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AN EQUILIBRIUM MODEL WITH FIRM-SPECIFIC DEMAND

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AN EQUILIBRIUM MODEL WITH FIRM-SPECIFIC DEMAND

    DOES WAITING MATTER?

    AN EQUILIBRIUM MODEL WITH FIRM-SPECIFIC DEMAND

    Sigbjørn Sødal

    Agder College, Norway

    This version: May 1, 1999

    Abstract We model a one-sector economy where a large number of firms undertake two-step

    irreversible investment ? patenting and production ? conditional on firm-specific demand

    processes and constant investment costs. The results include a criterion for whether it will be

    optimal to split the two investments in time. Typically, postponing the second investment is

    optimal if the demand is highly volatile or highly expected to increase, if the discount rate is

    small, and if the patent cost is small relative to the production cost.

Keywords: Irreversible investment, patents, uncertainty, markup pricing.

    JEL classification: C61, D92, D81.

    Acknowledgments: Thanks to Geir Asheim, Victor D. Norman and Stein Østbye for helpful

    comments.

Address:

    Sigbjørn Sødal

    Agder College

    Faculty of Economics and Social Sciences

    Tordenskjoldsgate 65, Postuttak

    4604 Kristiansand

    Norway

    Phone: +47 38 14 15 22

    Fax: +47 38 14 10 28

    E-mail: Sigbjorn.Sodal@hia.no

1. Introduction

    Products like cars, pharmaceutical drugs, aircraft, and computers require two quite different types of investment: R&D, marketing and other ancillary investments on the one hand, and wide-scale production on the other. In one sense or another, the first type of investment will represent a patent, while the second one implies to activate the patent. Technically, the patent may be thought of as an option, and the decision to produce as exercising the option. Since cost or demand variables may change, it is not always optimal to undertake wide-scale irreversible investment in production right after the patent has been acquired. For example, if learning could arise by stretching R&D over time or by external knowledge spillovers, costs may be saved by postponing production. Alternatively, and as we will focus on, the demand for a specific product may fluctuate conditional on its exact character: some products may gain popularity when the consumers get to know them well, whereas other products lose. The development of demand may also be affected by information that is spread prior to production ? either by spillovers related to the R&D stage or by deliberate marketing efforts. The demand may also change if something occurs to the good itself, as will indeed be true in the industry that is used to exemplify our model.

    Dynamic effects of any such kind imply, more generally, the possibility of an increasing wedge between the (net present) value of sales and the cost of production, thereby also gains from holding on to the patent instead of exploiting it massively right away. There will, however, also be a cost of such waiting, since the obtained revenue must be discounted more heavily the longer production is postponed. Our objective is simply to discuss whether the gains from waiting may exceed the costs from waiting in a stylistic one-sector model where patenting and production activities are summed up in two completely irreversible investments. Labor in fixed supply will be the only production factor, and both investment costs will be fixed, implying that gains from waiting will stem from dynamics on the revenue side.

    The model seems the most appropriate for hi-tech industries, but it will be illustrated with a more classical decision problem as discussed, for example, by Varian (1996, pp. 206-211): When is it optimal to cut a tree whose net value grows according to the deterministic function F, if future benefits are discounted at the constant rate ?? The net present value of a decision t

    -?Tto cut at time T is equal to eF, which by optimization yields the first-order condition T

    (/)/dFdTF??. This means that it is optimal to cut when the rate of growth equals the TT

    1

discount rate; i.e., when the marginal value of waiting further equals the marginal cost of

    waiting further. Below, Varian’s model is expanded in the following ways:

    1. There will be a large number of firms, each one manufacturing a specific tree containing

    homogeneous lumber.

    2. A fixed cost, A, of planting a unit-sized tree is included. Technically, this will be like

    acquiring a patent, so A can be thought of as a patent cost.

    3. The net value, F, will consist of two components, F=P?B, where P will be the value from tttt

    cutting the tree, and B is the related cost. Technically, B will be the cost of activating the

    patent, so more broadly it can be interpreted as a production cost. We will require B to be

    fixed, whereas P will fluctuate due to growth or quality changes. Although fluctuations in t

    P in this case have a physical explanation, we will often refer to the underlying process as a t

    demand process, since that seems more appropriate in many other applications.

    4. Firm-specific uncertainty will be allowed in the underlying process as some trees may

    happen to grow quickly, whereas other trees might even rot.

    5. There will be free entry, and the number of firms will be large enough to ensure zero

    expected profit from planting a tree. The large-group assumption combined with firm-

    specific uncertainty will enable the households to diversify investments and eliminate risk.

    Throughout the presentation we make reference to the new markup approach to irreversible

    investment proposed by Dixit et al. (1999). The model can be regarded as one type of

    equilibrium extension of the firm-level model of that paper.

    2. The model

    Consumers

    The instantaneous utility function of the representative, infinite-lived consumer is:

    uc?(1) ; tt

    where c is consumption at time t. The intertemporal maximization problem can be written as: t

    2

    ??????tt(2) such that ; maxUcedt?pcedtw?ttt0??ct00where ? is the time preference rate, ? is the interest rate (which will also be the discount rate),

    p is the unit price of lumber, and w is initial wealth. The first-order condition for optimum t0

    ????ttepe??yields (for all t), where ? is the marginal utility of wealth. In a steady-state t

    equilibrium with continuous saving and consumption, we must have , and a fixed unit ???

    price . Initial wealth equals the net present value of income, so we also have: p(/)?1?

    ??????tt(3) ; wwedtedt???0tt??00

    where w is the rate of income from labor, and ? is the net rate of income from the asset value ttof firms. In equilibrium, the net present value of income from firms is zero, so the rightmost

    term of eqn. (3) vanishes, and the consumer is left with income from labor. The labor market

    is perfectly competitive, and we choose labor as numeraire by setting w?1 for all t. t

    Firms

    The life cycle of the firm is based on two irreversible investments that are stretched over fixed

    time intervals. This is illustrated in Figure 1.

     (Consumption)TC

     T (Patenting) (Production)TAB

     Time

     EntryActivation

    T (Delay)

Figure 1. The life cycle of the firm.

    First, a fixed amount of labor, L, is needed to plant a unit-sized tree. The investment takes A

    place over the time interval T, during which L/T workers are continuously employed. We AAArefer to the start of the investment as entry, and to the complete effort as patenting. The

    investment implies the following net present cost:

    3

    TAtT????A(4) ALeTdtLeT???/()/1?. AAAA?0

    From then on the (quality-corrected) size of the tree will follow an independent, autonomous

    Markov process {X} ? more generally referred to as a demand process ? where X= 1. For t0 simplicity, the process is assumed to take off when patenting is initiated instead of when it is

    completed. (As long as T is fixed, the difference is a matter of scaling.) A

    Similarly, a fixed amount of labor, L, is needed to cut a tree regardless of size. This B

    investment is stretched over the time interval T, during which L/T workers are continuously BBBemployed. We refer to the starting point as activation, and to the total effort as production.

    The net present cost, evaluated at the time of activation, becomes:

    TBtT????B(5) BLeTdtLeT???/()/1?. BBBB?0

    Production brings about a stream of the consumer good as the tree is turned into something

    like firewood or paper. More generally, we may think of a firm with an activated patent as an

    operating factory with a certain capacity.

    Consumption takes place at a constant rate, and over the time interval T (? T). (The most CBreasonable assumption is probably to set T = T.) A tree of age T is X units tall, implying the CBTfollowing net present revenue from sales:

    TCtT????CPXpeTdtXpeT???/()/1?(6) . TTCTC?0

    ??TCPpeT??()/1?The initial value; i.e., the revenue from cutting immediately, becomes . 0C

    (Eqn. (6) assumes that the tree stops growing when cutting is initiated, but that could easily be

    modified. The revenue would still be proportional to X, but with a different factor. With Tuncertainty, it would also have to be calculated as an expectation.)

    As in the somewhat similar model by Dixit and Pindyck (1994, pp. 267-277), the law of large

    number applies, so the equilibrium number of firms with various levels of X will be constant, talthough the identity of the firms occupying each level will change. The goods are perfect

    substitutes, however, so only the sum will matter. The rates of entry and of activation will also

    4

be constant, labor demand will be constant, and labor will be fully employed. The net present

    profit accruing to a firm that makes entry at time zero and activates at time T, becomes:

    ??TePBA()??(7) . T

    The rightmost term of (7) follows as the cost of planting, A, is incurred at time zero. By

    -?Tcutting at time T there will be the net revenue P?B, which is discounted by the factor e T

    since it arises in the future. The patent cost is sunk at the time of activation, so (7) implies,

    more generally, the following timing problem: When is it optimal to incur the constant cost B

    * to obtain the fluctuating revenue P? The answer is: when a fixed P> B is reached for the first t

    time. (The only effect of waiting for a second time would be to obtain the same net revenue

    later. See McDonald and Siegel, 1986, and also the seminal work by Merton, 1973.)

    *Following Dixit et al. (1999), the markup from B to P can be described in elasticity terms:

    The expected and discounted value of investing as soon as an arbitrary is reached, is PP?0

    -?TE[e](P?B), where E is the expectations operator, and T is now the first hitting time from P 0to P; i.e., the net revenue at the time of activation is reduced by the expected discount factor.

    The larger is P, the longer it takes to reach it, so the discount factor can be expressed

    equivalently as a strictly decreasing function in P. By such a transformation, which effectively

    embodies dynamics and uncertainty into the discount factor, the expected net present value to

    be maximized (with respect to P) can be written as:

    (8) ; QPPBA()()??

    -?Twhere Q(P) ? E[e] is the expected discount factor when going from the constant P to 0PP?QP()?1 for the first time. The initial value is , since there will be no discounting if 00

    activation takes place immediately.

    As the profit function, (8), shows, the investment decision of the firm will be analogous to the

    pricing decision of a static monopolist, with P acting like a price variable, Q(P) like a downward-sloping demand curve, B like a constant marginal cost, and A like a fixed cost.

    *Therefore, as in the static model, the optimal P is determined by a markup involving the

    *elasticity of Q with respect to P. The discount factor function, Q(P), and the markup, P?B, are illustrated in Figure 2. The innermost curve is analogous to a marginal revenue function,

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    *and the optimal P is found by setting marginal revenue equal to marginal cost. See Dixit et al.

    (1999) for details.

    P

    Q(P)

    *P

    B

    P0

    Q + Q/(dQ/dP)

    0,0Q*1Q

    Figure 2. Optimal investment.

    The following result is obtained by maximizing (8) with respect to P:

     has reached a fixed markup over B Proposition 1. The optimal time of activation is when Ptaccording to the formula:

    **()//PBP??1?(9) ;

    where ? = ?(dQ/dP)/(Q/P) is the magnitude of the elasticity of the discount factor with respect

    *to P, and this is evaluated at the optimal P.

The elasticity is a measure of dynamics, or, more precisely, of fluctuations relative to

    discounting. If ? is large, there will be much more discounting by waiting for a higher P ?

    either because the discount rate is high or because P moves slowly. (It can be shown that ? is tindependent of P, which must be the case for the investment rule to make sense.) 0

    Proposition 1, which is the main result of Dixit et al. (1999), doesn’t address the question of

    whether waiting will be optimal, as P is not determined. In our context, the answer to this 0

    question will follow from the free entry condition that we will turn to next.

    6

    Equilibrium

    The rate of entry will increase until all expected profits are eliminated, implying that (8) must

    be zero in equilibrium. Using eqn. (9), this implies:

    *QK??()/?1(10) ;

    *where Q is the optimal discount factor, and K = B/A is the production cost relative to the

    * patent cost. For waiting to be optimal we must have Q< 1, and thus the following result: Proposition 2. Entry and activation will be split in time if:

    . (11) ???1K

    Proposition 2 can be explained as follows: If Q(P) is too elastic, the dynamic effects will not be strong enough. Then the model collapses, so to speak, to a trivial static one. The optimal unit

    cost becomes A+B, which in equilibrium will be equal to the value of sales. In such cases the

    difference from a static model with homogeneous goods, perfect competition and constant

    returns to scale is mainly that our model applies perpetually, but time is really irrelevant.

    The elasticity requirement for waiting to be optimal is stronger the smaller is K, which means

    that we should expect waiting to be more likely the larger is the production cost relative to the

    patent cost. This will indeed be true in the examples below, but note that there may exist some

    cases where a higher K makes waiting less likely. The reason for this is that ?, in general,

    depends on Q (or P), which again will depend on K in steady-state. Note also that the exact equilibrium rates of entry and of activation cannot be determined

    without turning to specific demand processes. This is because some patents may never be used

    if the process is highly stochastic. In case all patents are known to be used, the accumulated

    labor demand of each firm will be L+L, and the common rate of entry and of activation AB

    simply becomes L/(L+L), where L is the (fixed) rate of labor supply. AB

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3. Examples

    The geometric Brownian motion

    Suppose that the (independent) demand processes are geometric Brownian:

    (12) (? < ?). dXXdtXdz????ttt

    It follows from eqn. (6) that this leads to similar independent P-processes. As shown by Dixit tet al. (1999), the expected discount factor with this process is:

    ?QPPP()(/)?(13) ; 0

    where ? is the positive root (exceeding unity) of the following quadratic equation in x:

    12???xxx()????10(14) . 2

    Thus, the discount factor will be analogous to a static demand function with constant elasticity,

    ? = ?. The shape of Q(P) in Figure 2 fits in well with this example, and (11) takes the

    following simple form:

    (15) ???1K.

    From (15) we can conclude that waiting will be more likely the smaller is the patent cost

    relative to the production cost; i.e., the larger is K. To simplify the sensitivity analysis with

    respect to ?, let us assume T = T, since then we have K = L/L, which is independent of ?. ABBAIt can be shown that ? is increasing in ?, but decreasing in ? and ?. Hence, the discount factor is more elastic, and waiting is less likely the larger the discount rate, but the discount factor is

    less elastic, and waiting is more likely the higher the drift or the volatility of demand.

    Figure 3 plots two sets of values for ? as a function of ?, both sets assuming ? = 0.04. The limit results are of particular interest. First, ? ? 1 if ? ? ?, so waiting will be optimal for all values of K if the volatility is high enough. Second, ? ? ? when ? ? 0 if ? ? 0, as that removes all possibilities for gains from waiting. However, ? = ?/? (> 1) if ? = 0 and ? > 0, so uncertainty is no requirement for waiting to be optimal as long as K is large enough.

    8

    ?

    5?0=

    4

    3

    ?0,02=2

    1

    0

    00,050,10,150,2

    ?

Figure 3. The elasticity of the discount factor for geometric Brownian motions (? = 0.04).

    21???All patents will be used if , as shown by the following argument: If P is geometric t2

    Brownian with drift ? and volatility ?, then, by Ito’s lemma, lnP is arithmetic Brownian with t

    21???drift and volatility ?. According to Dixit (1993, p. 56), an arithmetic Brownian 2

    motion will hit any higher value than the current one (with probability one) if the drift is

    positive. Since the ln-function is strictly increasing with no upper limit, P will also reach any t

    *21???higher value if ; in particular, P will be reached when starting from P. (In the 02

    opposite case, the difference between the rate of entry and the rate of activation will obviously

    *be larger the smaller is the probability of ever hitting P.) The model contains no externalities, so waiting will also be socially optimal if (15) holds. Let

    us conclude by illustrating this point when ? = 0 and ? > 0. In that case all trees will be cut when having reached a certain age. We consider a possible optimum where waiting is not

    optimal, and where, for simplicity, the rate of entry is scaled to one. Then ask whether costs

    could be saved at a particular instant by pushing entry slightly into the past, but without

    changing the rate of consumption. For the latter to hold, there must be a marginal reduction of

    ? trees, corresponding to marginal growth before cutting. This yields the marginal gain

    ?(A+B). Due to discounting, there will also be a marginal cost of the change, ?A, since

    planting is now taking place earlier. Waiting is optimal if the gain exceeds the cost; i.e., if

    ?(A+B) > ?A. This inequality is just a re-arrangement of (15) for the chosen parameters.

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