Journal of Economic Dynamics and Control 9 (1985) 457-476. North-Holland

INTEREST RATE, DEMAND AND INPUT PRICE UNCERTAINTY AND THE VALUE OF FIRMS

George FRENCH Tulane University, New Orleans, LA 701 IS, USA

Received March 1985, final version received April 1986

In an infinite-horizon inventory model, an increase in interest rate uncertainty increases the value of a firm which has positive value. An increase in input price uncertainty increases the value of the firm. If decisions are made before the realization of demand uncertainty, increased uncertainty about an additive demand shock reduces the value of a price-setting firm with a concave value function, and leaves unchanged or increases the value of a quantity-setting firm. If decisions are made after the realization of demand uncertainty, an increase in the uncertainty of an additive demand shock increases the value of the firm.

1. Introduction

This paper is a partial equilibrium analysis of the effect of increasing uncertainty on the value of firms. One of the many aspects of the economics of uncertainty to which economists have devoted their attention is the question of how an increase in uncertainty affects the well-being of economic agents. The partial equilibrium approach to this question generally leads to the conclusion that agents prefer more risk to less, holding the mean of a distribution constant. For example, Waugh (1944) points out that a reduction in price increases consumers’ surplus more than an equal price increase would reduce it. Oi (1961) shows that a competitive firm’s profits are increased if an existing distribution of output price is replaced by one with the same mean but a higher variance. More recently the literature on search theory has come to similar conclusions. Using the measure of risk proposed by Rothschild and Stiglitz (1970), conditions have been given under which a consumer searching for the lowest price or a worker searching for the highest wage are made better off by an increase in the risk of the distribution being searched. [See, for example, Lippman and McCall (1976).]

As Samuelson (1972) points out, one must not make too much of these results. He points out that one cannot take the Waugh and Oi results together and conclude that the economy could ‘pull itself up by its bootstraps’ through ‘manufactured uncertainty’. Thus, in a world of unchanging tastes, technology, and endowments, the constant price competitive equilibrium is Pareto-optimal

01651889/85/$3.30@1985, EIsevier Science Publishers B.V. (North-Holland)

458 G. French, Parameter uncerraing~ and the value a/ thefinn

and could not be Pareto-dominated by any contrived scheme of price uncer- tainty. Nevertheless, the arguments underlying these results are impeccable. We must simply confine the results to their proper sphere, that of analyzing how some group in society benefits by a change in the degree of dispersion of some parameter affecting its behavior.

This paper examines the effect of an increase in the uncertainty of various parameters on the profits of a risk-neutral firm which maximizes expected discounted profits over an infinite horizon. An increase in uncertainty is taken to mean a mean-preserving spread of a distribution, in the sense of Rothschild and Stiglitz (1970). Input prices, the interest rate, and a demand index are i.i.d. random variables which are realized in each period. It is shown that under a variety of assumptions regarding the appropriate treatment of stockouts, the time required for production, and the timing of decisions, an increase in the uncertainty of interest rates increases the value of the firm if the value of the firm is positive. Even more robust is the finding that an increase in the uncertainty of spot input prices increases the value of the firm. The effect of an increase in demand uncertainty depends in general on the functional form of the relationship between the demand index and the realized price or quantity. In the special case of additive shocks, we get an interesting breakdown of results. Regardless of assumptions regarding the treatment of stockouts or the time required for production, an increase in demand uncertainty increases the value of the firm if price and production decisions are made after the realization of current demand uncertainty (of course, next period’s demand index is still unknown). If decisions are made before the realization of the demand index, the result depends on whether the firm is a price-setter or a quantity-setter. If the firm sets a quantity to sell and then lets the price adjust to sell that quantity, an increase in demand uncertainty leaves the value of the firm unchanged or increases it. If the firm sets a price and then sells what it can at that price, then an increase in demand uncertainty reduces the value of the firm, provided the firm has a concave inventory value function.

Several recent papers have used the production-inventory model to examine questions regarding pi-&flexibility and production smoothing. See, for exam- ple, Abel (1985), Amihud and Mendelson (1983), Blinder (1982) and Reagan (1982). The difference between the models used in these papers illustrates the fact that there are several reasonable ways to model the intertemporal profit- maximization problem faced by a firm. We now turn to a discussion of some of these modelling issues.

It is natural to assume that the firm may hold inventories of its ‘finished product at some cost. This paper and all of the papers mentioned allow for possibility. It is equally natural to assume that the firm may hold inventories of inputs to production. This paper shares with most other papers in ,the inventory literature the defect of not allowing for this possibility. It is not clear what the ‘natural’ assumption is regarding the appropriate treatment of

G. French, Parameter uncertainty and the value oj thefirm 459

stockouts. If demand exceeds the available physical stock of the good, is the excess demand lost forever, or is it possible for the firm to sell the good short (that is, accumulate backorders) and fill the excess orders out of future production? Reagan follows the first approach, Blinder the second, while Abel considers both possibilities. Another question concerns the length (in time) of the production process. Should we regard output as being produced instanta- neously, or is there a lag between the time at which production commences and the time at which output becomes available for sale? The first approach is followed by Blinder, while Reagan, Amihud and Mendelson, and Abel follow the second approach. It is apparent that there is no ‘correct’ way to model the stockout or the production lag question. The appropriate model would depend on the product under consideration. The results in this paper can be shown to be robust to model specification as regards these issues: the impact of increased uncertainty of a parameter on profit does not depend on whether excess demand is lost or backordered, or on whether production is instanta- neous or time-consuming. The results are also independent of whether the firm has market power or is a perfect competitor.

Issues of model specification which affect the results of the paper regarding demand uncertainty concern whether decisions are made before or after the realization of demand uncertainty, whether the firm is a price-setter or a quantity-setter, and the functional form of the relationship between the random demand index and the realized price or quantity. Let us consider the last question first. Let h(p, a) denote demand as a function of price p and a random variable a, and let p(q, a) denote the demand price as a function of sales q and the random variable a. If h or p are strictly convex functions of a, then an increase in the spread of a would increase expected demand (price) at any price (sales level), and should increase expected discounted profits. The opposite should be true if h or p are strictly concave in a. Subject to a few qualifications, both statements are true, and unsurprising. It is of more interest to consider a purely neutral increase in the uncertainty of a, one which would leave expected demand (price) at any price (sales level) unchanged. This would be the case if we took a mean-preserving spread of u and let, say h( p, a) = u(p) + a. In this case the effect of increased uncertainty of demand will depend on the timing of decisions and on whether the firm is a price-setter or a quantity-setter.

The appropriate model of the timing of decisions depends on the environ- ment in which the firm is assumed to operate. On the one hand, the firm might be regarded as knowing current demand conditions and knowing the normal or expected level of future demand. When demand is high relative to normal, the firm draws down inventories, and when demand is low it builds up inventories in order to avoid future higher marginal cost production. In this view, the firm is speculating on the future value of its inventory holdings. A model in which decisions are made after the observation of i.i.d. demand

J.E.D.c. 1)

460 G. French. Parameter uncertain(v and the value of the firm

curves would capture the essence of this problem. A neutral increase in demand uncertainty would increase the value of the firm in such a model. Intuitively, since the firm observes demand before making decisions, increased dispersion of demand increases the opportunities for profitable inventory speculation.

On the other hand, the level of expected demand facing the firm might be taken to be constant. In this case the firm’s price and production level respond to passive inventory fluctuations caused by the fact that demand is not known with certainty. In this setting the appropriate model is one in which decisions are made before the realization of demand uncertainty. As regards empirical evidence, Carlson (1973, p. 85) estimates that the period of production in two thirds of two-digit U.S. manufacturing is less than one month, and is of the opinion that inventory models should allow the firm some leeway in changing production after the observation of current demand.

If decisions are made before the realization of uncertainty, we must dis- tinguish between quantity-setting and price-setting behavior. If the firm sets a quantity to be sold and lets its price be the random demand price necessary to sell that quantity, then a neutral increase in uncertainty would leave expected revenue unchanged, and would not affect any other components of the firm’s payoff. It would therefore have no effect on the value of the firm. (It is noted later that the value of the firm would increase if non-negativity constraints on price become binding.) If, however, the firm sets a price and then meets the demand at that price, an increase in demand uncertainty would not only affect the revenue function, but would increase the dispersion of next period’s starting inventory stock, for any given choice of this period’s price and production level. Under assumptions sufficient to guarantee concavity of the inventory value function, this increased dispersion would reduce expected discounted profit.

The next section describes the notation and the general form of the model that will be used. Section 3 gives the basic propositions necessary to derive our results, namely a set of sufficient conditions for an increase in the uncertainty of a parameter to increase, decrease, or leave unaffected the value of the firm. Section 4 gives the results on the effect of increasing uncertainty of spot input prices and of interest rates on the value of the firm. Section 5 describes the results concerning increases in demand uncertainty, and section 6 contains concluding comments.

2. Model and notation

We use the following terms and assumptions:

(A.l) Let a, denote a random component of demand, revealed to the firm each period, either before or after the firm must make its decisions. Let a2

G. French, Paramerer uncertainty and the value of the firm 461

denote the random one-period interest rate, revealed to the firm each period before any decisions are made. Let u3,. . . , u, denote random spot input prices, also revealed each period before decisions are made. Each random variable a,, i = 1,. . . , n, is i.i.d., atomless, has c.d.f. Fi and is contained in the interval [Li, Ui], with Li > 0 for i = 2,. . . , n. Since the random variable a, will appear frequently in the paper, we will often drop the subscript: ‘u’ denotes ‘a,'.

(A.2) Let y denote the amount produced, and let c(y, u3,..., a,) be’ the minimum cost of producing y given the input prices us,. . ., a,. For fixed 03,. . ., a,,, c is a twice continuously differentiable, increasing, convex function of y with c(0) = 0. We will abuse notation and let c’ > 0 and c” 2 0 stand for the first and second partial derivatives of c with respect to y. In some cases y will be assumed to be instantly available for sale; otherwise it will be assumed that there is a one-period lag between the decision to produce y and the time at which ‘y is available for sale.

(A.3) Let p denote price and q denote quantity sold. When we consider a firm that sets price, we will let h( p, a) denote the quantity demanded at price p and demand index u. When the firm sets a quantity to sell, we will let p( q, a) be the demand price associated with the quantity q at demand index a. Assume

(9 h, < 0, h,>O, (ii> p4 < 0, Pa ’ 0,

where the subscripts indicate partial derivatives.

(A.4) Let x, denote the inventory stock or backorder level with which the firm starts period t. If excess demand is lost permanently, x, 2 0 is required. If demand can be backordered, then x, can be positive or negative. Given x,, the demand index u,, production y, and price p, or sales q,, a new inventory level X t-1 is determined by the transition equation

X,-l = qx,, Y,, P,, 4,, 4. (1)

The functional form of the transition equation will depend on whether excess demand is backordered or lost, on whether production is instantaneous or takes one period to become available, and on whether the firm is a price-setter or a quantity-setter.

(A.5) Let the function g(x) give the cost of opening a period with inventory or backorder level x. g is assumed strictly convex with g(0) = 0. When

462 G. French, Paramerer uncertain[y and the value of the firm

backorders are allowed the domain of g is the real line. If excess demand is lost, the domain of g is the non-negative half-line.

(A.6) The random variables a,, . . . , a,, are independent.

(A.7) There is no distinction made between real and nominal interest rates; the general price level is assumed constant. One dollar loaned today becomes 1 + a2 dollars next period with certainty. The firm is risk-neutral. Ai of time t, the firm has an expectation of the value of all future profits starting from time r + 1. The firm is indifferent between this prospect and that of receiving the expected amount with certainty. Therefore the discount rate it uses to de- termine present value is

b = l/(1 + a2).

Obviously b is an i.i.d. random variable with a distribution induced by that of a*.

(A.8) The firm’s instantaneous or current period payoff, s, equals sales revenue less production cost less the inventory or backorder penalty cost. Let the (ex post) current period payoff be denoted

= sales revenue - c(y,, ajr,. . . , a,,) - g(x,), (2)

where the form of the revenue function depends on whether the firm w[\ price or quantity, whether excess demand is lost or backordered, and on whether production is available for sale immediately or after a lag.

(A.9) The f?rm knows x,, a,,,. . ., a,,, and possibly al,. Based on these state variables the firm chooses (J+, y,) or (qr, y,) to maximize expected discounted profit bver an infinite horizon. Given that the firm enters a period with inventory level x and proceeds optimally for the rest of time, its expected discounted profits are V(x). V is called the value function. If the firm does not know al when it makes it choice, the value function satisfies the optimality equation

V(x)= E a*.....a.

G. French. Parameter uncertainty and the value of thefirm 463

where b is defined in (A.7). The firm’s objective function is inside the square brackets. We have assumed that the firm is a price-setter. If it sets q instead, the optimality equation is obtained by replacing all the p’s in (3) with q’s. If the firm knows a, before making decisions, the optimality equation becomes

In the case where a, is known before decisions are made, price-setting and quantity-setting are equivalent. Since it is easier to work with q as a choice variable, we will assume that the firm sets q in this case. The notation Ea2,... (I indicates that we are integrating the expression that follows with respect to thl probability measures 4, i = 2,. . . , n, of the random variables a,, . . . , a,. We will often use the notation E, and E,, to distinguish expectations with respect to different distributions’of the random variable a,.

(A.lO) lim, - m (nyBObi) = 0 for all realizations of the random sequence (bt)Eo- This assumption guarantees that V(x) is finite for all x. The assump- tion would be satisfied if, for example, there were some e > 0 such that b, < 1 - e for all t, with certainty.

(A.ll) Throughout the paper, &* will denote a mean-preserving spread of the distribution function 4 of the random variable ui, in the sense of Rothschild and Stiglitz (1970). F,* represents a ‘riskier’ or ‘more uncertain distribution of ui than does &. The firm’s value function will depend on the distributions of all the random variables a,, . . . , a,. We use the notation V6 and V$ to indicate the value function when ui has the distribution Fi and Fi*, respectively.

3. Uncertainty and the value of firms: Preliminary results

Suppose that the firm makes decisions before observing the demand index, so that (3) is the optimality equation. We consider first the effect of a mean-preserving spread of the interest rate or of one of the input prices on the firm’s value.

Proposition I. Let the firm choose p and y before observing a,. Suppose that, holdingfixed x, y, p, a,, u2,. . ., ujel, u~+~,.. . , a, we have

a2 -q = x,y,p,a, ?...> [ ( 0,) + bv#(x, Y, P, a,)] > 0,

464 G. French, Parameter uncertainy and the ~wlue of lltejinn

forsomej> 1, where aand VF, aredefinedin (2) and (3). Then V,.(x) > V,(x) for all x (see A. I I in section 2 for clarification ).

Proof. If this is true, then E,,(s + bV<) is strictly convex in aj, for fixed values of x, y, p, a*, . _. , ajel, aj+l,. . . , a,,. It follows that maxP,.v E,](m + bV,), being the upper envelope of a family [parameterized by (p, y)] of conbex functions of aj, is again a strictly convex function of uj for fixed x, a 2,. - -, aj-1, aj+l,. . -, a,. Integrating out the remaining random variables, we find that

E a* . . . . . a ,-,, =,+,, . . . . =,

max E(r+bVC) P.S =1 1

is again a strictly convex function of uj, for fixed x. Now, from (3),

v,(x) = E E max E(r+bV,,) 3(1* , . . . , a,-1.a,+ , , . . . , a, p.)‘ a,

<E E To* * . . . , a , - , . aj+ 1. . -- , a,

maxE(r+bV<), (5) p,.,’ a,

where the inequality is by a theorem in Arrow and Intriligator (1981, p. 216). This theorem is an application of Jensen’s inequality. Apply this inequality again to the V5 appearing on the right-hand side of (5), and so on recursively. This leads to V5( x) -K VF,’ (x) by definition of VT.

The reader should note that Proposition 1 is also valid if the firm sets q and y rather than p and y: the argument is identical The next proposition concerns the effect of a mean-preserving spread of the demand index on a firm which makes decisions before the realization of q.

Proposition 2. Let the firm choose p and y before observing aI. Suppose that holding fixed x, y, p, a2,. . . , a,,, we have that the function P + bVr, is strictly convex in ul, where n and Vr, are us in (2) and (3). Then Vr,.(x) > V,,(x). If IT + bV,, is strictly concave in a, for fixed values of the other variables, then Vr,.(x) -C V,(x). Zf R +‘bVr, is linear in uI for fixed values of the other variables, then V,(x) = V,.(x).

Proof. Suppose T + bV,, is strictly convex in ui for fixed values of the other variables. Then by (3),

VF,(x) = E max E(s+ bVF,) a2 ,.... a, p..v F,

< E max E(a+bVF,), a~ . . . . . a, p..v F,*

G. French, Parameter wrcerroing and the ma/ue of rhefim 465

by the same theorem mentioned in Proposition 1. Applying this inequality recursively to the VF, appearing on the right-hand side, we get V,(x) < V,.(x). If 7~ + bV,, is strictly concave in a1 for fixed values of the other variables, then the same argument would give V,(x) > V,.(x). Suppose finaIly that 91 + bV,; is linear in aI. Then

77 + WI =fFp) +kF,(z)Q,,

where f and k are constants which depend on all the other variables Z=tx, y,p,a,,..., a,) and on the distribution Fi. In this case

The second equality is by definition of a mean-preserving spread. Hence by (3),

Applying this equality recursively to the I$, term appearing on the right-hand side yields VF,( x) = V,,(x).

Finally consider a firm which makes decisions after the realization of the demand index. In this case the optimahty equation is (4), and we have:

Proposition 3. Let the jinn make decisions after the realization of a,. Suppose that holding fixed x, y, q, al,. . . , ajel, a,,,, . . . , an,, the function T + bV, in (2) is strictly conuex in aj for some j = 1,. . . , n. Then V,(x) < V,,(x).

Proof. By (4)

V,(x) = E E ‘I aI ,..., a,-1.a,+ ,‘.... a,

max(a+bV,) p,.l

<E E y a I..... *,-,.*,+ ,...., OI

ma(a+bVF,), *,.I

466 G. French. Purameter uncertainty UIKI the cdue oj rheJirm

and apply the inequality recursively to the VF, on the right-hand side. The inequality is obtained as in Proposition 1: since r + bV,, is strictly convex in aj, so is the function obtained by maximizing over q and y and so is the function obtained by integrating out a,, . . . , aj- i, uj+i,. . . , a,.

4. Increased uncertainty in interest rates and input prices

In this section we show that in the context of the model we have con- structed, the firm prefers more variability of the interest rate and of input price to less. The only qualifier we must add is that in order for the firm to like increased interest rate uncertainty, the value of the firm must be positive. Mathematically, these results are a simple application of Propositions 1 and 3 in the previous section.

Proposition 4. Let Fz* denote a mean-preserving spread of the distribution F, of the interest rate u2. If the value of the jrm VF,( x) is positive for all economically relevant x, then V,:(x) > V,,(x) for these x, regardless of whether the $rm makes decisions before or after the realization of demand uncertainty, and regardless of the speci’cution of the model with regard to stockouts or the time required for production.

Proof. If the firm makes decisions before the realization of demand uncer- tainty, its objective function is given on the right side of (3). Holding fixed x, p, y, al, a3,. . . , a,,, we have

a2 1 2

aa: 7r+ i -vF, = 1 +a, i 0 + a2J3

VF, > 0 if VF, > 0.

Applying Proposition 1 gives the result. If decisions are made after the realization of demand uncertainty we compute the same second partial deriva- tive and appeal to Proposition 3.

The intuition behind the result and the reason for the requirement of non-negative value is seen in fig. 1, which graphs the present value of a payment of m > 0 dollars one year from today as a function of the interest rate u2. The convexity of the expression m/(1 + a2) indicates that a one percent increase in the interest rate would reduce the present value of the future income by less than a one percent reduction in the interest rate would increase it. Therefore ,increased variance or uncertainty in interest rates increases the average present value of future income, and conversely makes the

G. French, Parumeter uncerraing and the value of the firm 461

In

1 +a2

Fig. 1

present value of future debt more negative. Note that VF2, which is required to be positive by Proposition 4, is a function of the inventory level x. What is required is that for all economically relevant x, that is, for all x which might occur in the stochastic steady state arising from the firm’s optimal policies, the present value of all future income V,(x) should be positive. This is necessary because the inequalities in Propositions 1 and 3 must be applied recursively. The requirement of non-negative value means that the interest rate result should be regarded as applying to firms in industries in which there are barriers to entry.

The result concerning increased uncertainty of input prices is simply an application of Propositions 1 and 3 to the well known fact that production cost functions are concave in input prices.

Proposition 5. If 4*, i = 3, . . . , n, is a mean-preserving spread of the distribu- tion F;. of an input price aI, then V,.(x) > V,(x). This result is valid regardless of whether the firm makes decisions before or after the realization of demand uncertainty, regardless of the specijication of the treatment of stockouts or the length of production, and regardless of whether the firm is a competitor or has market power.

J.E.D.C:- E

468 G. French. Parameter uncertainty and the value of thejm

Proof. If decisions are made before the realization of uncertainty, hold fixed x, p, Y, ~1, a,, a3, ~~-1, ai+l,. . . , a,,- Then

;( r+bV,,)=- .-&& Y,Us Y..., a,,)>*O, !x( I I

where ?z + bvF is obtained from the right side of (3). Applying Proposition 1 gives the result. The starred inequality is not dependent on any of our assumptions A.l-A.ll, but is a general property of cost functions. See Varian (1984, pp. 44-45) for a proof. The same property of cost functions in conjunction with Proposition 3 gives the result for the case when decisions are made after the realization of demand uncertainty.

5. Increases in demand uncertainty

5.1. Decisions made before the realization of uncertainty

5.1.1. The quantity-setting-firm

If production is instantaneous and excess demand is lost, the transition equation (1) becomes

x,+~ = m=t@ x, +Y, - 4,), (6)

and ‘sales revenue’ in (2) is

Pb+,+h 4A 4n4x,+y,, 4,). (7) If excess demand is lost but production takes one period, (1) is

and revenue is

PbWx,, 4,h 4~ntx,, 4,).

If demand can be backordered, (1) is

(9)

X t+1 =x,+y,-4,, 00) and revenue is

P(4v a,)q,. (11)

G. French, Parameter uncertainty and the value oj thejirm 469

If we assume that

Ph 4 = 44 + a* 04

that the firm chooses q from the interval where d(q) r 0, and that the distribution of ‘u’ we consider is always such that d(q) + a > 0 with certainty (we discuss the effect of relaxing this assumption after proving the proposi- tion), then we have:

Proposition 6. If the jirm sets q and y before the realization of demand uncertainty, if demand shocks are as in (12), and if the constraint p( q, a) 2 0 is

neuer binding, then VF,.(x) = V,,(x) f or all x (where F1* and F1 are as in assumption A.11).

Proof.’ The optimality equation is (3) with ‘q’ substituted for ‘p’. Hold fixed x, 4, Y, a2,. . . , a, and calculate (a ‘/a~$)[8 + bVF,] under specifications (6) and (7),(8) and (9), or (10) and (11). In all three cases this second partial is zero since a ‘p/da: = 0. Proposition 2 then gives the result.

The increase in uncertainty leaves E( 7~ + bV) unchanged for given values of the state variable and choice variable because a, only enters s + bV through the sales revenue function; the linearity in a, of this function together with the assumption that p (q, a) 2 0 is non-binding means that expected sales revenue is unchanged. If the firm ever selected q such that there was a positive probability that p(q, a) = 0, then sales revenues as a function of a would be as in fig. 2. In this case arguments identical to those in section 3 would imply that VF,, > VF1. Intuitively, when the constraint p(q, a) 2 0 is binding, ‘down- side’ demand risk doesn’t reduce revenue any further, while favorable demand risk increases it. Hence increased dispersion of ‘a ’ would increase expected revenue, and we would proceed as in the proof of Proposition 2.

A note on the perfectly competitive firm. If we consider a perfectly competi- tive firm which sets a production level and a sales level in advance of the realization of a random market price, and which then sells what it has committed to sell at that price, we get the same result. Simply define p(q, al) = a, as the market price. The second partial mentioned above, ( a2/Ja:) [n + bV], will again be zero, and the firm will be indifferent between mean- preserving spreads of the market price. The well-known Oi (1961) result regarding the competitive firm’s preference for greater price dispersion (i.e., regarding the convexity of the profit function in prices) does not apply in this situation since the firm does not know the price when it makes the decisions. This result will be relevant when we consider firms which make decisions after the realization of demand uncertainty.

410 G. French, Paramerer uncertainty and the wlue oj rhe jrnt

sales revenue

a

Fig. 2

5.1.2. The price-setting firm

Suppose the firm sets a price and a production level before the realization of demand uncertainty. If production is instantaneous and excess demand is lost, (1) is

x,+,=m~[0,x,+~,-h(p,,a,)l, 03)

and sales revenue in (2) is

p,min[x,+y,, h(prv~,)l.

If production takes one period to become available for sale and excess demand is lost, (1) is

X 1+1 =y,+max[O,x,-h(p,,a,)l,

and sales revenue is

G. French, Parameter uncertainq and the value o/thefmr 471

If demand can be backordered, (1) is

x I+1 =xr+Y,-h(P,d 07)

and sales revenue is

PN PI? 4. (18)

We will suppose that

h(p,a)=u,-u,p+a, (19)

that p is chosen from the interval where ui - u2p 2 0, and that h( p, u) > 0 with certainty, under any distribution of u that we consider. This implies

i.e., sales revenue is a strictly concave function of price, for all realizations of the demand index. These assumptions are enough to guarantee that the value function is strictly concave for the model specification (17) and (18) in which demand can be backordered. We will not go through a demonstration of the concavity of the value function in this case. The demonstration depends crucially not only upon the concavity and convexity assumptions about the underlying functions, but also upon the specification (19) which ensures that a2h/ap2= 0 and that a2h/apau=0. Also crucial is the assumption that there are no stockouts. If, on the contrary, excess demand were lost, then changes in the firm’s price affect the probability of a stockout. For this reason, when there is no backordering special assumptions regarding the distribution of the random component of demand must be made to guarantee the unique- ness of optimal policies, the ‘expected’ signs of comparative statics derivatives (for example that ilp/ax < 0), and most importantly for our purposes, the concavity of the value function. This has been known since the work of Zabel (1970.1972).

Suppose under the specification (13) and (14) or (15) and (16) that the distribution of the random term is such that the value function is in fact concave. Note that the sales revenue functions (14) and (16) are both concave functions of u for fixed p under the assumption of additive shocks, lost excess demand and positivity of h. As fig. 3 illustrates, ph(p, a) =p(ul - u2p + a) increases at the rate p as a increases until h( p, a) equals the available stock x [in (16)] or x +y in (14). At this point further increases in u do not add to sales revenue, given the fixed p.

412 G. French, Parameter uncertainty and the value of rhefirm

sales revenue

Fig. 3

We now have:

Proposition 7. If the firm chooses p and y before the realization of demand uncertainty, and if demand is as in (19), then in the model specification (17) and (18) in which demand is backordered, or in the specification (13) and (14) or (15) and (16) if the valuefunction is strictly concave, then Vr,*(x) I V,(x). The inequality is strict for any x such that the firm chooses p and y such that the probability of a stockout is less than one.

Proof. The optimality equation is (3). T + bVr, is concave in a, if we hold fixed x, p, y, a,, . . ., a,,. This is so because a1 enters only into the sales revenue function and the bV,, term. Sales revenue is concave or linear in a, as fig. 3 shows. As for the bV, term,

-$[bVr,(T(x,p, y,a))] =Oifa,: stockoutoccurs, 1

= b( i3h/aa,)2V”( T( e)) < 0 (?l)

if a,: no stockout.

G. French, Porometer uncertoing ond the value of thejrm 413

We follow Proposition 2, recursively applying the inequality

to the right side of (3). According to (21), this inequality will be strict when the initial inventory is such that p and y are chosen so that stocking out is not a certainty. The proof is complete.

As mentioned, the reason the price setting firm dislikes even a neutral increase in demand uncertainty is that next period’s inventory becomes more variable for any given values of the choice variables. When the value function is concave this reduces expected future profits because the expectation of a concave function is reduced by increased dispersion of its argument.

5.2. Decisions made after the realization of uncertainty

In this section we show that a firm which makes decisions after the realization of demand uncertainty is made better off by increased demand uncertainty. Consider the problem

maxP(q,a)q-c(y,a,,...,a,)-g(x)+bV(x+y-q), .v. 9

(24

subject to

y>O and 420, or Osqlx, or O<q<x+y,

which is an expanded form of the right-hand side of (4). p(q, a) is assumed to be as in (12). Also assume that

a2[P(q*a)ql <O, Va

aq2

The form of the sales constraint depends on .whether excess demand is lost, or whether production is instantaneous (0 I q I x + y) or lagged (0 I q I x). Since the current reward p( -)q - c( y,) - g(x) is jointly concave in x, q and y and the transition equation is linear, the value function V is concave, and is in fact strictly concave since g is assumed strictly convex.

If we write down the first-order conditions for the above problem and differentiate them with respect to a, we obtain

a4 C” - bv” -= aa (bJV)~- for an interior solution with both q and y positive.

474 G. French, Parameter uncertainty and the value of thejirm

If y = 0 and q is interior we still get aq/aa > 0, and if q is on the boundary of the feasible region, 6’q/& = 0. Hence aq/aa 2 0, under any assumption regarding backordering or timing of production we have considered. A mod- ified version of the argument used in section 3 gives

Proposition 8. If the firm sets q and y before the realization of demand uncertainty, solving the problem (22) with p(q, a) as in (IZ), then VJ(x,) r b,(x).

Proof. Fix x, a*, . . . , a, and let q and y be functions of a,, namely the optimal values of q and y given x, ai,. . . , a,. Differentiate

a(x,q(al),Y(al),al,..., a,) + bV&-‘b, da,), rh), ~1)

with respect to a,:

+ + 4,) = mh 4

aa1 aa1 4h).

This is an envelope result. Differentiating again,

a++ bVF,)

l

aq alp alp 1

ap a4 aa: =q caqaa, 1 -+g +j--po.

1 1

The inequality holds since the first two terms are zero by (12), while the third term is non-negative by A.3 and the fact that aq/aa, 2 0. Now proceed as in section 3. In the optimality equation, eq. (4)

+ bV&-(-))I

where

z= (x, a, )..., a,).

(23)

Apply the inequality recursively to the VF, on the right-hand side. This gives VF,<X> s V,,(x).

Note that the inequality in the proposition is strict for any x such that there> is a positive probability that q will be chosen in the interior of the feasible set. Note also that the proposition is valid for a perfectly competitive firm as well.

G. French, Parameter uncerrainty and the value of thefirm 475

Simply define p(q, a> = ul, the market price of the firm’s product, and note that dq/aa 2 0 in the competitive case as well. Proposition 8 can then be applied, and indicates that Oi’s result easily generalizes to a multi-period setting.

6. Conclusion

In an infinite-horizon inventory model of firm behavior, we have analyzed the effect of increased uncertainty about various parameters on the value of firms. We have concluded first that interest rate uncertainty increases the value of the firm if the value of the firm is positive. This is in essence a consequence of the fact that the expression for the discount rate l/(1 + r), is strictly convex in r. Second, due to the concavity of the cost function in input prices, increased input price uncertainty increases the value’ of firms. Third, increased uncertainty regarding an additive demand shock increases the value of a firm which makes decisions after the realization of uncertainty. If decisions are made before the realization of demand uncertainty, an increase in demand uncertainty increases or does not affect the value of a quantity-setting firm, but reduces the value of a price-setting firm which has a concave value function.

Some limitations of the analysis must be mentioned. First, one can quarrel with the assumption of risk neutrality made throughout the paper. Second, the partial equilibrium nature of the analysis must be borne in mind. In particular, even though the analysis formally applies to a perfect competitor as well as to a firm with market power, any change in expected value of competitive firms should be offset by entry and exit to and from the industry. Third, we do not consider uncertainty about whether inputs will be available on time and in adequate amount, or how productive they will be.

Finally, this paper does not pursue the possibility of analyzing redistributive effects (between borrowers and lenders) of changes in the variability of interest rates suggested by fig. 1. It would be interesting to explore this issue.

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