20. Investment

Post-1980s, most developed economies' investment/GDP fell, even as the interest rate (cost of capital) fell and investment became cheaper. Alternative explanations with no finance background: (1) reached an upper limit of new technology; (2) firms have stronger market power, less incentive to invest efficiently; (3) intangible assets are more important, not in traditional investment calculation. We view investment from a finance angle. Hayashi's (1982) marginal-\(q\) model (simplified).

20.1.1 Setup. A firm uses technology \(A\) and capital \(K\) to produce a numeraire good of value \(\pi(A,K)\) (\(A\) subject to random shocks); depreciation \(\delta\in(0,1)\), discount factor \(\beta\in(0,1)\); adjustment cost \(C(K',A,K)\); capital-good price \(p\) this period, \(p'\) next. Investment \(I=K'-(1-\delta)K\). Denote \(\pi_K\equiv\partial\pi/\partial K\), \(C_{K'}\equiv\partial C/\partial K'\), \(V_{K'}\equiv\partial V/\partial K'\).

20.1.2 Firm's maximization.

$$V(A,K,p)=\max_{K'}\pi(A,K)-C(K',A,K)-p(K'-(1-\delta)K)+\beta\mathbb{E}[V(A',K',p')\mid A] \tag{20.1}$$

The f.o.c. w.r.t. \(K'\):

$$C_{K'}(K',A,K)+p=\underbrace{\beta\mathbb{E}[V_{K'}(A',K',p')\mid A]}_{\text{marginal }q} \tag{20.2}$$

The RHS is marginal \(q\) (the marginal benefit of investing one more dollar); (20.2) says invest until MC = MB.

20.1.3 Simplifying assumptions. Assumption 1 \(\pi(A,K)=AK\) (20.3); Assumption 2 \(C(K',A,K)=\frac\gamma2\left(\frac{K'-(1-\delta)K}{K}\right)^2 K\) (20.4); Assumption 3 \(A'=\rho A+\varepsilon'\) (20.5) (\(|\rho|<1\), \(\varepsilon'\sim\mathcal N(0,1)\)). Guess \(V(A,K,p)=Kq(A)\) (20.6). Rewriting (20.2) with (20.4), (20.5):

$$\frac IK=\underbrace{-\frac p\gamma+\frac\beta\gamma\mathbb{E}[q(A')\mid A]}_{\equiv z(A)} \tag{20.7}$$

Plugging into (20.1) verifies \(V=Kq(A)\) solves this dynamic program (\(q(A)=A-\frac\gamma2 z(A)^2-pz(A)+\beta[z(A)+(1-\delta)]\mathbb E[q(A')\mid A]\)). The key conclusion:

$$\frac IK=-\frac p\gamma+\frac\beta\gamma\mathbb{E}[q(A')\mid A] \tag{20.8}$$

Testable prediction: from (20.8), specify \(\left(\frac IK\right)_{it}=\alpha_{it}+\frac\beta\gamma\mathbb E[q(A')\mid A]+\boldsymbol\psi_i\cdot\mathbf X_{it}+\varepsilon_{it}\) (20.9), predicting

$$\boldsymbol\psi_i=\mathbf 0 \tag{20.10}$$

But this comes from many simplifying assumptions and isn't ready to test directly; even if they hold, marginal \(q\) is unobservable (we see only average \(q=V/K\), generally unequal), small firms' average \(q\) is hard to observe, and the derivation has no financial friction (with friction marginal \(q\) includes frictions, not necessarily the same form).

20.2 Fazzari et al. (1988). Cash flow is an important factor determining investment, since internal financing is cheaper than external; a firm with cash flow has more internal financing and can invest more cheaply. Empirical test: Prediction 1 — cash flow has a positive effect on investment; Prediction 2 — firms more constrained on internal financing (more cash-short) rely more on cash flow, their investment reacting more strongly. Data: 1970–1984 public manufacturing firms (Value Line), four classes by cash shortage (Class 1: dividend/income $<0.1$ for \(\ge10\) years, most cash-short; …; Class 4: others, least short; Figure 20.1 summary stats). Results (Figure 20.2, \(Q_{it}\) = market/book value proxying average \(q\)): cash flow (and lagged) has a significant positive effect (confirms Prediction 1); Class 1 has stronger cash-flow effects than Class 4 (confirms Prediction 2); seemingly contradicting (20.10), but (20.10) comes from a too-simplified model and isn't really denied. Comments: classes are not comparable in capital size; Class 1 firms are smaller, with market price reflecting value less accurately → \(Q_{it}\) suffers more attenuation bias → Class 1's \(Q\) coefficient more biased toward 0 → its cash-flow coefficient more biased upward, so Prediction 2 need not hold.

20.3 Kaplan-Zingales (1997) rebut: the conclusion that high investment-cash sensitivity means more financially constrained is wrong. Theory: K-Z define financial constraint by the wedge between internal and external capital cost (broad; almost all firms could be constrained). A one-period model: the firm chooses \(I\) to solve \(\max_I F(I)-C(E,k)-I\) (20.11) s.t. \(I=W+E\) (20.12) (\(F'>0,F''<0\); external-fund deadweight cost \(C(E,k)\) convex in \(E\), \(k\) the wedge measure). Substituting gives the f.o.c.:

$$F_I(I)-C_E(E,k)=1 \tag{20.13}$$

Total-differentiating with \(dk=0\):

$$\frac{dI}{dW}=\frac{C_{EE}(E,k)}{F_{II}(I)-C_{EE}(E,k)} \tag{20.15}$$

With imperfect markets \(C(E,k)>0\), \(\frac{dI}{dW}>0\); with perfect markets \(C=0\), $=0$ — so investment-cash sensitivity is positive only when the market is imperfect and the firm constrained. With \(dW=0\), \(\frac{dI}{dk}=\frac{C_{Ek}}{F_{II}-C_{EE}}\) (negative if \(C_{Ek}>0\)). But the Fazzari requirement that sensitivity decrease in internal funds, \(\frac{d^2I}{(dW)^2}<0\), reduces (using a third-order condition \(F_{III}=C_{EEE}\)) to

$$\frac{F_{III}(C_{EE}^2-F_{II}^2)}{(F_{II}-C_{EE})^3}<0 \tag{20.16}$$

which need not hold (it depends on the relative curvature of \(C\) and \(F\)). Empirically: using the same sample, they collect detailed info from SEC filings on 49 low-dividend firms Fazzari classed as Class 1, classify each into groups 1–5 by qualitative reports + quantitative statements (constraint measures: liquidity access, dividend restriction, unrestricted retained earnings, cash/capital ratio; Figure 20.3 ordered logit shows these are significant, justifying the 5-group classification); Figure 20.4 shows the cash-flow coefficient has no decreasing pattern from never-constrained to probably-constrained — so the financial-constraint conclusion based on high investment-cash sensitivity doesn't hold with better-measured constraints.

Lenzu-Manaresi (2019) propose a more fundamental approach: directly compute the wedge in the firm's profit-maximization f.o.c., rather than estimating proxies/indicators — directly showing whether a firm has a wedge (financial constraint). Needs: marginal-borrowing-cost data; estimation of the marginal revenue product of capital \(MRP^K\) (using the production-estimation toolkit).

Theory: heterogeneous firms maximize the PV of future cash flows to risk-neutral shareholders, with randomness from firm-specific productivity \(\omega_{i,t}\); each period after observing \(\omega_{i,t}\) the manager decides to repay or default and exit, \(\bar\omega\) the endogenous default threshold; without default, the manager chooses next-period capital \(K_{i,t+1}\), labor, intermediate input, and financing (bank debt \(B_{i,t+1}\), internal cash or new equity); on default the creditor takes over and liquidates after the current period at cost \(X\ge0\) fraction of assets. Lenders offer a single rate \(\tilde r_{t+1}\) to a group of similar firms. The credit limit, as in Kiyotaki-Moore (1997): \(B_{i,t+1}\le\lambda_{i,t}K_{i,t+1}\) (lower \(\lambda\) = smaller pledgeable fraction). The resulting f.o.c.:

$$\rho\int_{\bar\omega}^\infty[MRP_{i,t+1}^K-(r_{t+1}+\delta)]\,d\Phi(\omega_{i,t+1}\mid\omega_{i,t})=\psi_2^K(K_{i,t},K_{i,t+1})+\rho\int_{\bar\omega}^\infty\psi_1^K(K_{i,t+1},K_{i,t+2})\,d\Phi(\omega_{i,t+1}\mid\omega_{i,t})+\underbrace{\chi_{i,t}(1-\lambda_{i,t})}_{\equiv\tau_{i,t}\text{ Gap}}$$

where \(\rho\) is the risk-neutral discount, \(\delta\) the capital discount rate, \(\Phi\) the conditional cdf, \(\psi^K\) the adjustment cost (\(\psi_j^K\) the derivative w.r.t. the \(j\)th argument); \(\tau_{i,t}\equiv\chi_{i,t}(1-\lambda_{i,t})\) = the shadow cost of funds, closing the gap between the marginal gain and cost of raising one more dollar of debt. The \(\tau_{i,t}\) of interest depends on the adjustment cost \(\psi^K\) and the borrowing constraint \(\lambda_{i,t}\).

Empirics: advantages — theoretically justified, estimable for small non-public firms. Using Italian data to estimate each firm's gap:

$$\hat\tau_{i,t}=\rho\left(1-\hat{\mathbb{P}}(\text{Exit}_{i,t+1}\mid\mathcal F_i)\right)\cdot\left[\widehat{MRP}^K_{i,t+1}-(r_{t+1}+\delta_i)\right]$$

Results (Figure 20.5, percentage deviation from target capital): many firms don't invest at the optimum; under-investors are more constrained (higher \(\tau\) more severe); marginal \(Q\) positively correlated with the gap, the gap negatively correlated with financing availability (age/size); relationship lending lowers the gap as the relationship lengthens (Figure 20.6); the extensive margin matters — firms gaining credit access for the first time have a discontinuous drop in \(\tau_{i,t}\) (Figure 20.7). Comment: the crucial assumption is that financial constraint works through quantity rationing (can't borrow at the specified rate). But lenders might raise the rate high enough to block borrowing (price rationing); if we accept price rationing as a constraint, L-M omits it — a firm predicted as unconstrained could actually be price-rationed.

20.5 Rauh (2006). Exploits the discontinuous drop in internal funding from mandatory defined-benefit (DB) pension contributions to study cash flow's effect on investment (controlling endogeneity). Institutional detail: when the pension fund's market value $>$ liability no contribution is required; when $<$ liability it is; the amount follows a complex non-linear function, generating a sharp discontinuity around the threshold (Figure 20.8: mandatory contribution = max of the minimum funding contribution and the deficit-reduction contribution). Design: treat the discontinuity as an exogenous negative cash-flow shock, removing the endogenous relation between funding status and investment opportunities. Specification:

$$\frac{I_{i,t}}{A_{i,t-1}}=\alpha_t+\alpha_i+\beta_1 Q_{i,t-1}+\beta_2\frac{\text{Non-Pension Cash Flow}_{i,t}}{A_{i,t-1}}+\beta_3\frac{Z_{i,t}}{A_{i,t-1}}+\boldsymbol\gamma'\mathbf X_{i,t}+\varepsilon_{i,t} \tag{20.17}$$

\(Z_{i,t}\) the mandatory pension contribution, \(\beta_3\) the parameter of interest. Results (Figure 20.9): average \(Q\) significant and robust; the exogenous cash-flow variation \(Z_{i,t}\) significantly affects investment even with higher-order powers of funding variables (coefficient negative, since \(Z\) measures the cash-flow drop, so the cash-flow effect on investment is positive). Comment: the percentages of capex and pension contribution from a kernel regression should be discontinuous at the threshold, but Figure 20.10 shows them continuous → problematic; Bakke-Whited (2012) note Rauh's results come mainly from heavily underfunded firms (a small, sharply different fraction). Remark 20.1: such reduced-form estimation only describes an effect, with no stance on efficiency; papers like Lenzu-Manaresi (2019) have a clear notion of efficiency.

20.6 Zwick-Mahon (2017). Exploits the tax-based cash-flow shock from bonus depreciation. Bonus depreciation is a government tool to spur investment by accelerating the depreciation schedule; under a \(\theta\) bonus schedule (\(\theta\in(0,1)\)), the firm can depreciate

$$z=\theta+(1-\theta)z^B$$

of the total value in period 0 (\(z^B\) the period-0 depreciation fraction without the bonus; the bonus applies once upon purchase). Longer-duration investment benefits more (early depreciation = an instantaneous cash-flow spike, positive NPV since the tax benefit isn't discounted). Data: 120,000 public and private firms, with policy variation at the industry-year level. DiD: \(\ln I_{i,t}=\alpha_i+\beta z_{N,t}+\boldsymbol\gamma'\mathbf X_{i,t}+\delta_t+\varepsilon_{i,t}\) (\(z_{N,t}\) = the present value of one average dollar deduction for eligible investment, \(N\) a four-digit NAICS code); identifying assumption: parallel trend. Results (Figure 20.12, the odds ratio \(\ln\frac{\mathbf P_N(I>0)}{1-\mathbf P_N(I>0)}\) = extensive margin): both intensive and extensive margins are significantly positive; small/medium firms respond much more strongly (weighted elasticity 27% higher than large firms, Figure 20.13).

It is hard to tease out the internal-external wedge's effect on investment because of selection bias: internal financing is likely correlated with investment opportunities (firms with more internal funds want to invest more even counterfactually). Whited (1992) considers two Euler equations — without and with a borrowing constraint — testing which matches the data better.

Model: the value function \(V_{i0}=d_{i0}+\mathbb{E}_0\left[\sum_{t=1}^\infty\left(\prod_{\tau=0}^{t-1}\beta_{i\tau}\right)d_{it}\right]\) (20.18) (\(d_{it}\) after-tax dividend). The firm chooses \(I_{it}\) (or \(K_{it}\)) each period to maximize value, subject to: capital accumulation \(K_{it}=I_{it}+(1-\delta)K_{i,t-1}\) (20.19); a cash-flow constraint giving the dividend

$$d_{it}=(1-\tau)[F(K_{i,t-1},\mathbf N_{it})-\mathbf w_t\cdot\mathbf N_t-\psi(I_{it},K_{i,t-1})-i_{t-1}B_{t,t-1}]+B_{it}-(1-\pi_t^e)B_{i,t-1}+p_{it}I_{it} \tag{20.20}$$

(\(\tau\) corporate tax, \(F\) production function, \(\psi\) real adjustment cost, \(B_{it}\) net debt, \(i_{t-1}-\pi_t^e\) real rate, \(p_{it}\) real capital-good price); non-negative dividend \(d_{it}\ge0\) (20.21) (multiplier \(\lambda_{it}\)); a transversality condition (20.24) preventing infinite debt. Without the constraint, the Euler equation w.r.t. \(K_{i,t+1}\):

$$\beta_{it}\mathbb{E}_t\left[\frac{1+\lambda_{i,t+1}}{1+\lambda_{it}}\Big[F_K-\psi_K+(1-\delta)\big(\psi_I+\tfrac{p_{i,t+1}}{1-\tau}\big)\Big]\right]=\psi_I(I_{it},K_{i,t-1})+\frac{p_{it}}{1-\tau} \tag{20.22}$$

(discounted expected future marginal benefit = current marginal cost); the f.o.c. w.r.t. \(B_{it}\) is (20.23). Adding the borrowing constraint \(B_{it}\le B_{it}^*\) (multiplier \(\gamma_{it}\)), the \(K_{it}\) Euler equation becomes (20.25) with the factor \(\frac{\beta_{it}}{1+\gamma_{it}+\mathbb E_t[\lambda_{i,t+1}]}\). Assuming \(F_K=\frac{\eta Y_{it}-\mu C_{it}}{K_{i,t-1}}\) (20.26), \(\psi=\frac\alpha2\left(\frac{I_{it}}{K_{i,t-1}}-\nu\right)^2 K_{i,t-1}\) (20.27), substituting gives the expectation-error equation (20.28) (with \(\Lambda_{it}=1-\frac{1+\lambda_{i,t+1}}{1+\lambda_{it}}\), firm/time FE). Empirics: estimate \(\Lambda_{it}=c_0+c_1 DAR_{it}+c_2 DAR_{it}^2+c_3 COV_{it}+c_4 COV_{it}^2\) (a second-order polynomial of the debt-asset ratio \(DAR_{it}\) and interest coverage \(COV_{it}\)); use GMM testing \(\mathbb{E}_t[e_{i,t+1}X_t]=0\) (the model says \(e_{i,t+1}\) is orthogonal to time-\(t\) info, \(\ne0\) rejects the model), with and without the restriction \(\Lambda_{it}=0\) (equivalent to \(\lambda_{i,t+1}=0\), unconstrained). Results (Figures 20.14, 20.15, with/without bond ratings): with \(\Lambda_{it}=0\) the model is easily rejected, without it it isn't → firms are financially constrained; the model is rejected more clearly for firms without bond ratings → they face more serious constraints. Remark 20.2: the structural approach's advantage is better economic intuition and clearer interpretation; its disadvantage is the difficulty of justifying modeling assumptions and empirically measuring some structural parameters.

Matvos-Seru (2014) empirically quantify the trade-offs of the internal capital market: the benefit is avoiding time-varying external-financing costs; the cost is efficiency loss from "corporate socialism." Empirically they use exogenous oil-price shocks to construct exogenous variation in productivity dispersion among conglomerate divisions, identifying the parameters.

Model: each firm's manager has per-period utility \(u_t\) ((20.29)) with the core term

$$u_t=\sum_j z_{tj}k_{tj}-\underbrace{\lambda\sum_j(z_{tj}-z_t^*)k_{tj}}_{\text{corporate socialism}}-\sum_j I_{tj}-(\text{investment/financing/cash cost terms})$$

where \(z_{tj}\) is division \(j\)'s productivity at \(t\), \(z_t^*=\frac1n\sum_{j=1}^n z_{tj}\) the cross-sectional mean; \(\lambda\sum(z_{tj}-z_t^*)k_{tj}\) measures the disutility of corporate socialism (the manager dislikes some divisions outperforming the average). It also includes investment costs (\(\phi_0,\phi_1,\phi_2\)), external-financing costs (\(c_0,\dots,c_8\), depending on TED state \(\zeta_t\)), and a cash-holding penalty (\(j_0,j_1\), \(p_t\) cash). The manager solves \(V(\mathbf s_t,\boldsymbol\sigma,\boldsymbol\theta)=\max_{\boldsymbol\sigma}\mathbb E_t[\sum_{\tau\ge t}\beta^{\tau-t}u_\tau]\) (20.30), s.t. capital accumulation and cash transition. The goal: estimate the parameter vector \(\boldsymbol\theta\) from real data.

Two-step estimation: Step 1 (policy function) estimate \(I_{tj}=\max\{0,Q_2(\mathbf s_t;\boldsymbol\beta_f)+e_{I_{tj}}\}\) (20.31), \(f_t=Q_2(\cdot)+e_{f_t}\) (20.32) (\(Q_2\) a second-degree polynomial), and state transitions: capital (20.33), division productivity \(z_{t+1,j}=G_s(z_{tj})+\varepsilon\) (20.34) (\(G_s\) a 9-knot linear spline, \(z\) proxied first by \(Q\) then ROA), TED \(\zeta_{t+1}=\alpha_\zeta+\beta_\zeta\zeta_t+\varepsilon\) (20.35), cash (20.36). Step 2 (utility parameters) simulate state sequences forward with the policy functions (\(e=0\)) → plug into (20.29), (20.30) for the implied \(\hat V^{(j)}(\boldsymbol\theta)\); perturb the policy functions (\(e\) drawn from the empirical distribution) for \(\hat V^{(j)(h)}(\boldsymbol\theta)\); define the profit from deviating, \(g=\max\{0,\hat V^{(j)(h)}-\hat V^{(j)}\}\), ideally with expectation 0 (value already maximized); Monte-Carlo average \(g(\boldsymbol\theta)^2\), solve \(\boldsymbol\theta=\arg\min_{\boldsymbol\theta}g(\boldsymbol\theta)^2\). Remark 20.3: the estimation depends on hard-to-justify assumptions — (1) managers really have the (20.29)-form utility; (2) firms actually maximize value per (20.29).

Results: the estimated \(\boldsymbol\theta\) (Figure 20.16, \(Q\) and ROA productivity measures); out-of-sample forward simulation gives Figure 20.17 (evolution of a diversified firm's excess value relative to stand-alones): conglomerates are always below stand-alones (the efficiency loss from corporate socialism = the dark side of internal financing); in high-bank-rate periods (high overall credit risk) conglomerates have higher relative value (though still negative) — the benefit of internal financing when external financing is more costly = the bright side.