household's demand for capital cannot be increased by incentives to saving in the present model, since this demand is constrained by the rate of time preference.

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