imperfect capital markets and the theory of investment

The Review of Economic Studies, Ltd.

Imperfect Capital Markets and the Theory of InvestmentAuthor(s): Lucien FoldesSource: The Review of Economic Studies, Vol. 28, No. 3 (Jun., 1961), pp. 182-195Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2295947 .

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Imperfect Capital Markets and the

Theory of Investment 1. Introduction and Summary

In most statements of the theory of the firm which take account of the time factor it is assumed, explicitly or implicitly, that firms can borrow and lend as much money as they wish at the rates of interest for various periods prevailing in the loan market. Since this assumption plays a large part in the analysis which follows, it will be called " the basic assumption ". With its aid certain inmportant results on the character of the firm's optimum production plan, the valuation of assets and the length of the average period of production can be established. The main purpose of this paper is to consider how these results must be modified when there is imperfect competition in the loan market and the basic assumption does not hold.

It is surprising that writers on imperfect competition in the product and factor markets have paid little attention to this question, despite the existence of obvious imperfections of competition in the loan market. The situation in the real world is complicated by the distinctions between equity and loan capital of various kinds and the difficulties of comparing rates of interest when risks differ, whereas the usual theoretical models simply speak of borrowing and lending and deal with uncertainty in a summary way. Never- theless, it is clear that many " borrowers " raise rates against themselves when they obtain more funds, while in other cases there are economies of scale in financing. On the " lending" side, it may not be possible to invest unlimited sums at constant rates; and even if lending rates are constant, they may be lower than either marginal rates of return in the firm or marginal borrowing rates, and therefore irrelevant to the firms' planning. It is sometimes supposed that the results obtained when the basic assumption holds remain true with trivial (though generally unspecified) modifications when it fails, but in general this is not so.

The main results of the paper, and some implications, will be set out in this section, the remainder of the paper being devoted to further discussion and proof. Before stating the results, some definitions are needed. We distinguish two meanings of the word discounting. It may refer either to the arithmetical operation of multiplying a sum of money by a discount factor such as (I + r)-t or e-t (where t is time and r, s are rates of interest). It may also refer to the assessment of the present value of a future sum, the value being defined as that present compensation for the hypothetical loss of the future sum which would leave the owner's utility unchanged. The word discounting will always be used in the former sense, the word valuation in the latter. The concepts can be used interchange- ably when the basic assumption holds, but not in general.

* I should like to thank Dr. George Morton for our very helpful discussions of the questions which are the subject of this paper, and Mr. J. Black and others who read an earlier draft and made valuable suggestions.

182



IMPERFECT CAPITAL MARKETS 183

Some concepts for describing flows of cash into and out of the firm are also required. By dividends we shall mean any withdrawals of cash from the firm by its owner(s). For simplicity it will be assumed that the owner lives entirely on his dividends from the firm under consideration and does not keep cash balances, so that dividends and consumption are equal and non-negative at all times.

Net receipts are defined as follows:

Net receipts Receipts from sales

- Payments for purchases of goods and factors (excluding dividends)'

= Dividends

- Net increase in indebtedness (I.1)

The purpose of this definition is to separate cash flows due to purchases and sales from those due to loan market transactions.

In Section II, a simplified model of the firm will be described, and the results which follow are subject to the assumptions made there (except that the restriction to two points of time is inessential). In particular, it will be assumed that the firm's aim is to maximise the value of a utility function which depends only on the dividends paid in various periods, subject to constraints imposed by the possibilities of production and of borrowing and lending.

We now recall certain propositions which are true when the basic assumption holds, and set out in each case the modifications required when this assumption fails, and some of their implications. The standard results for perfect capital markets are established in Irving Fisher, " The Theory of Interest ", and J. R. Hicks, " Value and Capital ".

(1) The firm's optimum plan-i.e., that which maximises utility subjects to the constraints-is the plan which maximises the discounted value of net receipts, the discount rates being the market rates of interest for the appropriate periods. The discounted value of dividends is also maximised.2 The production plan, i.e. the plan for net receipts, can be determined independently of preferences among dividend streams, since dividends can be modified separately by borrowing and lending at constant rates of interest; thus im- patience for consumption need not hamper the growth of the firm. It is possible to make precise computations for production without specifying the form of the utility function. Further, since the discount rate is independent of the internal allocation of funds, individual receipts and payments within the production plan can be discounted separately, and the return from individual projects assessed separately (provided they are otherwise independent). These results are not upset by the occurrence of indivisibilities in production, and the optimum plan always occurs in a range of diminishing marginal returns to investment in the firm.

I If the finn keeps cash balances additions to them are best regarded as investments in the business in the same way as, say, additions to stocks. They should accordingly be subtracted from receipts to obtain net receipts. The results in the text are unaffected.

2 By definition, the difference between these two sums is a constant, namely the initial debt, (less any initial cash balance), and less the discounted value of changes in indebtedness. This difference cannot be changed by borrowing or lending followed by repayment, because the discount rate is equal to the rate paid on loans, so that the discounted value of a repayment is equal to the loan, but of opposite sign.



184 REVIEW OF ECONOMIC STUDIES

These results are entirely altered when the basic assumption fails (though in some cases they can be rescued by weaker assumptions). We can no longer refer to " the " rate of interest for a period; instead, there is a supply curve of funds to the firm, and a demand curve for borrowing from it. The firm's optimum dividend, production, and borrowing and lending plans are determined simultaneously and are interdependent. If economies of scale in production and in the loan market are ruled out, it remains true that there is some discount rate such that the optimum plan maximises discounted net receipts and dividends; however the rate is itself determined simultaneously with the optimum (like the dual prices in a programming problem) and therefore cannot generally be found independently in advance and used for planning purposes. If there are economies of scale, the optimum may be such that discounted net receipts or dividends are not maximised at any discount rate whatever, which further limits the use of discounting techniques in planning. It is no longer certain that the optimum is at a point of diminishing returns.

It is worth while to point out some of the consequeinces of these results. In the first place, they introduce a number of complications into the application of business planning techniques such as mathematical programming which are not encountered when the basic assumption holds. The present values of particular future receipts and payments are interdependent, and the yields of individual projects cannot be evaluated separately. Indivisibilities in production give rise to new complexities in planning when the capital market is imperfect. Finally, the dependence of the production plan on preferences for dividends means that the best production plan cannot be found unless a utility function is specified (except where the period of the plan is so short that dividends can be neglected). This gives rise to obvious difficulties.

The conventional assumption that the firm maximises profits, or discounted net receipts, gives the theory of the firm a greater degree of simplicity and definiteness for purposes of prediction than is found in the theory of the household, with its unspecified utility function. Unless some approximations to actual preferences for dividends can be obtained, these advantages are lost when the basic assumption is dropped. On the other hand, the change in assumptions makes the theory consistent with certain observations which are incompatible with a perfect capital market. One is that dividend policies often affect a firm's growth, its price and output policy and its requirements of funds and thereby its receipts and payments of interest.' Another point is that an improvement in a firm's investment opportunities may lead to a reduction in dividends in the early years, despite the fact that the firm is better off, a phenomenon which cannot be satisfactorily explained when the capital market is perfect (see Sections IV and V).

It has already been mentioned that, when the basic assumption fails, costs and revenues cannot in general be discounted separately because the discount rate depends on the optimum; moreover in some cases the discounting method leads to incorrect results whichever rate is used. Thus the separate cost and revenue curves displayed in texts on imperfect competition-at least the long period curves-cannot in general be drawn up, and we have the amusing result that the usual analysis of imiperfect competion in the product and factor markets requires the assumption of a perfect capital market.

(2) When the basic assumption holds, the capital value of a stream of net receipts- i.e. the compensating payment-can be assessed by discounting at market rates of interest, and is independent of preferences among dividend streams. When the assumption fails, the capital value of a stream of net receipts (or of dividends) cannot in general be found by

1 The supply and demand schedules of funds themselves may also shift as the firm grows. This feature is not taken into account in our model, but could be introduced.




discounting at the marginal internal rate of return, or at the average rate of return which could be obtained if a compensating payment were invested outside the firm, or indeed at any rate which can be found without first knowing the capital value. This value must be assessed by applying the usual Hicksian compensation theorem, and simplified methods are only applicable in special cases.

Given a perfect capital market, assets which yield money income (but no direct personal services) should therefore have the same value to all potential buyers, at any rate if com- plementary resources are available to all of them at the same prices. In an imperfect market, the shape of the income stream may appeal to some people more than to others, so that demand prices may vary.

(3) If the basic assumption holds, the average period of production, i.e. of the stream of net receipts, as defined by J. R. Hicks (see Section VI below) increases when the market rate of interest falls. (Hicks assumes a single rate for all periods). When the loan market is imperfect, the average period of net receipts does not necessarily increase when the supply of loans to the firm rises, or the demand for loans from it falls. If we add cash flows due to borrowing and lending transactions to net receipts, and consider the average period of the dividend stream, the corresponding proposition is false also. The reasons for these results are set out in Section VI.

The rest of the paper presents an extended discussion of these points. In Section II, a simplified model of the firm is described. In Section III, the case when the basic assumption holds is illustrated geometrically. Sections IV-VI consider the problems under the three headings above, in the case where the basic assumption fails.

II. A Simplified Model One of the main assumptions made in this section is that market transactions and

dividend payments occur at two points of time only. Its purpose is to allow the use of diagrams. The limitations which it entails are not very serious, since in the case when the basic assumption holds, general proofs are mostly available in the literature; while in cases where the assumption fails we shall be chiefly concerned to show that certain theorems are not true, and for this purpose simple examples suffice. The main consequence of the restriction to two points of time is that complementarity cannot be considered.

The following assumptions specify our model: (1) Uncertainty: the firm makes single-valued estimates of future values of the

relevant variables-i.e. not probability distributions-and acts on these estimates as though they were certain.

(2) Time: market transactions, including borrowing and lending, and the payment of dividends occur at two times only, to and t1. Productive processes take place at inter- vening times. The firm possesses some assets at to, but all assets (including earnings between to and t1 after repayment of debts) are to be distributed and consumed by tl.

(3) Aims: the firm's aim is to obtain the most desirable stream of dividends (consumption), preferences being expressed by some utility function

U - u [co, cl] (Co, Cl > O) with the usual properties.'

1 If assets remained at t1, their value to the firm would presumably depend on the dividends which they could yield at later dates, and so the firm's plan would in general extend to infinite time; alternatively, arbitrary valuations of closing assets would have to be introduced. Neither of these refinements is indispens- able for our analysis. In passing, it may be mentioned that the assumption that utility depends only on future dividends does not preclude the possibility that the firm's owners value the growth of the firm " for its own sake "; such a preference (at least in certain versions) can be regarded as a preference for deferring dividends, possibly to infinite time.




(4) Opportunities: the relationships between dividends (c), net receipts (y) and net increases in indebtedness (b) are obtained from the definition (1.1) as

ct = yi + bi (i- = ,2)

The values of y and b may be positive or negative, while c is non-negative. Oppor- tunities for investment in the firm are given by a function

Yi = f(yo)

This function is monotonically decreasing. The value of yo for yi = 0 is the sum which could be realised if the firm were closed down at to, so that the function depends on the firm's initial assets. It will be assumed that this value of y0 is non-negative (though exceptions might arise in the case of a firm bound by past contracts), so that the graph of the function passes through the non-negative quadrant (see Diagram 1). Cases where the graph is convex upwards-i.e. where there are no indivisibilities-will be of special interest.

I X

Figure I

Turning to opportunities 'in the loan market, the possibility of buying or selling equity capital is ignored, and all loans to or by the -firm are assumed to be repayable at tl. There are no debts outstanding at to. Opportunities are defined by

b, = g(bo)



IMPERFECT CAPITAL MARKETS

This function also decreases monotonically, with derivative

g' (b) < -I

corresponding to a non-negative rate of interest. The values of bo and b1 are of opposite signs, and the graph of the function passes through the origin (see Diagram 2). Here also convex functions will be of interest. In the special case where the basic assumption holds we have

b, = -b(1 + r) =-bo/

where r is the rate of interest, and p =- 1/(1 + r) is the discount factor. The discounted equivalent of any flow variable, say y, with values (yo, Yl) is then

Yo + 13Y

b,

Figure 2

In a diagram, if Yo is measured horizontally and Yi vertically, a line of slope -(1 + r) is drawn through the point (Y0, yi), and the intercept with the horizontal axis gives the discounted sum. For example, in Diagram 1 the slope of AB is -OB/OA = -(1 + r), and the discounted equivalent of the combination at the point C is OA. Lines of equal

187

1(0WILrW)




slope lying higher to the right correspond to greater discounted sums. Thus the point of greatest discounted net receipts on the opportunity line fin Diagram I is the point C wheref reaches the highest discount line.

For some purposes we wish to express directly the relationship between cl and c0, taking investment opportunities inside and outside the firm together. For this purpose we define the function

Cr - c (co) (co c, > 0)

J ikit

f

la t6a t

Figure 3

which gives for each c0 the greatest attainable cl. This function may be dcrivcd geometrically fromf and g as follows:' Select a point S off in Diagram 1, place the origin of Dia- gram 2, through which g passes, on S, and trace out g with S as origin. This amounts to adding all possible values of bo and g (bo) to one combination S [yo, f(y0)], obtaining the set of all possible dividend plans (c0, cl) for fixed S. Repeat the process for all points off. To obtain the highest value of cl for each c0 draw the envelope of all these curves. The restriction of the envelope to the non-negative quadrant is b. The optimum dividend plan (see Diagram 3) is given by the point Q where b reaches the highest indifference curve u, and the corresponding net receipts are given by S on f. The broken line connecting S and Q is g5, the graph of g with S as origin. The horizontal distance between Q and S is bo, the vertical bl. Since transformations through the firm and through the loan market are alternatives, the slopes of 0 and gs at Q and off at S must be equal (if the derivatives exist). In this way, dividends, net receipts and borrowing are simultaneously determined.

I I am. indebted to Mr. J. Black for this elegant construction.




III. Special Consequences of the Basic Assumptioni

The situation when the basic assumption holds is illustrated in Diagram 1. The g curve is now a straight line of slope -(1 + r), such as the line AC. Applying the method of construction just described, we obtain 0 by drawing the highest line of this slope which passes through any point off, and restricting it to the non-negative quadrant. This is the segment BA. Any dividend plan, for instance G, can be chosen on this line. The corresponding production point is always at C. The optimal character of C can also be described in another way: as investment in the firm rises, i.e. yo falls and Yi rises, the rate of return (either for a lumpy or a marginal change) exceeds r only as long as a higher discount line of slope -(1 + r) can be reached, i.e. up to C. The gap between C and G is filled by borrowing bo = FJ = KG and repayment b, =-KC = -KG (1 + r).

This construction illustrates the main properties of the production plan. First, the optimum C is independent of the dividend plan G. Secondly, C is at the point of greatest discounted net receipts. Thirdly, the value of the business to its owner, as assessed by the compensation rule, is equal to the discounted net receipts OA, which is also equal to the discounted value of dividends. If the business were taken away, and all opportunities of earning more than r on any funds available therefore lost, a compensating payment equal to OA would just enable the former combination of dividends G to be attained by investment at the market rate. However the compensating payment is independent of the point G which is chosen, and therefore of the utility function.

If the firm's investment opportunities improve (f moves outwards) while the rate of interest remains unchanged, the straight line k is shifted outwards but its slope remains the same. The effect on the dividend plan is analogous to the effect of an increase in income with constant prices in the theory of consumption. If neither c0 nor cl is an inferior good, both increase; thus a reduction in immediate consumption when investment opportunities improve will usually be a sign of imperfect competion in the loan market.

Finally, we prove a property of the optimum C onf which will be required below. The point C is characterised by the tangency of the highest discount slope to f. Now if f is not convex, we consider the closed point set lying on and below f-the production set- and form its convex hull.' Whatever the rate of interest, C must lie in the boundary h of the convex hull, since a tangent to any point off not in h would cutf somewhere and therefore could not be the highest tangent of that slope. This incidentally implies thatfis convex at C and that diminishing (or constant) marginal returns to investment in the firm prevail.

IV. Production Planning when the Basic Assumption fails

It will be sufficient to consider the case where the firm can neither borrow nor lend, so that f and f coincide. Suppose first that c is strictly convex (see Diagram 4 below) and that it reaches the highest attainable indifference curve u at C. The point C fixes both dividend and investment plans, which are interdependent. Clearly i and u have at least one common tangent at C which cuts neither curve, in this case the line A CB. The slope of this line gives an internal rate of interest, and the discounted equivalent of the dividend stream is at its maximum at C if that rate of interest is used. However, the internal rate, like any other purely internal price, is in general found as part of the optimum plan, and therefore cannot be used to find it. (Exceptions may of course occur if c is linear, or approxi-

1 The convex hull of a set is the smallest convex set containing it. In two dimensions it can be drawn by connecting pairs of points of the set by straight lInes until the enlarged set is convex, i.e. until all further connecting straight lines fall inside it or coincide with its boundary.



REVIEW OF ECONOMIC STUDIES

mately linear, over a range and it is known in advance of planning that the optimum lies in that range). Further, it is shown below that the discounted sum obtained in this way is not generally equal to the compensation value of the business.

In cases where f is not convex over some ranges, the optimum may not lie in the boundary of the convex hull of i, and in this case the optimum is not such as to maximise discounted dividends, no matter what discount rate is used. For example in Diagram 3, where the optimum is at Q, a tangent at Q cuts q at other points. A point like R has a higher discounted value (using the discount rate given by the slope at Q), but lies on a lower indifference curve. There is no discount rate whatever for which Q is the point of greatest discounted dividends. This difficulty is important because non-convexities (indivisibilities) occur frequently in investment planning.

It may happen that the basic assumption does not hold, (and that it is not possible to find any special weaker sufficient conditions for the validity of the discounting approach which hold for the problem in question) but that the firm nevertheless decides to make its plans on the basis of a " reasonable " rate of discount which it sets itself. This amounts to assuming that the indifference curves are straight lines, at least over a range.

It is easily shown that, on the assumptions of this section, an improvement in investment opportunities (an outward movement of O) can lead to lower consumption at to, quite apart from inferiority. The details are left to the reader. The other consequences of the present analysis which were indicated in Section I follow immediately from the remarks above.

Ct

B\

X \

C

J

4 U"

Co

0 E A F P Figure 4:

190




V. Asset Valuation

The capital value of an asset or of a business at a specified date may be defined as the payment, at that date, which would just compensate the owner for the loss of the asset or business. Thus the theoretical problem of valuation is solved by applying the Hicksian compensation theorems and it obviously makes no difference in principle whether the " commodities" are money and perishable consumers' goods, as in Hicks' work, or dividends at different points of tinme. However, in discussions of the latter problem, valuation is often defined as the discounting of future sums at an appropriate rate of interest. It is in fact more satisfactory to define valuation in ternms of compensation and then to prove that when the basic assumption holds, compensation can be assessed by discounting. For the two-dimensional case, this was done in Section III. When the basic assumption fails, the correct compensating payment for the loss of a business depends on potential earnings outside that business. Thus the present value is in general different from expected net receipts or dividends of the business discounted at the marginal internal rate of return. Moreover, the compensation cannot in general be assessed by discounting dividends at the average external rate at which the compensation would be invested, even if that rate is constant and can therefore be found independently of the amount of compensation. This is illustrated in Diagram 4, which for simplicity depicts a case where borrowing and lending happen to be zero at the optimum point C on b. The external (lending) rate is assumed to be that given by the line GC (or by the parallel line FK), which is below the marginal internal rate. The value of C, discounted at the external rate, is OG. However, this sum would over-compensate the owner, since it could be invested to yield a point like J, which is preferable to C. The correct comn- pensation is OF, which could be invested to yield K, which is indifferent to C. This is more than the sum OA obtained by discounting at the marginal internal rate.

We therefore conclude that the value of a business cannot in general be assessed by discounting its dividends (or its net receipts) at internal or external rates of interest, or indeed at any other rate that can be established without first carrying out the valuation by other methods. It remains true in all cases that the optimum plan is that which maximises the value of the business to its owners, and conversely. However, it is not possible to use this property for planning, since the value of the business to the owner depends on the optimum plan (and on other external opportunities), not the other way about.

VI. Interest and the Period of Production

A famous proposition, suggested by Bohm-Bawerk and proved (in a different form) by J. R. Hicks, (Value and Capital, pp. 219-222 and 326-328) asserts that a firm's period of production is longer, the lower the rate of interest. This theorem presupposes that the firm can borrow and lend as much as it wishes at the going rate of interest; the present section investigates the consequences of dropping this assumption.

Hicks' proof of the theorem depends on his definition of the period of production, which in some ways may seem arbitrary. The reader is referred to Hicks' work for a discussion of this definition. For our purposes it will simply be accepted because it is the only known definition which produces the required result even when the basic assumption holds. Hicks defines the average period P(x) of a flow variable xt taking on different values at different times t as

n n

P(x) = E tPt xt/E; Ptxt (VI. 1) t=O t=O

where ,B = (1/1 + r*) is the discount factor corresponding to some constant rate of interest



REVIEW OF ECONOMIC STUDIES

r* which is taken as the base for comparisons. The rate is assumed to be the same in all periods. Only a finite number of periods is considered. The values of the xt's are a function of the rate of interest r regarded as a variable. In comparing the values of P(x) corresponding to different values of r (and therefore in evaluating dP(x)/dr) only the xt's are allowed to vary, while the terms Pt, being weights of an index, are kept constant. Using this rule, Hicks shows by calculus methods that the average period P(y) of net receipts for an optimal production plan rises when r falls, i.e. dP(y)/dr < 0. For the two-dimensional case, we shall give an alternative proof which holds for finite changes in r.

For later reference, we first establish how the average period of net receipts changes when Yo and y1 change, without specifying whether the values of the variables are optimal in any production plan. Under our assumptions, we have

P(y) = Yo + Y

Differentiating this expression, we find that P(y) increases or remains constant when y, increases (more investment in the firm) if and only if

dyo0 V Yo -Y1 d > 0 (VI.2)

We now show that if the curve yi = f (Yo) is convex throughout, monotonically decreasing, i.e. f'(yo) < 0, and passes through the positive quadrant, i.e.f (0) > 0, then P(y) increases or remains constant as yo falls and yi increases. Since the curve is convex, any point y = (Yo, Y1) will lie below or on a tangent drawn at any other point y = (Yo, Yi), i.e.

Y + f'(Yo) (Yo - Yo) >

Putting yo = 0, j = f(0), we have

Y1 -f'(Y) Yo > f(0) > 0

Dividing by the negative term f'(y) = dyl/dyo and changing signs, we have condition (VI.2) above. (If the derivative f'(yo) does not exist at some points, the same argument applies to the upper and lower derived numbers at such points; the limits must exist since the curve is monotonic). Thus P(y) is non-decreasing with y, at all points on f, and therefore over any finite range also.

If f is not convex, an increase in Yi can be associated with a fall in P(y), (still assumingf(0) > 0). This is illustrated in Diagram 5. Taking yo < 0, y9 > 0, the condition (VI.2) becomesf'(yo) > Yl/Yo. We note thatf'(yo) is the slope of thef curve, while yl/yo is the slope of a ray through the origin to a point (yo, Yi), and that both are negative. Paying attention to sign, it is clear that the condition (VI.2) is not fulfilled between points A and B in the diagram, so that a rise in yi is associated with a fall in P(y).

We now turn to the main problem, and show that, when the basic assumption holds, the value of P(y) for an optimal plan rises or remains constant when r falls. Suppose first thatf is convex. The optimum C inf is determined by the tangency of a line of slope -(1 + r); if r falls, the tangent is flatter, hence the new optimum C is further left than C (or C - C), i.e. 9, > y1; owing to convexity, P(y) > P(y). Iff is not convex, consider the boundary h of the corresponding convex hull. It has been shown that both C and C must be in h. We can therefore substitute h for f in the above proof.

We now drop the basic assumption, and redefine Hicks' problem accordingly. First it is no longer possible in all cases to refer unambiguously to a rise or fall in market rates

192




Figure 5

of interest. There is a supply schedule of funds to the firm, and a demand schedule for borrowing from it. For simplicity, we consider only borrowing (i.e. bo > 0, b, < 0); the analysis of lending is similar and is left to the reader. We also confine the discussion to cases where borrowing becomes unambiguously cheaper or dearer; if g is the new function we say that borrowing is cheaper if

g(bo) > g(bo) for all bo > 0 with the strict inequality holding for some bo. Dearer borrowing is defined correspondingly.

Secondly, the constant ,B 1/(l + r*) appearing in Hicks' formula for P must be redefined, since it is no longer possible to speak of " the " rate of interest. An obvious way is to define it as the slope off in some plan which is taken as a " base " for comparisons, i.e. P = l/f'(yo*), where yO* is the optimum value in the base plan. Alternatively the slopes of g or k could be taken-if all three exist, all must be equal; if none exists, a sup- plementary rule is required to fix 3 within the limits of indeterminacy. In fact, it turns out that in two dimensions f cancels out of all the results.'

'For more dimensions, we can define Pt = -ayo/ay and write Pt for Pt in the formula for P. This allows for different rates of return in different periods. These problems will not be considered here.




Thirdly, we may wish to redefine the variable x whose average period P(x) is to be measured. There are two possibilities. We can take x = y = net receipts, as above. Alternatively, it may be argued that under the new assumptions the distinction between investment in the firm and loan transactions loses its significance, so that we should set x = y + b c, i.e. that the average period of consumption P(c) should be considered. These problems will be considered in turn.

We first consider the effect of cheaper borrowing on P(c); although not very interesting in itself, the discussion is a useful preliminary. Since c0 and cl are non-negative, the condition for P(c) > P(c), namely

Co + C1 5 Co + Ci

can be written j Cl

co co In other words, we require the conditions in which the proportion of consumption

at t1 to that at to rises when borrowing becomes cheaper. Now a simple diagram for the case when the basic assumption holds shows that, for a firm which borrows but does not lend in the initial optimum position, a fall in interest has an income effect which tends to raise both cl and c0 if neither is inferior, and a substitution effect which tends to increase c0 at the expense of cl. Thus the above inequality is unlikely to hold even in this case. The same is true when the basic assumption fails. If we redefine the income effect as that which occurs when s and b have the same slope for each cl at which both curves are defined, and suppose that the actual fall in borrowing costs is such that is flatter than q for each cl, the same statements apply. The only complication is that, although q must be to the right of 0 when borrowing costs fall, it need not in general be flatter for each cl. Even convexity off, g and g, (and hence of fi and $) is insufficient to ensure this, unless we also assume that g is flatter than g for each b1 (i.e. for each repayment level).

We turn now to the average period P(y) of net receipts, and assume initially that all the curves are convex, so that P(y) rises if y, rises. One might expect investment in the firm, and hence Yi, to rise on account of lower total borrowing costs, but this does not necessarily happen, because margirnal borrowing costs may be higher for some values of bo. We might exclude this by assumirng that g must be flatter than g for each bo. To exclude the complication indicated in the previous paragraph, we might also assume that this was true for each bl. But apart from this, the substitution effect in consumption might lead to a fall in cl, thus tending to reduce b1 or Yi or (if the derivatives exist) both to some extent. The fall in Yi on this account might be so great as to lead to a net reduction in Yi and a fall in P(y).1 Even if further assumptions about preferences are added to ensure an increase in cl, everything is again uncertain if the convexity assumption is dropped, since a rise in Yi may then be associated with a fall in p(y).2

' Construct a diagram as follows: Draw convex f, g and 0. Let S be the original optimuin on f, Q that on b and g,, and recall that the slopes at S and Q must be equal. Now draw convex g and + subject to the restrictions on the relationship of g to g indicated in the text. Thus s is higher and flatter than + for all c0 and cl. Let T be the point on s wlhere the slope equals that off at S. Depending on the indifter- ence curves, the new optimum Q may be above or below T on +; if it is below, the corresponding S on f must be below S, so that Yi falls, and vice versa.

2 In Diagram 5, an increase in Yt from A to B reduces P(y). It remains to check that A and B can be optimum positions in a case satisfying the restrictions indicated in the text. To show this, suppose that there is absolute preference for consumption at t1 (horizontal indifference curves), and that the firm can borrow at zero interest just enough to push investment in the firm to A, but that borrowing costs become infinite thereafter. Then A is optimal on f; B may be optimal in the same way, with somewhat greater borrowing. It is easily checked that the special conditions are satisfied.




We therefore conclude that the average period of production does not necessarily rise as the cost of borrowing falls, and that rather strong additional assumptions are required to produce this result. The nature of these assumptions indicates why matters are different when the loan market is perfect.

London. LUCIEN FOLDES.



imperfect capital markets and the theory of investment

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