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The effects of future output price uncertainty and wage uncertainty on a firm’s investment decision are examined in this paper, by assuming the competitively risk-neutral firm maximizes the expected value of the sum of discounted cash flows. We find that the optimal investment behavior is such that the expected proportional growth rate of investment is invariant over time, although there exists a tradeoff between the effects of the two uncertainty on firm’s investment because the shift in output price has positive effects on firm’s investment, whereas the shift in wage has negative impacts on firm’s investment. And what’s more important, fluctuations in output price and wage are correlated so that changes in output price tend to be accompanied by changes in wage.

During the past decades, many scholars studied the effects of uncertainty on a firm’s input and output decisions. Regarding to this topic, one can see references we enumerate at the end of the paper and etc. Within those papers, considerable results and conclusions have been achieved. For example, Hartman [

Besides, Lucas and Prescott [

Ultimately, it’s worth mentioning that the shortcoming of Pindyck [

The major contributions, in brief, of this paper relative to existing literatures are, on the one hand, an explicit solution is derived and firm’s investment behavior is analyzed qualitatively and quantitatively rather than just qualitatively, on the other hand, both price and wage uncertainty are taken into account reasonably and simultaneously rather than consider the price uncertainty only. What’s more important, so far to our knowledge, the question we discussed in the paper is still open. As we expected, our conclusion is consistent with that in Abel [

In this section, we will introduce the model again in detail for the sake of intactness, although it’s a special case of the model in Pindyck [

The firm is risk-neutral and competitive with strictly quasi-concave production function as well as with convex costs of adjustment function. But for simplicity and operability, production function is adopted by CobbDouglas type, namely

where, and are respectively capital stock and labor, and denotes technology. Without loss of generality, we normalize to unit one hereafter, i.e.,. It’s obviously that satisfies the conditions of quasi-concave, i.e., And the convex costs of adjustment function is adopted by the form as is the purchase price of one unit capital(equipment), is the constant elasticity, and is the gross investment. As many literatures suggested previously, confirms with and, for any. Note that the convexity of the adjustment costs function implies that it is more costly to increase capital stock quickly than slowly.

Capital accumulation equation at time t is defined by

here, is the depreciation rate.

The stochastic behavior of output price p and of wage rate w at time t follows geometric Brownian motion, that is

where, and are two Wiener processes with zero mean and unit variance, are all non-zero constants. As a matter of fact, and can be regarded as inflation rates. Besides, we set

here, is the instantaneous correlation coefficient for the two Wiener processes and. is the expectation operator.

For a risk-neutral and competitive firm, its objective is to maximize the expected present value of cash flow by selecting labor and investment. Mathematically, the optimal problem can be described as

subject to constraints in (1)-(4). Of course, discount rate is set to be constant here.

Remark 1. In Pindyck [

subject to (1) and, where, is a general production function with strictly quasi-concavity and is a general adjustment-cost function with convexity, and are two functions respectively with respect to. In the general case, Pindyck did not obtain an explicit solution for the optimal investment, but analyzed the firm’s investment behavior qualitatively by using stochastic phase diagram approach instead. However, our model will collapse to the the special case expressed in Abel [

Remark 2. We argue that it is not nonsense to assume that the wage rate follows a stochastic process in reality. Also, it’s feasible in practice. For example, Hartman [

It’s not difficult to prove that, reduced by hereafter, obeys formula

Straightforwardly, the left-hand side of (5) can be regarded as the total mean return desired by the firm over the time interval dt, if the firm requires a mean return rate. On the contrary, the right-hand side of (5) is the whole return expected by the firm, which is the sum of cash flow and expected capital gain or loss. In other words, optimality means the desired mean return is equal to the expected return.

Now, we will derive the expression of the optimal investment rate step by step. To begin, we present the following Proposition 1.

Proposition 1. The firm’s optimal rate of investment can be expressed as

where

It is clear to see from (6) that the optimal rate of investment does not depend on, but increases with, decreases with, since.

Proof. First of all, we claim that Itô’s Lemma is applicable under the hypothesis. Thus, applying Itô’s Lemma to calculate the capital gain or loss, we have

Inserting (1)-(4) into (8), meanwhile recalling that and for any standard Wiener process, we get

(9)

Substituting (9) into (5), we obtain

By simple computation, it yields

Notice that, is the marginal revenue product of capital.

Next, differentiating the right-hand side of (10) with respect to, there holds

An economic interpretation can be seen straightly from (12), i.e., the optimal investment rate is such that the marginal cost of investment equals the marginal valuation of capital. Plugging (11) and (12) into (10), we have

where

and as in (7). (14)

It’s well known that we can rewrite (12) and (13) as a nonlinear second-order partial differential equation, despite such equations are unsolvable in general case. But fortunately, an explicit solution can be derived herein due to our assumptions. Note that we can strictly prove that there exists a to satisfy the nonlinear second-order partial differential equation given by (12) and (13) together. The argument process, however, is not the emphasis in the paper. Hence, we omit the details here. Now that the solution exists and taking the form of (12) and (13) into account, we guess the form of as

where, and are all undetermined coefficients.

In terms of (15), it yields

Lastly, applying method of undetermined coefficients, and combining (13) as well as (16)-(18), it’s not difficult to get

where

as in (7),and as in (14).

Following (12) and (19), we see immediately that

, as in (7) (20)

Obviously, some results can be obtained directly from (19) and (20). The firm’s value , for instance, is a linear function of the capital stock. The optimal rate of investment It does not depend on, but increases with, decreases with, since.

We will analyze the qualitative effects of price and wage uncertainty on investment in this section, for given the current level of output price, and of wage rate.

Proposition 2. If

, then the increases in uncertainty of output price and of wage rate together will have positive effects on the optimal investment rate. Conversely, if

, then the increases in the two uncertainty will have negative effects on the optimal rate of investment. Particularly, there will be no effects of shifts in uncertainty on the investment so long as

. Here,

are respectively the fluctuations in uncertainty, which associate with, correspondingly.

Proof. As a matter of fact, in order to analyze the effects of uncertainty on investment behavior, we just need to determine the effects of uncertainty on

, because relies on only. To this end, taking the expression of into account, we only require focusing on the change of

. By simple analysis, we see the changes in uncertainty of output price and of wage rate together will result in a rising in the optimal investment rate if

i.e.,

. Similarlythe changes in the two uncertainty will lead to a falling in the optimal rate of investment if Specially, as long

. Specially, as long asthen there will be no effects of changes in uncertainty on the investment behavior.

Remark 3. Certainly, the conclusion above will hold as long as, but no matter the marginal adjustment cost function is convex, concave, or linear.

Subsequently, we will explain the effects of uncertainty on investment. To achieve this goal, it’s a good choice to simulate the idea in Abel [

Here, we used (2)-(4) to get the second equality. Naturally, the expected present value of the marginal revenue product of capital is

. Inserting (21) into this integral directly, we see that the integral is actually equal to, this indicates is the expected present value of the marginal revenue product of capital. Hence, we can say that the uncertainty affects investment by means of influencing the expected value of future marginal revenue product of capital.

Finally, the dynamic behavior of investment will be explored at the end of this section.

Proposition 3. The optimal investment behavior is such that the expected proportional growth rate of investment(non-zero in general) is constant over time.

Proof. Applying Itô’s Lemma to, it yields

Substituting (2)-(4) into (22), it’s easily to obtain

Applying Itô’s Lemma to (20), there holds

Substituting (2)-(4) into (24), it’s easily to get

So far, from (23) and (25), we observe that both the expected growth rate of marginal revenue product of capital and the expected proportional growth rate of investment are constants over time, neither they depends on output price or wage. Simultaneously, it’s clearly to see the two expected growth rates are zero in the case without uncertainty.

The objective of this paper is to analyze the effects of uncertainty on a competitively risk-neutral firm’s investment decision. By referring the methods presented in Abel [

ifwhereas the fluctuations in uncertainty will lead to negative impacts on the optimal rate of investment if

, and there won’t have any effects of fluctuations in uncertainty on the investment so long as

, where

are changes in uncertainty of output price and of wage rate, respectively.

The expected growth rate of marginal valuation of capital and of investment are both invariant over time. And moreover, by observing (23) and (25) carefully, we see that the expected growth rate of marginal valuation of capital multiplies the elasticity of investment with respect to marginal valuation of capital, , is equal to the expected growth rate of investment when the marginal adjustment costs function is linear, i.e.,. As to the convex marginal adjustment costs function and the concave type, the relationship between the two expected growth rates are uneasy to determine, which is unlike the case demonstrated in Abel [

We remark that it’s unnecessary to confine the production function to Cobb-Douglas type, and strictly speaking, the argument presented in the paper can be applied to any production function obeys homogeneous of degree one. Besides, we derived our results without considering whether the marginal revenue product of capital is convex, concave or linear, which is different from Pindyck [

[

At last, we mention the major contribution of this paper. For one thing, an explicit solution is derived and firm’s investment behavior is analyzed qualitatively and quantitatively rather than just qualitatively. For another thing, both price and wage uncertainty are examined simultaneously since fluctuations in output price tend to be accompanied by fluctuations in wage, rather than consider the price uncertainty only. What’s more important, so far to our knowledge, the question we discussed in the paper is still open.

The author would like to express his sincere thanks to the anonymous referees for their helpful comments and suggestions, which greatly improved the presentation of this paper. The author also would like to thank the editors for their help.