Appendix B: Investment decisions
In determining the level of investment, each firm considers whether two conditions are met: (1) a shortage of production capacity is expected, and (2) the investment promises to yield a positive profit. Regarding (1), we follow Doms and Dunne (
1998) who developed “the simulation models that best fit the observed capital adjustment patterns”.
44 Regarding (2), our boundedly rational firm employs the investment criterion developed by Minsky (Minsky
1975; Minsky and Kaufman
2008): The firm compares the construction cost of capital stock with the present expected value of quasi-rents that flow from the new capital stock.
First, we derive the desired level of capital stock, denoted by
\(O^*_t\). In determining the volume of new construction of capital stock, each firm forms expectations regarding the demand for its products,
\(\hat{X}_{f,t}\), and the profit rate,
\(\hat{\Pi }_{f,t}\), over the investment horizon. Based on the realized volume of sales,
\(X_{f,t-1}\),
\(\hat{X}_{f,t}\) is formed adaptively with an inertia
\(\delta _x\):
$$\begin{aligned} \hat{X}_{f,t} = (1-\delta _x)X_{f,t-1} + \delta _x \hat{X}_{f,t-1}. \end{aligned}$$
(B.1)
Let
\(q_t\) be the total cost of one unit of capital stock whose construction is initiated in period
t. Assume that at the beginning of period
t, a firm owns
\(O_{t-g-m}\) operational factories of age
m (
\(m=0,1,\ldots , U\)). Note that the unit cost of
m periods of old capital stock is
\(q_{t-g-m}\). During period
t, for each vintage of capital stock, the firm pays interest
\(r^L_t q_{t-g-m}(1-m/U) O_{t-g-m}\) plus principal repayments on the long-term debt,
\(q_{t-g-m}O_{t-g-m} /U\). Thus, the firm pays the sum of depreciation costs,
\(Dep_t=\dfrac{1}{U}\sum _{m=0}^{U-1} q_{t-g-m}O_{t-g-m}\), to the bank as principal repayments. The long-term debt balance at the beginning of period
t is given by
$$\begin{aligned} L^{\ell }_t=\sum _{m=0}^{U-1}\Bigl (1-\frac{m}{U}\Bigr )q_{t-g-m}O_{t-g-m}. \end{aligned}$$
(B.2)
Consequently, we can see that the firm pays
\(r^B_t L^{\ell }_t\) as an interest payment.
Let
\(K_{t-1}\) denote the capital stock operating in period
t. We assume that completed capital stock will continue to operate for
U periods with a constant production capability. After
U periods, it is shut down and disposed of with zero disposal value. Thus,
\(K_{t-1}\) stock has been completed in the most recent
U periods:
45$$\begin{aligned} K_{t-1}=\sum _{m=0}^{U-1} O_{t-g-m}. \end{aligned}$$
(B.3)
Regarding condition (1), we first obtain the target amount of the starting investment,
\(O^*\), by following the flexible accelerator model. According to this model, a firm adjusts capital stock toward its desired level by a constant fraction of the difference between the desired and actual levels of capital (Jorgenson
1971). The desired level of capital depends on input prices. As stated in Sect.
2.2.2, an investing firm invests
\(\phi \) units of output in each period during the gestation period, i.e.,
g periods, to build one unit of capital stock. To obtain the estimated cost of capital stock
\(\hat{q}_t\) in period
t, first we obtain the actual cost
\(q^g_t\) when the gestation period is
g. If
\(g=1\), then
\(q^1_t=\phi p_t(1+r^L_t)\), where
\(p_t\) is the cost of investment goods. Recursively, we have
\(q^i_t=(q^{i-1}_{t+i-1}+\phi p_{t+i-1})(1+r^L_{t+i-1})\) for
\(i \le g\). Thus,
\(q^g_t=\phi \sum _{m=1}^g p_{t+m-1} \prod _{\tau =1}^g (1+\hat{r}^L_{t+\tau -1})\), where
\(\hat{r}^{L}\) denotes the expected lending rate. Substituting
\(P_t\) for
\(p_{t+m-1}\) for all
\(m=1, \ldots , g-1\), and
\(r^{L}_t\) for
\(\hat{r}^L_{t+m-1}\), the estimate
\(\hat{q}_t\) is obtained as
$$\begin{aligned} \hat{q}_t = \phi P_t\sum _{m=1}^{g} {(1+r^{L}_t)^m}. \end{aligned}$$
(B.4)
The firm then estimates the user cost of capital according to
\(\hat{q}_t\). The user cost of capital includes the cost of inputs and interest payments during the gestation period, as well as during operational periods. Because the capital stock operates for
U periods, the depreciation cost is
\(\hat{q}_t/U\). To obtain the part of the interest payment that is included in the user cost, we assume that our boundedly rational firm naively expects the current interest rate to continue over the next
U periods. The remaining debt in relation to the initial value of capital stock decreases as time passes, giving
\(\frac{U-\tau }{U}\) (
\(\tau = 1,2, \ldots , U\)) over
\(\tau \) periods after its completion. Thus, the firm expects to pay
\(r_t^L \hat{q}_t \sum _{\tau =0}^{U-1} \frac{U-\tau }{U}\) in total over the operational periods, which simplifies to
$$\begin{aligned} r^L_t \hat{q}_t(U+1)/2. \end{aligned}$$
(B.5)
Consequently, the expected average interest payment is
\(r^L_t \hat{q}_t(U+1)/(2U)\). For future reference, we note here that the firm owes
$$\begin{aligned} \frac{\hat{q}_t(U+1)}{2U} \end{aligned}$$
(B.6)
of the long-term loan balance on average for each unit of capital stock. By adding the depreciation cost, the estimated user cost of capital is
$$\begin{aligned} \hat{v}_t=\frac{\hat{q}_t}{U} \left( 1+ r^L_t \frac{U+1}{2}\right) . \end{aligned}$$
(B.7)
Then, based on
\(\hat{v}_t\), the firm determines the desired capital–labor ratio, denoted by
\(k^*\), by minimizing the average cost
\(\hat{v}_t K+w_t N\), subject to
\(AK^{\alpha }N^{1-\alpha } \ge 1\). From the first-order condition,
$$\begin{aligned} k^*_t=\frac{\alpha w_t}{(1-\alpha )\hat{v}_t}. \end{aligned}$$
(B.8)
Let
\(K^d\) be the optimum level of capital stock for firm
f at time
t if the firm incurs no friction in adjustment.
46 Given that the profit condition is satisfied, the desired level of capital is derived using the expected demand
\(\hat{X}_t\) and the desired capital–labor ratio
\(k^{*}_t\) as the following:
$$\begin{aligned} K^d_t=\dfrac{1}{A}(k^*_t)^{1-\alpha } \hat{X}_t. \end{aligned}$$
(B.9)
Meanwhile, if no new investment is undertaken in the current period, the available capital stock in period
g will be
\(K^0_{t}={\sum \nolimits _{m=1}^{U-1}}O_{i,t-g+m}\). Then,
\(O_t^*\) can be expressed as the difference between the desired and actual capital stock, i.e.,
\(O_t^*=K^d_t-K^0_t\).
Regarding condition (2), our boundedly rational firm uses the investment criterion developed by Brainard and Tobin (
1968), Tobin (
1969), Minsky (
1975), and Minsky and Kaufman (
2008): It compares the replacement cost of capital stock with the present expected value of quasi-rents that flow from the new capital stock. For the sake of simplicity, we assume that the bank and firm collectively assess the profitability of the new investment according to the current quasi-rent.
47 We first obtain the replacement cost
RC of one unit of working capital by replacing the real cost
\(q_{t-g-m}\) with the current estimated cost,
\(\hat{q}_t\), in Eq. (
B.2). From Eq. (
B.5), by adding the interest payment during the operational period, the interest-augmented replacement cost (opportunity cost) of one unit of new capital stock is
$$\begin{aligned} RN_t = \hat{q}_t \Big (1 + r^{L}_t \frac{(U+1)}{2} \Big ). \end{aligned}$$
(B.10)
The replacement cost of the existing capital stock is
$$\begin{aligned} RE_t=\sum _{m=0}^{U-1}\Bigl (1-\frac{m}{U}\Bigr )\hat{q}_t O_{t-g-m}. \end{aligned}$$
(B.11)
Then, by adding the capital gain
\(RE_t -L^{\ell }_t\) the average total quasi-rent over
U periods is
$$\begin{aligned} A\Pi ^q_t = \frac{U\big (p_{f,t} X_{f,t}-w_{f,t}N_{f,t}+RE_t-L^{\ell }_t \big )}{\sum _{m=0}^{U-1} O_{t-g-m}} . \end{aligned}$$
(B.12)
Therefore, Tobin’s average
\(Q^T\) is defined by
\(Q^T=\dfrac{A\Pi ^q_t}{RN_t} \). Let
\(\Pi ^q\) denote the rate of quasi-rent net depreciation costs:
$$\begin{aligned} \Pi ^q_t=\frac{\big (p_{f,t} X_{f,t}-w_{f,t}N_{f,t}+RE_t-L^{\ell }_t \big )}{\hat{q}_t \sum _{m=0}^{U-1} O_{t-g-m}} -\frac{1}{U}. \end{aligned}$$
(B.13)
By simple computation, we can see that
\(Q^T > 1\) if and only if
\(\Pi ^q_t - r^{L}_t \frac{(U+1)}{2U} \ge 0\).
48 The second term of this inequality gives the average interest rate for the average debt balance for one dollar’s worth of capital stock from Eq. (
B.6). Here, we have assumed that the firm adaptively forms the expected rate of profit as the following:
$$\begin{aligned} \hat{\Pi }_t = (1-\delta _{\pi })\Pi ^q_{t-1} + \delta _{\pi } \hat{\Pi }_{t-1}, \end{aligned}$$
(B.14)
where
\(\delta _{\pi }\) is the inertia factor applied to the previous expected rate of return. Hence, the profitability condition is expressed as a positive net expected profit:
$$\begin{aligned} \hat{\Pi }^n_t=\hat{\Pi }_t - r^{L}_t \frac{(U+1)}{2U}>0. \end{aligned}$$
(B.15)
It seems reasonable to assume that the bank will also be reluctant to provide investment funds to build a new production facility beyond replacing capital stock at the end of the current facility’s operational life when the operation is unprofitable. Conversely, under uncertainty, it is difficult for a firm to downsize its scale of production when it holds a prospect of profitability. In terms of notation, we thus assume, for new investment demand, that
\(O^*_t \ge O_{t-U}\), if
\(\hat{\Pi }^n_t \ge 0\). Otherwise,
\(O^*_t < O_{t-U}\), regardless of the level of demand.
49
We now combine the two conditions to obtain our investment function, which is basically consistent with the (
S,
s) models used in Caballero and Engel (
1999) and Doms and Dunne (
1998).
50 More specifically, a firm only expands its production capacity when it is expected to be profitable, i.e.,
\(\Pi ^n_t \ge 0\), and the desired level of new investment,
\(O^*_t\), substantially exceeds the replacement level,
\(O_{t-U}\).
51 Thus, no investment is undertaken if it is unprofitable, and
\(O^*_t\) is substantially smaller than the replacement investment. Otherwise, the firm undertakes an investment that exactly replaces capital that has depreciated. Denoting the upward and downward stickiness of replacement demand by
\(\sigma _i^U \) and
\(\sigma _i^D \in (0,1)\), respectively, the level of demand for a new project is given by
$$\begin{aligned} O^d_t(\delta _i)= \left\{ \begin{array}{ll} O(\delta _i) &{}\quad \text{ if } O^*_t> (1+\sigma _i^U) O_{t-U} \text{ and } \hat{\Pi }^n_{t-1} > 0, \\ 0 &{}\quad \text{ if } O^*_t< (1-\sigma _i^D) O_{t-U} \text{ and } \hat{\Pi }^n_{t-1} < 0, \\ O_{t-U} &{}\quad \text{ otherwise. } \end{array} \right. \end{aligned}$$
(B.16)
Therefore, the larger the value of
\(\sigma _i^U\) or
\(\sigma _i^D\) becomes, the more likely it is that a firm maintains its production capacity at the current level. The first line of Eq. (
B.16) indicates that a firm expands its operational scale only when investment demand is sufficiently large and the profit rate is expected to be positive. When that is the case, we further assume that the firm slowly adjusts its starting investment, denoted by
\(O(\delta _i)\), according to
\(O(\delta _i) =(1-\delta _i)O^*_t+\delta _i O_{t-U}\), where
\(\delta _i\) denotes the inertia effect on an investment decision.