return capital production technologies. Successful capitalists leave bequests to their offspring, thereby permitting them to internally finance some fraction.
Economic Theory 11, 171±200 (1998)
Credit market imperfections, income distribution, and capital accumulationw Joydeep Bhattacharya Department of Economics, State University of New York at Bualo, Fronczak Hall, Bualo, NY 14260, USA Received: June 3, 1996; revised version: February 4, 1997
Summary. This paper builds a model in which the distribution of income matters for capital formation, and uses it to analyze the eects of a simple policy intended to create a more equal distribution of income on the severity of certain credit market imperfections and, through this channel, capital accumulation. A neoclassical growth model is developed in which some capital investment must be externally ®nanced, and external ®nance is subject to a standard costly state veri®cation (CSV) problem. In particular, some fraction of the population is ``capitalists'', who have access to risky but high return capital production technologies. Successful capitalists leave bequests to their ospring, thereby permitting them to internally ®nance some fraction of their own investment projects. However some external ®nance is also required. This is provided by ``workers'' who save out of labor income. As is well known, the greater the capability of capitalists to provide internal ®nance, the less severe is the CSV problem. Thus bequests mitigate credit market frictions and, in that sense, promote ®nancial market eciency and capital accumulation. However, they also perpetrate income inequality. The structure is used to show that a policy that taxes the bequests of capitalists, and transfers the proceeds to workers, necessarily reduces the steady state capital stock. Indeed, when this eect is suciently strong, these redistributive tax/transfer schemes can reduce the total (wage plus transfer) incomes
w
This is a revised version of Chapter 2 of my doctoral dissertation submitted to Cornell University. I thank Bruce Smith, Karl Shell, Tapan Mitra, Eric Fisher, Mark Guzman, Rupa Chakrabarti, and an anonymous referee for their helpful comments, and Helle Bunzel and Mark Ehlen for their assistance with the computations. I also thank seminar participants at the University of AÊrhus, the 1996 Summer Meetings of the Econometric Society in Iowa, the 1996 SEDC Meetings in Mexico City, the University of Southern California, and the Federal Reserve Bank of Cleveland for their comments. Research support from a Sloan Foundation Doctoral Dissertation Fellowship and the Center for Analytic Economics at Cornell University is gratefully acknowledged.
172
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of workers, as well as their welfare. Thus some simple policies intended to redistribute income can be highly counterproductive. JEL Classi®cation Numbers: E62, E25, O23, O41, E44. 1 Introduction This paper investigates one possible theoretical link between income distribution and capital formation, here a link that is mediated via credit markets. It builds a model in which the distribution of income matters for capital formation, and uses it to analyze the eects of a simple policy intended to create a more equal distribution of income on the severity of certain credit market imperfections and, through this channel, capital accumulation. In particular, a neoclassical growth model with altruism is developed in which some capital investment must be externally ®nanced, and external ®nance is subject to a standard costly state veri®cation (CSV) problem. Some fraction of the population is ``capitalists'', who have access to risky but high return capital production technologies. Successful capitalists leave bequests to their ospring, thereby permitting them to internally ®nance some fraction of their own investment projects. However some external ®nance is also required. This is provided by ``workers'' who save out of labor income. As is well known, the greater the capability of capitalists to provide internal ®nance, the less severe is the CSV problem. Thus bequests tend to mitigate credit market frictions and, in that sense, promote ®nancial market eciency, and capital accumulation. However, they may also perpetuate income inequality. This structure is used to study the eects of policies intended to redistribute income on the steady state capital stock, the level of real activity, the distribution of income, and the welfare of all agents in the economy. It is shown that a policy that taxes the bequests of capitalists and transfers the proceeds in a lump-sum manner to all workers necessarily reduces the steady state capital stock. Indeed, when this eect is suciently strong, these redistributive tax/transfer schemes can reduce the total (wage plus transfer) incomes of workers, as well as their welfare. Thus some simple policies intended to redistribute income can be highly counterproductive, at least in a comparison of steady states. Central to this result is the role played by bequests in permitting capitalists to augment their own internal ®nancing of investment, and thereby to mitigate the severity of the CSV problem1 . Indeed, ®nancial market frictions are often at the heart of models intended to study the relationship between income distribution and growth. For example, Banerjee and Newman [2] and Galor and Zeira [10] construct models in which the initial distribution of wealth aects investment and real activity in the long run. Limitations on the 1
See Kotliko and Summers [13] or Ito and Krueger [11] on the practical importance of bequests for capital accumulation.
Credit market imperfections
173
ability of agents to borrow in order to ®nance investment are crucial to these results. However, the ®nancial market frictions responsible for this lack of access to credit are not explicitly modelled. In this paper, a speci®c credit market friction is introduced into a model that is similar in many other respects to that studied by Galor and Zeira [10]. The presence of this friction does not necessarily limit the access of agents to credit, but it does imply that the level of parental bequests aects the terms under which agents can obtain credit. Agents with relatively large bequests can also provide a relatively large amount of internal project ®nance. In the presence of the CSV problem, this permits credit to be obtained at lower cost, thereby perpetuating income inequality. In addition, the ability to provide internal ®nance provides a social bene®t, in that it tends to attenuate the severity of the CSV problem. By this mechanism, bequests improve the operation of credit markets, which in turn fosters capital accumulation. Thus factors which lead to income inequality may also be conducive to capital formation. Of course, many countries pursue policies intended to generate a more equal distribution of income. A particularly common policy device is the taxation of bequests2 . The framework developed in this paper can easily be used to analyze the consequences of inheritance taxes for the long-run capital stock, the level of real activity, and the distribution of income. Suppose, for example, that the government taxes the bequests of capitalists and redistributes the revenue collected to workers as a lump-sum. The reduction in the post-tax bequests received by young capitalists reduces the amount of internal ®nance they can provide and hence ± other things equal ± worsens the terms at which they obtain credit. It also exacerbates credit market frictions. As a result, the (steady state) capital stock falls, as does the real wage paid to workers. Indeed, conditions are described under which the wages of workers fall enough so that the income redistribution program actually reduces the total (wage plus transfer) income of workers. Under somewhat stronger conditions, their welfare level can fall as well. Thus policies intended to redistribute income from capitalists to workers may well harm their intended bene®ciaries, at least in a comparison of steady states. The results obtained here are vaguely reminiscent of an earlier literature associated with Kaldor [12], and many others. In that literature, the savings rate of capitalists exceeds that of workers. As a consequence, redistributions of income intended to favor workers reduce the aggregate savings rate and, by implication, the long-run capital stock. Here capitalists have savings rates that are less than those of workers. However, redistributing income away from capitalists acts to worsen credit market conditions. This eect can reduce the steady state capital stock, even though the aggregate savings rate does not decline.
2
See the OECD [15] survey for a discussion of inheritance taxes, and the relatively high tax rates that are quite common in practice.
174
J. Bhattacharya
The remainder of the paper is organized as follows. Section 2 outlines the environment, while Section 3 discusses trade in credit and factor markets. Section 4 analyzes bequests and the distribution of income. Section 5 discusses the process of capital accumulation and characterizes steady state equilibria, while Section 6 contains some comparative statics results. Section 7 analyzes the consequences of policies intended to redistribute income and Section 8 concludes. The appendices contain proofs of some of the main results. 2 Environment The formal setting follows Diamond [8] and, more speci®cally, Bernanke and Gertler [3]3 . The model consists of an in®nite sequence of two-period lived overlapping generations, plus an initial old generation. Time is discrete and is indexed by t 1; 2; . . .. At each date a new generation, consisting of a continuum of agents with unit measure, is born. Within each generation, agents fall into two categories. The ®rst, constituting a fraction a of the population, are entrepreneurs or ``capitalists''; the remainder are ``lenders'' or ``workers''. Capitalists, as a class, are distinguished from workers in that they have sole access to one of the investment technologies described below. In each period, there is a single ®nal good, which may either be consumed, or invested in the production of future capital. Capital is used in production of the ®nal good and then depreciates completely. 2.1 Endowments and technology The ®nal good is produced using a constant returns to scale technology with capital and labor as inputs. If Kt is the time t capital stock and Lt is the time t quantity of labor employed, output at t is given by F
Kt ; Lt : Furthermore, let kt KLtt denote the capital-labor ratio and f
kt F
kt ; 1 denote the intensive production function. The function f
: is assumed to satisfy: f
0 0; f 0 > 0 > f 00 ; and standard Inada conditions. Capitalists are endowed with access to a stochastic linear technology that converts q units of the ®nal good at t into j units of capital at t 1: Each capitalist is endowed with exactly one such project. Ownership of these projects cannot be traded and projects are indivisible, requiring exactly q units of the ®nal good to operate. The amount of capital produced next period is a discrete random variable taking on one of two possible values; that is, j 2 fj1 ; j2 g with j2 > j1 04 . I refer to j j2 as the ``good'' state and j j1 as the ``bad'' state. Let Prob fj ji g pi ; i 1; 2: Note that
3
Indeed, the notation largely follows that of Bernanke and Gertler [3]. There is no conceptual diculty associated with increasing the number of possible return states. However in general, with more than two states, optimal contracts can have unusual properties. On the latter issue see Border and Sobel [4], or especially Boyd and Smith [6].
4
Credit market imperfections
175
project returns are i.i.d across agents, and that the probability distribution of returns is time-invariant. There is no aggregate uncertainty. With respect to information, all capitalists costlessly observe the realized return on their own projects. However, any other agent can observe the ex post return on a project only by incurring a ®xed cost of c units of capital5 . There is also a second, inferior technology for producing capital that is available to all agents, including workers. One unit of the ®nal good invested in this inferior technology at t returns r units of capital at t 1 with probability one. 2.2 Preferences The structure of preferences plays an important role in the analysis. The preferences of workers are simple: workers are risk-neutral, and care only about old period consumption. Capitalists also consume only when old. However, in contrast to workers, capitalists are also assumed to derive utility from leaving bequests to their ospring6;7 . More speci®cally, a capitalist consuming ct when old, and leaving a bequest with a real value of bt derives utility equal to U
ct ; bt min
ct ; bbt ;
b>0 :
1
This particular speci®cation of preferences implies that the indirect utility function of capitalists is linear in income. The latter feature plays an important role in the characterization of optimal ®nancial contracts. 3 Markets Two kinds of transactions occur in this economy. Capital and labor are traded in competitive factor markets, and investment funds are traded in conventional credit markets. We now describe trade in each of these markets. 3.1 Factor markets Young workers are endowed with one unit of labor, which they supply inelastically. They earn the competitive real wage rate xt at t. Competition implies that workers earn their marginal product, so that xt f
kt ÿ kt f 0
kt x
kt : 5
2
The assumption that monitoring consumes capital is employed by Bernanke and Gertler [3] and Boyd and Smith [5]. 6 The introduction of bequests as an argument of agents' utility functions has been used extensively in the literature on income distribution (see Andreoni [1], Galor and Zeira [10], and Lloyd-Ellis and Bernhardt [14].) As has been pointed out by these authors, this formulation achieves considerable simpli®cation over the assumption that capitalists care about their ospring's utility. 7 Allowing workers to leave bequests would not change the analysis in any substantive way.
176
J. Bhattacharya
Clearly, x0
: > 0 holds. It will also be convenient to assume that x00
: < 0
8kt :
A:1
Production functions of the CES type with elasticity of substitution no less than one satisfy this condition8 . Similarly, capital earns the rental rate qt at t: In equilibrium, of course, this equals the marginal product of capital, so that qt f 0
kt :
3
Since entrepreneurs have no labor endowment when young, clearly kt Kt =
1 ÿ a: 3.2 Credit markets Capitalists have two sources of funds; bequests left by the previous generation, if any, and loans from workers9 . Since workers do not consume when young, all of their ®rst period income is saved. Thus the per capita supply of savings is
1 ÿ ax
kt : The per capita demand for funds by capitalists is aq (gross of the bequests they have received). We assume that aq
1 ÿ ax
kt 8kt :
A:2
Assumption (A.2) implies that given the current period capital stock, the potential supply of savings is always at least as large as the demand for credit by entrepreneurs. Thus Assumption (A.2) rules out the possibility of credit rationing. Borrowers obtain credit by oering loan contracts to lenders. Such contracts must specify the amount to be borrowed, a set of state-dependent payments to the lender, and a speci®cation of how and when state veri®cation will occur. Given that q units of the ®nal good are needed to start a project, a borrower who has an inherited bequest of b must obtain
q ÿ b units of funds externally. In order to simplify the exposition, it is assumed that all borrowers require some external funds. Conditions under which this is an equilibrium outcome are speci®ed below (see Assumption (A.6)). It is assumed that randomized veri®cation of the state is feasible. Thus contracts specify a probability pt 2 0; 1 that monitoring will occur if the borrower announces that the low return state has occurred10 . In addition contracts specify the amount retained by the borrower in the bad state in the event of an audit, yta ; as well as the amount retained in announced state i if no audit occurs, yit : Since in state i the goods value of an investment is qt ji ; the 8
Assumption (A.1) is used to guarantee that the steady state is unique (see Proposition 3 below). This is standard in the version of the Diamond [8] model where agents save a constant fraction of their income. 9 This lending can be regarded as intermediated. See Diamond [7] or Williamson [18] on the role of intermediated lending in costly state veri®cation environments. 10 There is no reason to verify the state if the borrower announces that the high return state has occurred.
Credit market imperfections
177
payment to the lender in the event of an audit is qt ji ÿ yta ; and the payment to a lender in state i if no audit occurs is qt ji ÿ yit . A borrower with the ex post income level yt chooses a consumption level, ct ; and a bequest, bt ; to maximize U
ct ; bt min
ct ; bbt subject to borrowers choose ct bbt and ct bt yt . It is evident that at an optimum, b yt . Thus the utility of a borrower, hence, optimal consumption is ct 1b
as a function of yt ; is simply b=
1 byt ; and it is then apparent that borrowers are risk neutral. Borrowers announce contract terms, consisting of the vector
pt ; yta ; y1t , y2t to maximize their own expected utility, subject to a set of constraints which are now described11 . First, a borrower at t, who has received a bequest of btÿ1 ; must raise
q ÿ btÿ1 units of funds externally. Let Rt denote the (gross) market rate of return between t and t 1: Then announced loan contract terms must yield a lender an expected return of Rt
q ÿ btÿ1 . The appropriate expected return constraint is p1 fqt j1 ÿ pt
yta c ÿ
1 ÿ pt y1t g p2 fqt j2 ÿ y2t g Rt
q ÿ btÿ1
4
In addition, the borrower must have an incentive to announce truthfully when the good state has occurred. The incentive constraint implied by this requirement is
1 ÿ pt
qt j2 ÿ qt j1 y1t y2t :
5
Finally, the borrower's income must be non-negative, so that yta 0 y1t 0 y2t 0 ; must hold, along with pt 2 0; 1: Since a borrower's expected utility is simply proportional to p1 pt yta
1 ÿ pt y1t p2 y2t ; a borrower with an inherited bequest of btÿ1 chooses a vector
pt ; yta ; y1t ; y2t to maximize this expression, subject to (4), (5) and the non-negativity constraints. The following assumptions are maintained throughout the analysis. First, the investment technology of capitalists ± inclusive of monitoring costs ± yields an expected return superior to that on the commonly available technology, even if pt 1: More speci®cally, p2 j2 p1 j1 ÿ p1 c > rq :
A:3
rq > j1 :
A:4
Second, I assume that
11 The assumption that borrowers announce contract terms follows Williamson [18, 19] and Bernanke and Gertler [3].
178
J. Bhattacharya
Assumption (A.4) guarantees that some monitoring occurs in the low return state. Observe that, if workers utilize the commonly available technology, then the market rate of return satis®es Rt rqt1 : Then, under the assumptions (A.3) and (A.4), the following result is established by Bernanke and Gertler [3]. Proposition 1 An equilibrium contract has r
q ÿ btÿ1 ÿ j1 p2
j2 ÿ j1 ÿ p1 c
6
y1ta y1t 0
7
y2t qt1
1 ÿ pt
j2 ÿ j1 :
8
pt
Clearly an optimal contract minimizes expected monitoring costs while assuring lenders a return of rqt1 : The minimization of expected monitoring costs dictates that a borrower receives no consumption in the bad state. As is apparent from equation (6), an optimal contract speci®es an auditing probability that is a decreasing function of the bequest received by a borrower. By implication then, so is the ``interest rate'' implied by the contract. Bequests allow borrowers, who have no other young period income, to engage in some internal ®nancing of their investment projects. As emphasized by Bernanke and Gertler [3], the ability to commit some internal ®nance tends to mitigate the costly state veri®cation problem. Here this is re¯ected in the relation between bequests and monitoring probabilities. The higher the bequest any borrower receives ± and hence the more internal ®nancing that occurs ± the lower is the expected cost of monitoring, and the contractual rate of interest. It is this connection between bequests and the costs of state veri®cation that is at the heart of this analysis. 4 The distribution of bequests I now turn to the distribution of inherited bequests. As noted above, this has important consequences for the amount of monitoring that occurs, and hence for credit market conditions. Consider an agent with a inheritance of btÿ1 at date t: If successful, her second period income is given by y2t qt1
1 ÿ pt
j2 ÿ j1 : Given the assumptions on the preferences of capitalists, this agent will then consume a fraction
b 1b
of her income, and will leave a bequest of
1 1b
y2t to the next
generation. Note that expected future income and hence expected future bequests are higher the larger are inherited bequests; therefore the expected income of any agent depends on the success of her progenitors. By the same token, agents with poor project returns have no income and leave no bequests. Their ospring can then provide no internal ®nance for their projects. Let j index the successive number of times the good state has been drawn within a family. In other words, by date t, a family indexed by j has had j
Credit market imperfections
179
consecutive successful investments, preceded by at least one failure. Note that agents with j 0 are the children of unsuccessful parents, and hence receive no inheritances12 . In view of our previous observations, agents indexed by higher values of j; and who have successful investments of their own, will also have higher second period incomes than agents with lower values of j, and hence will in turn leave correspondingly larger bequests to their own children. jÿ1 denote the level of bequests received at date t by an agent whose Let btÿ1 parent is the family's j ÿ 1 th successive success. If successful, her second j j period income (from Proposition 1) is y2t qt1
1 ÿ pt
j2 ÿ j1 : She would j j 1 then leave a bequest of bt 1b y2t : Therefore, " # jÿ1 r
q ÿ btÿ1 ÿ j1 1 j :
9 bt q
j2 ÿ j1 1 ÿ p2
j2 ÿ j1 ÿ p1 c 1 b t1 Equation (9) expresses the level of bequests left by a capitalist, if successful, as a function of the bequests she received. It formalizes our earlier intuition that larger inherited bequests raise future income (in the event of a success), which in turn raise the amount of bequests left. 4.1 The Distribution of Income Capitalists receive positive second period consumption only if they experience a good return realization on their investment. Therefore, it will result in no confusion to delete the subscript indexing income by states. Moreover, incomes of capitalists are linked to those of their parents through the index j. For the sake of analytical convenience, I henceforth concentrate solely on steady states, and therefore suppress time subscripts. Then let y j denote the second period income of a capitalist who constitutes her family's j th successive success, in a steady state. Using Proposition 1, it is possible to formally describe the income of a capitalist who is the j th successive success in her lineage. Lemma 1 Let A
q
j1 ÿ j2 p2
j2 ÿ j1 ÿ p1 c ÿ rq j1 > 0 p2
j2 ÿ j1 ÿ p1 c 1 qr
j2 ÿ j1 >0 B 1 b p2
j2 ÿ j1 ÿ p1 c
10
11
and de®ne y 0 0. Then y j A By jÿ1 : 12
12
Clearly this is a riches-to-rags story. Presumably this could be modi®ed to some extent by relaxing the assumption that return realizations are i:i:d within a family. Also dierences in ability are not modelled. These would be interesting extensions to consider in future work.
180
J. Bhattacharya
Proof. Proposition 1 implies that y j q
1 ÿ p j
j2 ÿ j1 : Using (6) and jÿ1 1 jÿ1 1b y gives the desired result. ( b Equation (12) is a ®rst-order dierence equation which is depicted in Figure 1. Its solution describes the steady state income distribution among capitalists. In order to validate the assumption that some amount of external ®nance is required for all capitalists, or in other words, q > b j for all j; it is clearly necessary to make the following assumption. B 1b 1ÿB (A.6) is maintained throughout the analysis.
Figure 1. Distribution of income among capitalists.
Credit market imperfections
181
4.2 Remarks Notice that capitalists who receive positive bequests always have the option of investing their bequests in the commonly available investment technology (instead of the one they period are endowed with). If they do so, their second income is given by
1 1b
rqy jÿ1 : A capitalist with a bequest of
1 1b
y jÿ1
who invests in her own technology obtains an expected income of p2 y j : Hence, external funding will be sought by all capitalists i 1 j rqy jÿ1 ; 8j 1 : p2 y > 1b Using (13), this expression can be rewritten as A 1 A j 1ÿB > rq 1 ÿ Bjÿ1 8j p2 1ÿB 1b 1ÿB
14
A sucient condition for (14) to hold is stated in the following assumption, which I also henceforth maintain. p2
1 b
A:7 q< r
5 General equilibrium In this section I describe the conditions that determine the steady state equilibrium capital stock, under the assumptions made to date. I also describe the restrictions on factor prices required in order to satisfy these assumptions. Under Assumption (A.2), there is no rationing of credit. Hence the projects of all capitalists are funded. Since the expected return on these j units of capital are produced by capitalists, projects is j p1 j1 p2 j2 ; a gross of monitoring costs. Workers provide the external ®nance that capitalists require. Capitalists 1 y jÿ1 ; and hence require external of index j ÿ 1 receive a bequest of 1b 1 ®nance in the amount of q ÿ 1b y jÿ1 : Since the fraction of capitalists who
amount are the j ÿ 1 th successive success is given by p1
p2 jÿ1 ; the h aggregate P1 of funding required by capitalists is a q ÿ p1 j1
p2 jÿ1 external 1 1b
y jÿ1 : All worker savings in excess of this amount is invested in the inferior capital production technology, yielding r units of capital per unit invested. Since the hper capita saving of workers is
1 ÿ ax
k; this i form of P1 jÿ1 1 jÿ1 units of investment yields r
1 ÿ ax
k ÿ aq ap1 j1
p2 1b y
capital each period, in a steady state. It remains to describe the amount of capital consumed by the monitoring process in each period. As noted previously, a fraction p1
p2 jÿ1 of capitalists
182
J. Bhattacharya
have the index j ÿ 1; they are monitored in their second period (if unsuccessful) with probability p j : Let p denote the average probability with which any capitalist is monitored. Then p is given by p
1 X
1 ÿ p2 2
p2 jÿ1 p j :
15
j1 j Equation (6) gives an expression for the monitoring probability p given a
1 bequest level b jÿ1 : Substituting (6) into (15) and using b jÿ1 1b y jÿ1 ; the average monitoring probability satis®es h i 1 1 r q ÿ 1b y jÿ1 ÿ j1 X
1 ÿ p2 2
p2 jÿ1 p p2
j2 ÿ j1 ÿ p1 c j1 1 X
1 ÿ p2
rq ÿ j1 1
1 ÿ p2 2 r ÿ
p2 jÿ1 y jÿ1 p2
j2 ÿ j1 ÿ p1 c 1 b p2
j2 ÿ j1 ÿ p1 c j1
where the second equality, follows from (13). The values of A and B de®ned in equations (10) and (11) and hence the values y j depend on the relative price of capital, q, and therefore so does the monitoring probability p. I now characterize the equilibrium relationship between p and q. In order to do so, it is useful to de®ne the variables A~ and B~ by A~
j2 ÿ j1 A p2
j2 ÿ j1 ÿ p1 c ÿ rq j1 p2
j2 ÿ j1 ÿ p1 c q 1 r
j2 ÿ j1 B : B~ 1 b p2
j2 ÿ j1 ÿ p1 c q
Note that A~ and B~ depend on parameters alone. Using them, it is straightforward to characterize the equilibrium relationship between p and q. Proposition 2 De®ne the function p
q by p
q w ÿ vH
q
16
where
1 ÿ p2
rq ÿ j1 ; p2
j2 ÿ j1 ÿ p1 c 1
1 ÿ p2 2 r ; v 1 b p2
j2 ÿ j1 ÿ p1 c w
and H
q
1 X j1
p2 jÿ1 y jÿ1
A
p2 =p1
1 ÿ p2 B
Then, in a steady state equilibrium, p p
q.
p2 A~ p1 1=q ÿ p2 B~
17
Credit market imperfections
183
It will evidently be useful to know some properties of the function H
q. These are summarized in the following Lemma, which is stated without proof. Lemma 2 Suppose B < 1 holds. Then (a) H
q 0; 8q; (b) H
0 0; and (c) H 0
q 0; 8q. We are now prepared to describe the determination of the aggregate capital stock, in a steady state. Next period's capital stock, of course, is simply the sum of capital produced by capitalists, plus capital produced using the commonly available investment technology, less capital consumed in monitoring. Thus the steady state capital stock satis®es K ap1 j1 p2 j2 ÿ p1 cp
q r
1 ÿ ax
k ÿ aq a
1 X
p1
p2 jÿ1 b jÿ1
j1
18 Dividing both sides of (18) by
1 ÿ a; and noting that k K=
1 ÿ a; I obtain a p1 j1 p2 j2 ÿ p1 cp
q k 1ÿa " # 1 r X jÿ1 jÿ1 p1
p2 b
19
1 ÿ ax
k ÿ aq a 1ÿa j1 It will often prove useful to have a more compact representation of the equilibrium condition (19). To this end, I de®ne 1 M fa p1 j1 p2 j2 ÿ ap1 cw ÿ raqg 1ÿa 1 N fap1 cvg > 0 1ÿa and
Z
1 fap1 r=
1 b g > 0 : 1ÿa
Then (19) can be rewritten as k M
N Z H
q rx
k :
20
I now establish the sign of M. Lemma 3. Assumptions (A.3) and (A.4) imply that M > 0: As shown in Appendix (C), equation (20) gives the steady state capital stock as an increasing function of the capital rental rate. The rental rate aects the capital stock because it aects the income of capitalists and ± through this channel ± their bequests. These bequests determine the amount of internal ®nance available. Higher values of q lead to more internal ®nance being provided and, consequently, to lower monitoring costs.
184
J. Bhattacharya
Equation (3) implies that in steady states q f 0
k :
21
Equations (20) and (21) constitute the system of steady state equilibrium conditions. These equations are depicted in Figure 2. The following result is proved in Appendix C. Proposition 3 Suppose that rx0
0 > 1: Then equations (20) and (21) have a unique solution,
k ; q : 5.1 Remarks It is, of course, the case that the derivation of the equilibrium conditions was predicated on Assumptions (A.1) ± (A.7) being satis®ed. Many of these conditions involve either implicit or explicit assumptions on the magnitude of q: Thus, I have ± in eect ± assumed that the solution (k ; q satis®es various conditions. These are now stated explicitly. ÿ aq 1. Assumption (A.2) is equivalent to q 1 b 1 ÿ B
1 ÿ s It is easily shown that satisfaction of (22) is implied by Assumption (A.6).
15 An issue of some signi®cance is whether capitalists care about the level of pre-tax or post-tax bequests they leave for their ospring. For convenience, I assume that old capitalists care about the level of pre-tax bequests they leave. The alternative possibility would be an interesting extension for future work.
188
J. Bhattacharya
It is also assumed that all capitalists prefer operating their own projects to investing in the commonly available capital production technology. Repeating the reasoning underlying equation (19), this requires that A 1 ÿ
1 ÿ sBj p2 1 ÿ B
1 ÿ s h i 1 A > rq 1 ÿ
1 ÿ sBjÿ1 8j
23 1b 1 ÿ B
1 ÿ s The satisfaction of (23) is implied by Assumption (A.7). Finally, I continue to assume that credit is not rationed; Assumption (A.2) is sucient to imply this. 7.1 The steady-state capital stock I begin by describing the eects of the income redistribution program on the steady state capital stock. To foreshadow, a tax on bequests reduces the ability of capitalists to provide internal ®nance for their investment. As a consequence, the costly state veri®cation problem becomes more severe, with adverse implications for the capital stock. I now formalize this intuition. As before, in a steady state, next period's capital stock is simply the capital produced by capitalists ± not inclusive of capital lost in the state veri®cation process ± plus capital produced through the commonly available technology, less capital consumed by monitoring. Moreover, credit is not rationed and the projects of all capitalists are therefore funded. Since the j units of capital are expected returns on these projects is j p1 j1 p2 j2 ; a produced by capitalists, gross of monitoring costs. Workers provide the external ®nance that capitalists require. Capitalists of index j ÿ 1 now receive 1
1 ÿ sysjÿ1 ; and hence require external ®nance an after tax bequest of 1b 1 in the amount of q ÿ 1b
1 ÿ sysjÿ1 : Notice that the bequest tax imposed on all capitalists reduces the inheritances of the next generation, requiring them to borrow larger amounts externally than before. Since the fraction of capitalists who are the j ÿ 1 th successive success is given by p1
p2 jÿ1 ; the aggregate amount of external funding required by capitalists is h i P1 jÿ1 1 jÿ1 : Since workers do not consume when a q ÿ p1 j1
p2 1b
1 ÿ sys young, all of their savings in excess of this amount is invested in the commonly available investment technology, yielding r units of capital per unit invested as before. The per capita saving of workers is
1 ÿ ax
k plus the transfer T
q; s they receive from the government. As before, this form of investment yields h n oi P jÿ1 1 jÿ1 y r
1 ÿ ax
k
1 ÿ aT
p; q ÿ a q ÿ p1
1 ÿ s 1 s j1
p2 1b units of capital each period, in a steady state. Notice that the taxation of bequests causes capitalists to borrow larger amounts, ceteris paribus, than they otherwise would, and that workers ef-
Credit market imperfections
189
fectively supply all of the transfers they receive in capital markets. Thus, in the absence of the state veri®cation problem, this redistribution would have no eect on the aggregate capital stock. However, the taxation of bequests also reduces the amount of internal ®nance provided by capitalists, with adverse consequences for the severity of the informational friction. I now pursue the implications of this observation. Let p
q; s denote the average monitoring probability, as a function of q and s. Then the amount of capital consumed by the monitoring technology is, in the aggregate, p1 cp
q; s: Repeating our earlier reasoning, it is easy to show that p
q; s p21
1 X j1
p2 jÿ1 psj w ÿ v
1 ÿ sH
q; s ;
where w and v are as de®ned in Proposition 3, and where 1 X p2 A p2 A~ jÿ1 jÿ1 H
q; s
p2 ys ~ ÿ s p1
1 ÿ p2 B
1 ÿ s p1 1=q ÿ p2 B
1 j1 It will be useful to summarize some properties of the function H . Lemma 5 (a) H
q; s 0 ; 8q 0 (b) H
0; s 0 ; and (c) qH1
q; s H
q; s H
q; s p2 qA~ H2
q; s p1 B~ ÿ H
q; s H
q; s A~
24
25
From our previous observations, it is apparent that the steady state equilibrium conditions are given by (21) and k M
N Z ÿ N s H
q; s rx
k
26
where M; N ; Z are as de®ned previously. Equation (26) de®nes an upward sloping locus in Figure 4 as before, while (21) continues to de®ne a negatively sloped locus in the same ®gure. Thus (21) and (26) continue to have a unique intersection, denoted by q
s; k
s16 . The eects of an increase in s are depicted in Figure 5. Evidently, the locus de®ned by (21) is independent of s. It is also straightforward to verify that an increase in s shifts the locus de®ned by (26) downward. Therefore, an
16
It continues to be necessary that the steady state value of q satisfy the conditions stated earlier (see Section 5.1).
190
J. Bhattacharya
Figure 4. Steady-state equilibrium under the tax/transfer policy.
Figure 5. Increase in s.
increase in s reduces the steady state capital stock, and increases the rental rate. These eects are entirely a consequence of the fact that a tax on bequests reduces the ability of capitalists to provide internal ®nance for their capital investments.
Credit market imperfections
191
7.2 Discussion It is important to note that the results discussed above hinge crucially on the assumption that the process of monitoring utilizes some capital. In the current speci®cation, it is obvious that if no capital is consumed in the monitoring process, a reduction in bequests would have no eects on the steady state capital stock. However, it is possible to imagine alternative formulations of the model where monitoring does not consume capital, and where reductions in monitoring costs continue to adversely aect the steady state capital stock. For example, it would be straightforward to consider a model with a continuum of return states in which credit rationing is observed, and state veri®cation is nonstochastic. In such a model the equilibrium rate of return to savings would be a decreasing function of the quantity of internal ®nance provided (Boyd and Smith [5]). Under the assumption of a positive interest elasticity of savings, the result would be that bequest taxation lowers the equilibrium return to savings, and hence reduces the availability of funds, exacerbates the rationing of credit, and lowers the steady state capital stock. Thus the results on bequest taxation should not be regarded as being especially sensitive to the assumption that monitoring utilizes capital. 7.3 The incomes of workers If there were no changes in the capital stock, the income redistribution described above would clearly raise the steady state income of workers, and reduce that of capitalists (as a group). However, redistributing income from capitalists to workers reduces the capital stock, as we have seen, and this at least partially counteracts the eects on the income of workers. Indeed, the possibility arises that the redistribution is actually counterproductive, in that it causes the total income (and even welfare) of workers to fall. I now describe conditions under which this is the case. In order to simplify the discussion, I henceforth assume that the production function has the CobbDouglas form f
k Dk r ; r 2
0; 1. The speci®c analytical result concerns what happens when a small tax is imposed on bequests. Some global results are then illustrated by example. I now state the main result of this section. Proposition 5 De®ne the net of transfer income of workers to be I
s xk
s T q
s; s: Suppose that q
0 satis®es "
# 1 ca ; q
0 > max N =c r
1 ÿ r p21 B~
27
ÿ a p 0 1 where c 1ÿa 1b. Then I
0 < 0 holds. The proof of Proposition 5 appears in Appendix E. Proposition 5 states a sufficient condition for a small income redistribution to have counterproductive consequences. Notice also that this is merely a sucient condition. As will be apparent from the proof, this condition is generally much stronger
192
J. Bhattacharya
than would be needed in order for the redistribution under consideration to have negative consequences for the steady state income levels of workers17 . 7.4 Example 1 I now provide an example illustrating the previous results, and exploring the consequences of some higher rates of bequest taxation. The parameters of the economy are as follows: r 1:07, q 200, p1 0:2, p2 0:8, b 2, c 184:4, j1 1, j2 375, a 0:03, D 13, r 0:53. The steady state equilibrium (for s 0) is k 53:11 and q 1:12. The average income of all capitalists is 159.5. The ratio of average capitalist income to that of workers is 3.1. Figures 6a±c report the consequences of varying s from s 0 to s 0:9: Notice that as s increases, both p and k fall uniformly, as does the posttransfer income of workers. It is easy to verify that the average earnings of capitalists fall as well. 7.5 The steady-state welfare of workers Proposition 5 provides a condition under which a redistribution of income that ostensibly favors workers can actually reduce their steady state income levels. Of course this need not imply that such a redistribution reduces their level of steady state welfare which, under our assumptions, is simply given by W
s rq
sI
s: I now provide a sucient condition under which the steady state welfare of workers is also injured by a redistributive program purportedly intended to favor them. Proposition 6 Suppose that "
cr cr ; q
0 > max ~ ÿ 1 crr
1 ÿ r N
1 ÿ r
2r ÿ 1 ap2 B
2r
#
28
1
holds. Then W 0
0 < 0 holds.
17
Shammas, Salmon and Dahlin [17] in their treatise Inheritance in America make the following argument: ``The irony that accident of birth is the prime determinant of one's material situation in a society that considers free market competition to be the most rational and ecient method of making economic decisions . . . led reformers at the beginning of the twentieth century to push for progressive taxation. [On the other hand]. . .using arguments that date back at least to Ricardo, proponents of the unrestricted transmission of wealth . . . contend that this accumulation of capital is essential to investment. They point to studies showing that the auent devote a higher proportion of their lifetime resources to investment and a smaller percentage of it to consumption than do the rest of the population. Progressive taxation, . . .interferes with this process and, in their opinion, depresses economic growth. . .Thus, they maintain, the policy ultimately hurts the very people it was designed to help.'' (emphasis mine)
Credit market imperfections
193
Figure 6a. Steady-state monitoring probability as a function of s (Example 1). Figure 6b. Steady-state capital stock as a function of s (Example 1). Figure 6c. Post-transfer income of workers as a function of s (Examples 1).
It is easyÿto show that W 0
0 < 0 can hold only if r > 0:518 . When r > 0:5 2rÿ1 holds, 0 < r < 1 is also satis®ed. Then it is easy to see that (28) is necessarily a stronger condition than (27). This is to be expected, since an increase in s also increases q
s; which acts to partially oset the negative consequences of the reduction in income. 7.6 Example 2 I now present an example illustrating the eects of various rates of taxation for the steady state welfare of workers. Parameter values are as follows: r 1:09, q 100, p1 :4, p2 :6, b 4, c 245:8, j1 1, j2 475, a 0:02, D 12, r :54. The steady state equilibrium (for s 0 is 18
If r 0:5 holds, then qx
k f 0
kx
k is decreasing in k, and a reduction in the steady state capital stock itself increases the welfare of workers.
194
J. Bhattacharya
k 55:6 and q 1:05. The average income of capitalists is 201.1, and the real wage rate is 48.8. The ratio of average capitalist earnings to workers incomes is 4.1. Figures 7a±c report the equilibrium values of p and k; as well as steady state welfare at various values of s: As in Example 1, higher values of the tax rate uniformly result in higher values of p and lower values of k; as well as lower values of the steady state welfare of workers. It is easy to verify that the average welfare of capitalists falls at the same time. Thus the program to equalize the distribution of income has no desirable consequences, at least in a comparison of steady states. 8 Conclusion This paper develops a model in which the distribution of income matters for credit market eciency and, through this channel, capital accumulation. In particular, a neoclassical growth model with altruism is constructed in which some capital investment must be externally ®nanced, and external ®nance is
Figure 7a. Steady-state monitoring probability as a function of s (Example 2). Figure 7b. Steady-state capital stock as a function of s (Example 2). Figure 7c. Post-transfer welfare of workers as a function of s (Example 2) W
0 100:
Credit market imperfections
195
subject to a standard CSV problem. Some fraction of the population is ``capitalists'', identi®ed by their access to risky but high return capital production technologies. Successful capitalists leave bequests to their ospring, thereby permitting them to internally ®nance some fraction of their own investment projects. The higher the level of bequests, the greater the quantity of internal ®nance that investors provide. As is well known, the greater the capability of capitalists to provide internal ®nance, the less severe is the CSV problem. Thus bequests act to mitigate the consequences of the informational asymmetry, and therefore their presence serves to promote the eciency of capital markets. This is conducive to capital accumulation and growth. At the same time, bequests serve to foster and perpetuate income inequality. Therefore, factors which lead to income inequality may also be conducive to capital formation. The presence of these considerations has been shown to have strong implications for the consequences of policies intended to redistribute income from capitalists to workers. In particular, it has been demonstrated that a policy which taxes the bequests of capitalists, and transfers the proceeds in a lump-sum manner to all workers, necessarily reduces the steady state capital stock. This is entirely a consequence of the fact that the implied reduction in (post-tax) bequests necessarily harms the ability of capitalists to internally ®nance their own investment projects. As a result more monitoring occurs, and more resources are lost due to the presence of the CSV problem. Indeed, when this eect is suciently strong, it can occur that these redistributive tax/transfer schemes reduce the total (wage plus transfer) incomes of workers, as well as their welfare. Thus some simple policies intended to create a more equal distribution of income may well harm their intended bene®ciaries, and indeed may act to the detriment of all agents, at least in a comparison of steady states. The latter result is reminiscent, in many respects, of models that yield ``transfer paradoxes''. A transfer paradox arises when a transfer of income immiserizes the recipient. Galor and Polemarchakis [9], for example, demonstrated that transfers of income across agents could reduce the income and welfare of the recipient in the Diamond [8] model. In particular, in their model, if the savings rate of the agents receiving the transfer is below that of the agents making the transfer, then the transfer results in a lower aggregate savings rate, a lower level of investment, and hence a lower long-run aggregate capital stock than would be observed in its absence. The net eect can be harmful to the recipient of a transfer. While the same kinds of eects occur here, they are not a consequence of dierences in savings rates between agents. Indeed, in the present model, young capitalists and young workers have identical savings rates. However, the composition of savings between workers and capitalists still matters, because the CSV problem imparts advantages to internal over external ®nance. Thus income transfers matter when they alter the composition of internal and external ®nancing of investment. Such a result serves to emphasize the importance of credit market imperfections in analyzing the consequences of the distribution of income.
196
J. Bhattacharya
In conclusion, it is important that a caveat be recorded. The analysis in this paper has been entirely restricted to steady states. A study of dynamical equilibria in this context would raise the relatively dicult issue of analyzing the evolution of the distribution of income among capitalists through time, and of determining this evolution jointly along with the time path of the capital stock. However, it is important to note that an analysis of dynamical equilibria could quite easily alter some of the conclusions about the uniformly adverse consequences of income redistributions that have been obtained for steady states. Indeed, since the transitional eects of income redistributions for capital accumulation will typically be smaller than the long-run eects, it may turn out that some transitional generations of workers bene®t from the kinds of redistributions analyzed, even though these bene®ts are ®nitely lived. In such a case, of course, these programs involve a redistribution between generations as well as between dierent members of the same generation. Appendix A Proof of Proposition (2) Using (9) and the de®nition of p; I can write 1 X
1 ÿ p2 2 1 jÿ1 jÿ1
p2 rfq ÿ y g ÿ j1 p p2
j2 ÿ j1 ÿ p1 c j1 1b
1 1
1 ÿ p2 2
rq ÿ j1 X
1 ÿ p2 2 r=
1 b X
p2 jÿ1 ÿ
p2 jÿ1 y jÿ1 p2
j2 ÿ j1 ÿ p1 c j1 p2
j2 ÿ j1 ÿ p1 c j1
1
1 ÿ p2
rq ÿ j1
1 ÿ p2 2 r=
1 b X ÿ
p2 jÿ1 y jÿ1 p2
j2 ÿ j1 ÿ p1 c p2
j2 ÿ j1 ÿ p1 c j1
a:1
Substituting (13) into (a.1) yields p
~ 2 =p1
1 ÿ p2
rq ÿ j1
1 ÿ p2 2 r=
1 b A
p ÿ ÿ p2
j2 ÿ j1 ÿ p1 c p2
j2 ÿ j1 ÿ p1 c 1=q ÿ p2 B~
Rearranging terms in (a.2) yields the result.
a:2 (
B Proof of Lemma (3) By de®nition, M ap1 j1 p2 j2 ÿ ap1 cw ÿ raq : Rearranging terms yields
1 ÿ p2
rq ÿ j1 : M ap1 j1 p2 j2 ÿ rq ÿ p1 c ap1 1 ÿ p2
j2 ÿ j1 ÿ p1 c
Assumptions (A.3) and (A.4) imply that both terms on the right-hand side of the above equation are positive. (
Credit market imperfections
197
C Proof of Proposition (3) Assumption (A.1) and the condition of the proposition imply that the con^ dition k rx
k has two solutions; one with k 0; and one, denoted k, ^ > 0: Since k ÿ rx
k is a concave function of k; satisfying k^ rx
k ^ k ÿ rx
k > 0 holds 8k > k: Since M; N ; Z and H
q are all positive, k ÿ rx
k M
N Z H
q ^ For all k > k; ^ it is easy to verify that k ÿ rx
k is can hold only for k > k: increasing in k: Therefore, (20) de®nes an upward sloping locus, as shown in Figure 2. Clearly that locus also has a ®nite, positive intercept. Obviously the locus (21) is downward sloping and has the con®guration depicted in the same ®gure. It is then immediate that (20) and (21) have a unique intersection. ( D Proof of Lemma (4) Rewrite (20) as
~ Ap2 =p1 rx
k 1=q ÿ p2 B~
ap1 r kM N 1b
Since B~ is decreasing in b (see equation (11)) clearly an increase in b shifts the locus de®ned by (20) downward, as depicted in Figure 3. The locus de®ned by (21) is independent of b; therefore, k must fall, as claimed, while q rises. ( E Proof of Proposition (5) sp1 First note that ÿ aL
q; ps
1bH
q; s, and therefore I
s xk
s csH
q; 1 s where c 1ÿa
1b. Then,
I 0
s x0
k
@k @q csH1
: csH2
: cH
: @s @s
and I 0
0 x0
k
@k j cH
q; 0 : @s s0
This condition holds since, as s ! 0, H1
: @q @s and H2
: remain ®nite. Therefore a necessary and sucient condition for I 0
0 < 0 is that @k j > cH
q; 0 ; @s s0 which upon rearrangement is equivalent to ÿx0
k
ÿ
kx0
k @k k j > cH
q; 0 x
k @s s0 x
k
De®ne m N Z: Then, since that
@k @s js0
E:1
sH2
q;0ÿNH
q;0 1ÿrx
mÿN 0
k
mÿN sH
q;0x0
k=k ; it follows 1
198
J. Bhattacharya
n o
: H
: m HH2
: ÿN
1 @k j : k @s s0 k ÿ krx0
k mH1
:x0
k
E:2
Also note that, under the assumption of Cobb-Douglas production, kx0
k 0 x
k r: Using these observations in (E.1), one obtains that I
0 < 0 holds i ÿrfm HH2 ÿ N gx
k >c : k ÿ krx0
k mH1 x0
k
E:3
Furthermore, lemma 5 and the de®nitions of H and B~ imply that (E.3) is equivalent to the condition m c H
: k p2 B ÿ >c ÿr ÿN 1 ÿ p2 B p1 qk rx
k k ^ Now observe that rx
k ÿ r > 0 necessarily holds 8k > k: Furthermore, H
: p2 < a p1 trivially holds. Hence a set of sucient conditions for qk
I 0
0 < 0 is
cap2 p2 B > p21 k ÿr N >c rx
k
E:4
E:5
It is easy to check that (E.4) is equivalent to ca < q
0 : p21 B~ In addition, written as
k rx
k
1 x01
k
1ÿrq
0 is satis®ed. Therefore, (E.5) can be re-
1 < q
0 N =c r
1 ÿ r Thus I 0
0 < 0 holds if "
# 1 ca ; q
0 > max : N =c r
1 ÿ r p21 B~ E Proof of Proposition (6) Dierentiating W
s rq
sI
s clearly 1 @W 1 @q 1 @I W @s q @s I @s Thus, W 0
0 < 0 holds i
(
Credit market imperfections
199
I 0
0=I
0 > q0
0=q
0
F:1
Since I
0 x
k and x0
k
1 ÿ rq; equation (F.1) can be rewritten in the form 1 @k 1 @k 0 x
k cH
: <
1 ÿ r
F:2 x
k @s k @s Rearranging terms in (F.2) yields the equivalent condition that W 0
0 < 0 holds i
1 ÿ 2r
1 @k cH
: > k @s x
k
F:3
Note that clearly a necessary condition for W 0
0 < 0 is that r > 1=2: Suppose then that r > 1=2 holds. Substituting (E.2) into (F.3), one has that W 0
0 < 0 is equivalent to the condition
1 ÿ 2r
fm HH2 ÿ N gx
k >c k ÿ krx0
k mH1 x0
k
F:4
It is straightforward to verify that equation (F.4) can be rewritten as mp2 2r ÿ 1 c k 1 ÿ 2r qB~ ÿ 2 >c ÿr N 1 ÿ p2 B r rx
k r ap1 Clearly, then, a set of sucient conditions for W 0
0 < 0 is 2r ÿ 1 c 1 > p1 B~ r ap1 q
0 1 ÿ 2r k N ÿr >c r rx
k
F:5
F:6
k 1 x01
k
1ÿrq
0 : Substituting this relation into (F.6), and Note that rx
k solving (F.5) and (F.6) for q
0, one obtains that W 0
0 < 0 holds if " # cr cr ; : ( q
0 > max ~ ap21 B
2r ÿ 1 rcr
1 ÿ r N
1 ÿ r
2r ÿ 1
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