2. The model

The economy consists of a fixed unit mass of firms $j\in [0,1]$ which produce homogeneous output $y_{jt}$ and a unit measure continuum of identical households who consume output and supply labor.

Technology: The production function is as follows:

(2) \begin{equation} y_{jt} = A_t z_{jt}k_{jt}^\alpha n_{jt}^\nu{, } \ \alpha +\nu \lt 1, \end{equation}

where $k_{jt}$ and $n_{jt}$ indicates the idiosyncratic capital and labor employed by the firm $j$ , and $A_t$ is aggregate productivity. For each firm, the idiosyncratic TFP $z_{jt}$ follows a log-normal AR(1):

(3) \begin{equation} log\left(z_{jt}\right) = -\left(1-\rho ^z\right) \frac{\sigma ^{z^2}}{2\left(1-\rho ^{z^2}\right)} + \rho ^z log\left(z_{jt-1}\right) + \epsilon _{jt}, \quad \epsilon _{jt}\sim N\left(0,\sigma ^z\right). \end{equation}

Adjustment costs: The investment cost function includes two components: a direct cost $i_{jt}$ and a fixed nonconvex capital adjustment cost $\xi _{jt}$ paid in units of labor if the firm adjusts by more than a small proportion of their current capital stock $(|ak|)$ :

(4) \begin{equation} c\!\left(i_{jt}\right) = i_{jt} + \mathbf{1}_{\left(|i_{jt}|\gt ak_{jt}\right)}\cdot w_t\cdot \xi _{jt}, \quad \xi _{jt}. \end{equation}

Firm Optimization: I denote by $V^A\!\left(k_{jt},z_{jt};\;\Omega _t\right)$ , $V^{NA}\!\left(k_{jt},z_{jt};\;\Omega _t\right)$ , and $V\!\left(k_{jt},z_{jt};\;\Omega _t\right)$ $\equiv$ $E_{\xi _{jt}}\tilde{V}\big(k_{jt},z_{jt},$ $\xi _{jt};\;\Omega _t\big)$ the value functions of a firm with an active investment choice, without an active investment choice, and with expected draw of $\xi _{jt}$ . The aggregate state $\Omega _t = \left(A_t, \Theta _t,\mu _t\left(k,z,\xi \right)\right)$ where $\Theta _t$ is a vector comprising the stochastic discount factor and wage at time $t$ , and $\mu _t(k,z,\xi )$ is the distribution of firms. The value functions are as follows:

(5) \begin{equation} V^A\!\left(k_{jt},z_{jt};\;\Omega _t\right) = \max _{i,n} \left\{ y_{jt} - w_t n_{jt} - c\!\left(i_{jt}\right) + \mathbb{E}\!\left[\Lambda _{t,t+1} V\!\left(k^*_{jt+1},z_{jt+1};\; \Omega _{t+1}\right)\right] \right\}, \end{equation}

(6) \begin{equation} V^{NA}\!\left(k_{jt},z_{jt};\;\Omega _t\right) = \max _{i\in [-ak,ak],n} \Big \{ y_{jt} - w_t n_{jt} - c\!\left(i_{jt}\right) + \mathbb{E}\!\left[\Lambda _{t,t+1} V\!\left(k^C_{jt+1},z_{jt+1};\; \Omega _{t+1}\right)\right] \Big \}, \end{equation}

where the stochastic discount factor $\Lambda _{t,t+1}$ is derived from the household problem since households own all the firms. $k^C_{jt+1}$ and $k^*_{jt+1}$ are the constrained and non-constrained capital choices.

The firm will choose to pay the fixed cost if and only if $ V^{A}\!\left(k_{jt},z_{jt};\;\Omega _t\right) - w_t \xi _{jt} \gt V^{NA}\!\left(k_{jt},z_{jt};\;\Omega _t\right)$ . There is a unique threshold $\xi ^*\!\left(k_{jt},z_{jt};\;\Omega _t\right)$ at which the firm breaks even:

(7) \begin{equation} \xi ^*_t\!\left(k_{jt},z_{jt};\;\Omega _t\right) = \frac{V^{A}\!\left(k_{jt},z_{jt};\;\Omega _t\right)-V^{NA}\!\left(k_{jt},z_{jt};\;\Omega _t\right)}{w_t}. \end{equation}

If a firm draws a fixed cost $\xi _{jt}$ below $\xi ^*(k_{jt},z_{jt};\;\Omega _t)$ (which I denote as $\xi ^*$ for short), the firm pays the fixed cost and then actively adjusts its capital, otherwise it does not. The value function is:

(8) \begin{equation} V\!\left(k_{jt},z_{jt};\;\Omega _t\right) = -\frac{w_t \!\left(\xi ^*+\underline{\xi }\right)}{2} + \frac{\xi ^*-\underline{\xi }}{2\sqrt{3}\sigma _{\xi }} V^A\!\left(k_{jt},z_{jt};\;\Omega _t\right) + \left(1-\frac{\xi ^*-\underline{\xi }}{2\sqrt{3}\sigma _{\xi }}\right)V^{NA}\!\left(k_{jt},z_{jt};\;\Omega _t\right), \end{equation}

where $\underline{\xi }=\mu _{\xi } - \sqrt{3}\sigma _{\xi }$ is the lower bound of the fixed cost. The firm expects to pay the fixed cost when drawing $\xi _{jt}$ lower than $\xi ^*(k_{jt},z_{jt};\;\Omega _t)$ . With probability $\frac{\xi ^*-\underline{\xi }}{2\sqrt{3}\sigma _{\xi }}$ , the firm chooses to actively invest, otherwise it stays inactive. Therefore, the capital stock evolves by the law of motion:

(9) \begin{equation} k_{jt+1} = \left \{ \begin{array}{l@{\quad}l@{\quad}l} \left(1-\delta \right)k_{jt} + i^*_{jt} & &{\xi _{jt}\lt \xi ^*\!\left(k_{jt},z_{jt};\;\Omega _t\right)}\\[4pt] \left(1-\delta \right)k_{jt} + i^C_{jt} & &{otherwise}\\ \end{array} \right .. \end{equation}

Household optimization: Households’ expected utility is as follows:

\begin{equation*} E_0\sum _{t=0}^{\infty } \beta ^t \left ( \frac {C_t^{1-\eta }}{1-\eta }-\theta N_t \right ), \end{equation*}

subject to the budget constraint: $C_t + \frac{1}{R_t} B_t \leq B_{t-1} + w_t N_t + \Pi ^F_t$ . Here $\beta$ is the discount factor of households, $\theta$ is the disutility of working, $R_t$ is the real interest rate, $B_t$ is one period bonds, $w_t$ is the nominal wage, and $\Pi ^F_t$ is the nominal profits from all the firms. The first order conditions of consumption, labor, and bonds deliver:

(10) \begin{equation} w_t = -\frac{U_n\!\left(C_t,N_t\right)}{U_c\!\left(C_t,N_t\right)} = \theta C_t^\eta, \end{equation}

(11) \begin{equation} \Lambda _{t,t+1} = \frac{1}{R_t} = \beta \frac{U_c\!\left(C_{t+1},N_{t+1}\right)}{U_c\!\left(C_{t},N_{t}\right)} = \beta \!\left(\frac{C_t}{C_{t+1}}\right)^\eta. \end{equation}

3. Mean, variance, and the interest elasticity of investment

Benchmark calibration: I calibrate the benchmark model with mean and variance bundled (henceforth, bundled model) as in Khan and Thomas (Reference Khan and Thomas2008) $\Big($ the uniform distribution has support from 0 to an upper bound: $\xi _{jt}\sim U\left[0,\bar{\xi }\right]$ , so $\left.\mu _{\xi }=\sqrt{3}\sigma _{\xi }=\bar{\xi }/2\right)$ to hit the target investment moments. For fixed parameters, I choose the discount factor $\beta =0.99$ to match an annual interest rate of 4%., elasticity of intertemporal substitution $\eta =1$ for log utility, leisure preference $\theta =2$ to match a one-third working time share, capital exponent $\alpha =0.25$ and the labor exponent $\nu =0.60$ to match a labor share of two-thirds and decreasing returns to scale of 85%, quarterly capital depreciation $\delta =0.026$ , free capital adjustment region $a = 0.001$ , and persistence of idiosyncratic TFP shock $\rho ^z=0.95$ . For fitted parameters, I choose $\sigma ^z=0.05$ and $\bar{\xi }=0.6$ to match the average investment rate (10.5%), the standard deviation of investment rates (0.13), the spike rateFootnote ³ (17%), and the partial equilibrium interest elasticity of aggregate investment (−5), reflecting the empirical moments as measured in Zwick and Mahon (Reference Zwick and Mahon2017) and Koby and Wolf (Reference Koby and Wolf2020).Footnote ⁴

How to measure the interest elasticity? The partial equilibrium interest elasticity of aggregate investment is defined by how aggregate investment, as yielded by the collective decisions of all heterogeneous firms, responds to an unexpected real interest rate shockFootnote ⁵. For instance, −5 means when firms face an unexpected real interest rate cut of 1%, partial equilibrium aggregate investment increases by 5%. Quasi-experimental evidence in Zwick and Mahon (Reference Zwick and Mahon2017) and Koby and Wolf (Reference Koby and Wolf2020) suggests this interest-elasticity should be about −5.

My exact numerical exercise in Sections 3 and 4 is to have the economy start at steady-state and hit by a one-time unexpected drop in real interest rate in the first period to generate the real interest rate series $\{r_t\}_{t=0}^T=\{r^*, r^*+\Delta r, r^*, ..., r^*\}$ . I then feed the stochastic discount factor series $\{\Lambda _t\}_{t=0}^T=\left\{\frac{1}{1+r^*}, \frac{1}{1+r^*+\Delta r}, ..., \frac{1}{1+r^*}\right\}$ and the steady-state wage series $\{w_t\}_{t=0}^T=\{w^*\}$ into the partial equilibrium and solve for the aggregate investment series $\{I_t\}_{t=0}^T$ . The partial equilibrium interest elasticity is then calculated as $\partial log I_t/\partial r_t$ at time $t=1$ . More specifically, I choose $\Delta r = -0.25\%$ , therefore, a $\partial log I_1/\partial r_1=-5$ means such a one-time unexpected drop in the real interest rate boosts aggregate investment by 1.25%.

Upper bound $\bar{\boldsymbol\xi }$ and interest elasticity: In Figure 1, I plot the model’s interest elasticity as the choice of $\bar{\xi }$ varies from 0.025 to 1. First, the interest elasticity of aggregate investment is very sensitive to changes in the upper bound of the nonconvex adjustment costs.Footnote ⁶ Second, a relatively large value of $\bar{\xi }=0.6$ gives an interest elasticity of −5. Conversely, if $\bar{\xi }$ is smaller, the aggregate investment will be oversensitive to interest rate changes. In Khan and Thomas (Reference Khan and Thomas2008), $\bar{\xi }=0.0083/4$ , which will deliver an interest elasticity around −500. This will imply that lumpy investment is irrelevant for aggregate dynamics as in Khan and Thomas (Reference Khan and Thomas2008). However, their irrelevance result is not consistent with the joint dynamics of aggregate investment and the real interest rate over the business cycle as shown in Winberry (Reference Winberry2021).

Figure 1. PE interest elasticity over $\bar{\xi }$ . Note: In the benchmark model, the uniform distribution starts from 0 to an upper bound: $\xi _{jt}\sim U[0,\bar{\xi }]$ . It bundles the mean and the variance by $\bar{\xi }$ : $\mu _{\xi }=\sqrt{3}\sigma _{\xi }=\bar{\xi }/2$ . Therefore, increasing $\bar{\xi }$ increases both $\mu _{\xi }$ and $\sigma _{\xi }$ simultaneously.

Mean, variance, and interest elasticity: Now I depart from the benchmark calibration of $\bar{\xi }=0.6$ . Instead, I study two alternative groups of calibrations: One, fixing the variance $\sigma ^*_{\xi }=\bar{\xi }/\sqrt{12}=0.6/\sqrt{12}$ and varying the mean $\mu _{\xi }$ from 0 to $2\mu ^*_{\xi }=\bar{\xi }$ to show how the interest-elasticity changes and two, fixing the mean $\mu ^*_{\xi }=\bar{\xi }/2=0.6/2$ and varying the variance $\sigma _{\xi }$ from 0 to $2\sigma ^*_{\xi }=\bar{\xi }/\sqrt{3}$ , to show how the interest elasticity changes. I use an identical quasi-experimental real interest rate shock as the one in the bundled model experiment above.

The findings plotted in Figure 2 are very interesting. First, the interest elasticity is not solely determined by the expected size (mean) of the nonconvex cost. Unlike common claims that the interest elasticity is controlled by the expected size of the nonconvex cost, the uncertainty (variance) plays a role. In panel (a), even though the mean is set to be relatively large, when the variance approaches zero, the interest elasticity is massive. Given the fixed mean, the model hits the targeted interest elasticity when the variance is equal or larger to that of the bundled model. Second, the interest elasticity is not solely determined by the uncertainty (variance) of the nonconvex cost. In panel (b), even though the variance is fixed to be relatively large when the mean approaches zero, the interest elasticity is again massive. Given the fixed variance, the model hits the targeted interest elasticity when the mean is equal or larger to that of the bundled model.

Figure 2. PE interest elasticity over $\mu _{\xi }$ and $\sigma _{\xi }$ . Note: The variance-fixed model fixes the variance by choosing $\sigma ^*_{\xi }=\bar{\xi }/\sqrt{12}$ and the mean-fixed model fixes the mean by choosing $\mu ^*_{\xi }=\bar{\xi }/2$ . The two models are identical along both vertical dotted lines when $\mu _{\xi }=0.3$ and $\sigma _{\xi }\simeq 0.17$ .

What is the mechanism behind choosing the mean and the variance, respectively?

4. The mechanism

To further inspect the mechanism behind the differences between the mean and the variance, I demonstrate results from three models with three alternative calibrations: (1) the carefully calibrated bundled model (Bundled); (2) a mean-fixed model $\left(\mu ^*_{\xi }=\bar{\xi }/\sqrt{12}\right)$ with zero variance $\left(Zero\hbox{-}\sigma _{\xi }\right)$ ; and (3) a variance-fixed model $\left(\sigma ^*_{\xi }=\bar{\xi }/2\right)$ with zero mean $\left(Zero\hbox{-}\mu _{\xi }\right)$ .

A decomposition of the interest elasticity: I first show the decomposition of the interest elasticity in all three models in terms of both extensive margin and intensive margin investment in Table 1 following the equation below:

(12) \begin{equation} \frac{d\sum I_j}{dr}=\frac{d\sum _{EM} I_j}{dr} + \frac{d\sum _{IM} I_j}{dr}, \end{equation}

where $d\sum I_j$ is aggregate investment, $d\sum _{EM} I_j$ is aggregate extensive margin investment, and $d\sum _{IM} I_j$ is aggregate intensive margin investment. The carefully calibrated Bundled model has an interest elasticity of −5.1, 96% of the investment response is from the extensive margin, and 4% is from the intensive margin. The Zero- $\sigma _{\xi }$ model has an interest elasticity of about −600, which is almost entirely from the extensive margin. The Zero- $\mu _{\xi }$ model has an interest elasticity of −80, but which is mainly from the intensive margin.

This decomposition shows that the mean and the variance play different roles. Without a sizable mean, the response of the aggregate investment to the interest rate is mainly from the intensive margin. The intensive margin is much too sensitive to real interest rate changes, which delivers a falsely large interest elasticity of aggregate investment. Without a sizable variance, the response of aggregate investment to the interest rate is mainly from the extensive margin. The extensive margin is extremely sensitive to changes in real interest rates. A firm either chooses to pay $\mu _{\xi }^*$ and invest a lot or stay inactive when the real interest rate changes. Firms on the extensive margin choose to ”all-in” which creates the unrealistically large interest elasticity.

Table 1. A decomposition of the interest elasticity

Note: The Bundled model has the calibration that matches the micro investment moments. The Zero- $\sigma _{\xi }$ deviates by setting $\sigma _{\xi }$ to zero while all other parameters are unchanged. The Zero- $\mu _{\xi }$ deviates by setting $\mu _{\xi }$ to zero while all other parameters are unchanged. EM stands for the extensive margin and IM stands for the intensive margin.

Distributions of the extensive margin and the intensive margin: In Figure 3, I plot the interpolated distributions of the extensive margin (adjustment probability) and the intensive margin (investment rate conditional on adjustment) at the steady states using two-dimensional interpolation with respect to productivity and capital stock. Since the intensive margin distributions are not much changed across models, I only plot these for the Bundled model. Warmer and darker colors indicate higher investment rates and higher adjustment probabilities.

Figure 3. Distributions of the extensive margin and the intensive margin. Note: This figure shows the distribution of firms’ investment decisions at both the extensive margin and intensive margin conditional on their productivity and capital stock. Since the intensive margin distributions are not much changed across models, I only plot these for the Bundled model. We could decompose firms’ investment decisions in two steps. Take the Bundled model for example; for firms at Productivity Grid 40 and Capital Grid 30, between 15% and 30% of these firms, according to the extensive margin rules in panel (b), would invest positively by 10% to 20%, according to the intensive margin rules in panel (a), and for firms at Productivity Grid 10 and Capital Grid 35, between 45% and 60% of these firms, according to panel (b), would disinvest, according to panel (a). Aggregate investment of the economy is, therefore, an integration of the extensive margin multiplying the intensive margin over the entire distribution.

In panel (a) for the Bundled model, we see that higher productivity and lower capital firms invest more at the steady state. These firm will also invest more in response to the opportunity presented by the interest rate shock. In contrast, lower productivity and higher capital firms disinvest at the steady state. In panel (b), we see that the extensive margin distribution is layered from 0% probability of adjustment along the diagonal to higher probabilities away from the diagonal where productivity-capital mismatches are more severe. For the highest (lowest) productivity and lowest (highest) capital firms, the adjustment probabilities are larger than 75%. Conditional on draws of the nonconvex adjustment costs, a proportion of the high productivity and low capital firms to the right of the diagonal would then invest positively and the low productivity and high capital firms to the left of the diagonal would than disinvest, both according to the intensive margin rules in panel (a).

However, for the Zero- $\sigma _{\xi }$ model and the Zero- $\mu _{\xi }$ model, the extensive margin distributions are entirely different. The extensive margin in the Zero- $\sigma _{\xi }$ model shows a vertical sorting pattern that is sharply moving from a 0% probability of adjusting to almost a 100% possibility of adjusting. Even slight changes in interest rate would easily cause massive movements in the boundaries, boosting firms at the margins to dramatically change from 0% to >75% or vice versa. As a result, the extensive margin is extremely interest rate sensitive. In contrast, the extensive margin adjustment probability in the Zero- $\mu _{\xi }$ model is always higher than 45% and has much smaller variations. Changes in the interest rate barely cause movements in the boundaries - the extensive margin is not that sensitive to interest rate changes.

5. Implications of aggregate dynamics

To demonstrate the dynamics implications of the the mechanism behind the differences between the mean and the variance, I show the impulse responses of aggregate investment to aggregate TFP shocks in the same three alternative calibrations: (1) the carefully calibrated bundled model (Bundled); (2) the mean-fixed model $\left(\mu ^*_{\xi }=\bar{\xi }/\sqrt{12}\right)$ with zero variance (Zero- $\sigma _{\xi })$ ; and (3) the variance-fixed model $\left(\sigma ^*_{\xi }=\bar{\xi }/2\right)$ with zero mean $\left(Zero\hbox{-}\mu _{\xi }\right)$ .

My exact numerical exercises start the economy at steady-state, then impose an unexpected aggregate productivity shock with persistence $\rho =0.8$ : $\{A_t\}_{t=0}^T=\{A^*, A^* + a, A^* + \rho a, A^* + \rho ^2 a, ..., A^*\}$ . I feed the aggregate productivity series into the general equilibrium and solve for the aggregate investment series $\{I_t\}_{t=0}^T$ . To demonstrate the state-dependency and the non-linearity of the state-dependency due to the micro lumpiness, I solve for four scenarios $\{a\} = \{+5\%, -5\%, +10\%, -10\%\}$ as representations for $\{$ Small Boom, Small Recession, Large Boom, Large Recession $\}$ for all three models, respectively. I then calculate the impulse responses relative to the steady-state in absolute value in percentages $\left\{100\% \times \left|\frac{I_t - I^*}{I*}\right|\right\}_{t=1}^T$ .

The impulse responses are plotted in Figure 4. In panel (a) for the Bundled model, we first observe strong state-dependency: compared to a small recession, aggregate investment responds by 6% more (21.6% relative to 20.4%) in a small boom. Second, the state-dependency is non-linear due to the size of the TFP shock: compared to a large recession, aggregate investment responds by 11% more (43.9% relative to 39.5%) in a large boom. These results show how lumpy investment is consequential for macroeconomic analysis.

Figure 4. GE impulse responses to TFP shocks. Note: The economy starts at steady-state $t=0$ is hit by an unexpected aggregate productivity shock with persistence $\rho =0.8$ : $\left\{A_t\right\}_{t=0}^T=\left\{A^*, A^* + a, A^* + \rho a, A^* + \rho ^2 a, ..., A^*\right\}$ . Scenarios $\{$ Small Boom, Small Recession, Large Boom, Large Recession $\}$ for all three models have a corresponding TFP shocks $\{a\} = \{+5\%, -5\%, +10\%, -10\%\}$ , respectively. The impulse responses relative to the steady-state are absolute value in percentages $\left\{100\% \times \left|\frac{I_t - I^*}{I*}\right|\right\}_{t=1}^T$ .

However, this is not the case for either the Zero- $\sigma _{\xi }$ model or the Zero- $\mu _{\xi }$ model. In panel (b) for the Zero- $\sigma _{\xi }$ model, the state-dependency and the non-linearity of the state-dependency are both unusually strong as well as are the magnitudes of the impulse responses. Aggregate investment responds by 83% more(134% relative to 73%) and by 153% more (220% relative to 87%) in a small/larger boom relative to a small/large recession, respectively. This is all because the extensive margin is extremely sensitive to the countercyclical changes in real interest rates which smooth out the recessionary effects when the economy is in recessions. In panel (c) for the Zero- $\mu _{\xi }$ model, in contrast, there is almost no state-dependency or non-linearity of the state-dependency. Since the response of aggregate investment to the interest rate is mainly from the intensive margin, the countercyclical changes in real interest rates hardly generate any state-dependency in aggregate investment.