A. M. K. CHEUNG ET AL.

34

where

11

2

2

1

11 21

and 1

P

r

w

Discussions on this equation are in order. There are

three sets of variables forming the inputs to the formula.

The first set consists of the production technology

parameter

and the per unit labor cost w. The second

set consists of the adjustment cost technology parameters

and

measuring the significance and speed of

adjustment. These two sets of parameters are assumed to

be constant. The last set consists of state variables K and

P. The former is deterministic and the latter stochastic

with coefficients and

. Finally, the market required

return on the spanned source of uncertainty dz is given by

the riskless interest rate under the risk neutrality argument.

The exogenous commodity price P, which is the

fundamental source of value to the firm’s profit stream,

affects the firm’s present value through a composite

variable

defined above. Since the composite variable

appears in the two separate terms of the value function in

(13), it is useful to isolate the discussion of the influence

of

channeled through these two terms. The first term

is the product of

and K. Given that K is the existing

capital stock owned by the firm,

is naturally

interpreted as the total value contribution to the firm by

the exiting capital. Financial economists define

as the

marginal revenue product of the firm’s capital.

It is worth pointing out that

has a noticeable

format reminiscent of the present value of a perpetual

income stream under certainty. This perpetuity interpreta-

tion is consistent with the presumption that the business

is infinitely lived with its future risky stream of profit

discounted by a complete arbitrage free financial market.

It is now useful to compare

in this paper with the

earlier result derived by Rubinstein [2]. Rubinstein’s

model sets the standard methodology for firm’s valuation

problem in finance. Given an exogenous stochastic cash-

flow process for a business firm, an appeal to an efficient

financial market governed by a martingale pricing operator

is necessary and sufficient to produce a fair market value of

the firm’s cashflow. Rubinstein assumes a discrete

stationary random walk process, which is a discrete

counterpart of the geometric Brownian motion process for

our commodity price process with a zero drift.

Our perpetuity reasoning for

in this paper is

different but consistent with Rubinstein’s result.

The difference arises from the fact that the firm’s

production activity is endogenized and the technology

parameter

plays a role in producing the transformed

ex- pected growth of the commodity price via

22

1112 1.

The difference

between the required market return r and the expected

growth opportunity stands for the market net required

return used to discount the marginal revenue contribution

by the installed capital.

The consistency of the first term with Rubinstein’s

result also allows us to emphasize the contribution of the

second value component. The second term highlights the

presence of the adjustment cost parameters and

that, when combined with the production parameter,

further transform the expected growth of the commodity

price process. As it takes time and resource for the firm

to turn the raw capital into its ultimate production form,

the firm has earned an access to the future benefit accrued

by these new capital via the firm specific cost technology.

Such adjustment cost associated benefit is spread over

the indefinite future and the financial market discounts

those benefits stream through an appropriately adjusted

cost of capital. The result is the rational appearance of

the second value component.

Two special cases arise from limiting arguments that

would vanish the second term and reduce the present

value formula to the standard result where value arises

mainly from the firm’s production technology. The first

case corresponds to no adjustment cost incurred when

new capital is acquired (0

). The second case arises

when the adjustment cost function is linear in investment

(1

). Substituting either one of the these cases is

sufficient to reduce the second term of the value function

to zero. As discussed earlier, both cases correspond to a

situation where the firm’s size is indeterminate and the

intertemporal optimization problem has no interior

solution. The standard perpetuity formula in the finance

literature appears to thrive on the validity of these two

cases.

An additional disquieting feature of the valuation

formula begins to surface when one continues examining

the stochastic evolution of the valuation function G(K,P).

Whereas the commodity price process follows a simple

geometric Brownian motion with a constant volatility,

the resulting process for G is not a geometric Brownian

motion with a constant volatility. A causal observation1

of the functional form for G suggests this consequent

feature. Some lengthy algebraic developments are presented

in the Appendix B to verify this claim.

In that appendix it is also shown that either 0

or

1

would allow one to restore the geometric Brow-

nian motion representation for G. On the contrary, when

the firm possesses a significant convex adjustment cost

1The fact that β and γ appear more than once in the second term of the

ricing formula does not simplify that term immediately to zero when

evaluated at β = 1 or γ = 0. Some delicate limiting arguments are estab-

lished in Appendix B to justify that this second term reduces to zero.

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