2.1 Intermediaries

The banks take deposits from households and extend loans to the firm sector. We assume the latter follows a two-stage game whereby lenders post-contract offers that borrowers can choose to accept. ^{Footnote 10} This takes place in an anonymous spot market that leads to a sequence of static contracts, ^{Footnote 11} agreed at the end of period t, ahead of period $ t+1 $ production. In addition to the interest rate, the lender introduces a lottery ^{Footnote 12} that allows the lender to set the probability of loan approval. As shown below, this will be the device that allows the lender to separate borrowers by designing incentive-compatible, or self-selecting, contracts. Specifically, the lenders post-contracts $c_t^i =\{ \tau^i_t,x_t^i \}$ for $i\in\{s,r\}$ , where $\tau_t^i$ is the repayment rate, and $x_t^i$ the financing, or approval probability. We assume that the banks have access to a low-return technology, yielding return $r^*$ and implying that they need not lend all available funds. ^{Footnote 13}

Letting $ p_{t}^i $ and $ R_{t}^i $ denote the success probability and gross rate of return on capital of a type- i project respectively, and $ \Lambda_{t,t+1} $ the stochastic discount factor, the lender must set contract terms subject to individual rationality (IR) constraints

(1) \begin{align}\mathbb{E}_t \left[ \Lambda_{t,t+1} p_{t+1}^i \left( R_{t+1}^i - \tau_t^i \right) \right] \ge 0 , \quad i = r,s,\end{align}

which promise a weakly positive surplus to the firm, and subject to incentive compatibility (IC) constraints given by

(2) \begin{align}\mathbb{E}_t \left[ \Lambda_{t,t+1} p_{t+1}^i x_t^i \left( R_{t+1}^i - \tau_t^i \right) \right] \ge \mathbb{E}_t \left[ \Lambda_{t,t+1} p_{t+1}^i x_t^j \left( R_{t+1}^i - \tau_t^i \right) \right] , \quad i,j = r,s; i\ne j.\end{align}

That is, the value to each borrower of declaring their type truthfully must be weakly greater than lying. As is standard in these mechanism design problems, and straightforward to prove, the problem can be simplified by dropping two constraints. The relevant constraints are the safe IR and the risky IC constraints, which further are found will be always binding as the objective function is increasing in the repayment rates. We can write these constraints as follows: ^{Footnote 14}

(3) \begin{gather}\mathbb{E}_t \left[ \Lambda_{t,t+1} \right] \tau_t^s = \mathbb{E}_t \left[ \Lambda_{t,t+1} R_{t+1}^s \right], \end{gather}

(4) \begin{gather}\mathbb{E}_t \left[ \Lambda_{t,t+1} p_{t+1}^r \right] \tau_t^r = \mathbb{E}_t \left[ \Lambda_{t,t+1} p_{t+1}^r R_{t+1}^r \right] - \mathbb{E}_t \left[ \Lambda_{t,t+1} p_{t+1}^r \left( R_{t+1}^r - \tau_t^s\right) \right] \frac{x_t^s}{ x_t^r} . \end{gather}

It further follows from these constraints that $x_t^r \ge x_t^s$ (see Appendix E), so risky-project firms are always weakly more likely to be funded than those with safe projects. The intuition is that in order to pay higher repayment rates, the banks must offer a higher probability of being approved for finance. The banks solve

(5) \begin{align}\mathcal{V} \left( c^s_{t-1},c^r_{t-1} \right) = \max_{c^s_t,c^r_t} & \left\lbrace \lambda x^s_{t-1} \left( \tau_{t-1}^{s} - r^*\right) + \left(1-\lambda \right) x^r_{t-1} \left( p_{t}^r \tau^r_{t-1} - r^* \right) + \mathbb{E}_t \left[ \Lambda_{t,t+1} \mathcal{V}_{t+1} \left(c^s_t,c^r_t\right)\right] \right\rbrace\notag\\\mbox{s.t.} & \quad 0 \le x^s_t \le x^r_t \le 1 \notag \\& \quad \lambda x^s_t + \left(1-\lambda\right)x^r_t \leq \bar{x}_t, \end{align}

and subject to constraints (3) and (4). The inequality constraint (5) is a feasibility constraint where $\bar{x}_t \le 1$ is the maximum proportion of firm applications that can be approved. This is determined in general equilibrium and will be less than one if the number of possible loans the bank can make is less than the number of firms seeking funds, in which case it is the ratio of the loan supply to the loan demand. When this ratio is greater than unity, $\bar{x}_t$ is bound at one. When constraint (5) is slack, rather than lending all available funds, banks invest a portion of their capital in a low-return asset or technology. Equations (3) and (4) allow $ \tau_t^r $ and $ \tau_t^s $ to be substituted out of the problem, leaving only $ x_t^r $ and $ x_t^s $ to be chosen. For these, the solution to the bank’s problem gives

(6) \begin{align}& \mathbb{E}_t \left[ \Lambda_{t,t+1} \left(p_{t+1}^r R_{t+1}^r - r^* \right) \right] = \varrho_t - \psi_t \frac{1}{1-\lambda} + \varphi_t^r \frac{1}{1-\lambda}, \end{align}

(7) \begin{align}& \mathbb{E}_t \left[ \Lambda_{t,t+1} \left( \left(\lambda + \left(1-\lambda \right)p_{t+1}^r \right) R_{t+1}^{s} - r^* \right) \right] = \varrho_t + \varphi_t^r - \varphi_t^s ,\end{align}

where $\varrho_t$ is the Lagrange multiplier on the feasibility constraint, $\varphi^s_t$ and $\varphi^r_t$ those on $x^s_t$ and $1-x^r_t$ , respectively, and $\psi_t$ is the Lagrange multiplier on $x^r_t-x^s_t$ . These first-order conditions are also subject to Kuhn–Tucker conditions that include zero lower bounds on the four Lagrange multipliers: ^{Footnote 15}

(8) \begin{align} {\varphi}^s_t , {\varphi}^r_t, \varrho_t, \psi_t \ge 0 . \end{align}

Due to these four inequality constraints, it is possible to identify four regimes that depend on parametrization and macroeconomic conditions, including pooling and separating equilibria, and the credit rationing of safe projects. A financial crisis, or credit crunch, will be characterized by banks storing a portion of available capital rather than using it to fund productive firms. Analysis of these regimes is given in Section 3 below. We turn now to the firm sector.

2.2 Firms

When firms draw their type at the end of the period, they apply for external finance for which they may or may not be successful; if firms are successful in securing funds, they purchase k units of capital ready for production in the following period, otherwise we assume they must exit. Of the funded risky projects, a proportion $1-p_t^r$ will fail before production begins. Success probability $p_t^r \in [0,1]$ follows the AR(1) process:

(9) \begin{align}p_t^r =\left(1-\rho_p\right) \bar{p}^r + \rho_p p_{t-1}^r + \varepsilon_{p,t} .\end{align}

If the firm fails, then the capital is lost completely. Let firm type be denoted $i \in \{c,s,r\}$ for corporates, safe-, and risky-project holding firms, respectively. A successful funded project requires k units of capital that is converted into $ \omega^i_t k $ productive units, where we assume $\omega_t^r > \omega_t^c = \omega_t^s = 1$ . The firm hires $h_t\left(\omega_t^i\right)$ units of labor and produces output using

(10) \begin{align}y_t \left(\omega_t^i\right) = z_t \left[\omega_t^i k \right]^\alpha \left[ h_t\left(\omega_t^i\right)\right]^{1-\alpha},\end{align}

where aggregate technology $z_t$ follows the stationary stochastic process:

(11) \begin{align}\log\left(z_t\right) = \rho_z \log\left(z_{t-1}\right) + \varepsilon_{z,t}.\end{align}

Capital depreciates at $\delta$ , so although a fixed input k is required for production, the capital remaining after production will be $ \omega_t^i \left(1-\delta\right)k $ . The value of a successful funded type- i firm can therefore be written as

(12) \begin{align}V_t^i = \max_{h_t\left(\omega_t^i\right)} \left\lbrace y_t \left(\omega_t^i\right) - W_t h_t\left(\omega_t^i\right) - \left(\tau_{t-1}^i - \left(1 - \delta\right) \omega_t^i\right) k + V_{t} \right\rbrace , \end{align}

where $ W_t $ is the market wage rate and $ V_{t} $ the ex ante value of a firm, prior to drawing its type, given by

(13) \begin{align}V_{t} = \mathbb{E}_t \left[ \Lambda_{t,t+1} \left(\eta V_{t+1}^c + \left(1-\eta\right) \left(\lambda x_{t}^s V_{t+1}^s + \left(1-\lambda\right)x_{t}^r p_{t+1}^r V_{t+1}^r \right) \right) \right] . \end{align}

The solution to the firm labor demand implies the real wage will equal the marginal product of labor for all firms

(14) \begin{align}W_t = \left( 1-\alpha \right) \frac{y_t \left(\omega_t^i\right)}{h_t\left(\omega_t^i\right)},\end{align}

where it follows that output per worker ${y}_t^i/h_t^i$ and the efficiency capital-labor ratio $\omega_t^i {k}/h_t^i$ will be equal across all firms, using superscripts for convenience. We can then write the gross return on capital used in the previous section as

(15) \begin{align}R_t^i \equiv \alpha \frac{y_t^i}{k} + \left(1-\delta \right) \omega_t^i , \end{align}

where the total surplus is $ \left(R_t^i - \tau_{t-1}^i\right) k$ and noting that the gross return on efficiency units of capital, $\alpha \frac{y_t \left(\omega_t^i\right)}{\omega_t^i k} + \left(1-\delta \right)$ , is equal for all firms. It follows that $R_t^r = \omega_t^r R_t^c = \omega_t^r R_t^s$ .

As firms can make profits in equilibrium, in the absence of costs of entry, new firms would enter until it is possible for banks to allocate all funds to firms holding risky projects, charging a higher lending rate and excluding the firms holding safe projects entirely. ^{Footnote 16} To prevent this, we introduce a small fixed cost of entry. Any unfunded firms will be liquidated and must repay the entry costs to operate in the period that follows. To pay the entry costs, firms sell equity to households. Under this assumption, new firms will enter until the expected discounted profits $V_t$ , given by equation (13), equals an exogenous fixed cost F. This condition is verified in the solution to the household problem, which we turn to now.

2.3 Households

The representative household faces the usual labor supply and consumption-savings decision, but with an additional portfolio choice problem. The household can choose to either deposit savings $S_t$ at a bank, purchase bonds, $B_t$ , or purchase equity in new firms, $ E_t $ , to solve

\begin{align}\max_{\substack{C_{t+s},H_{t+s}\\S_{t+s},B_{t+s},f_{t+s}}}\mathbb{E}_t \sum_{s=0}^\infty \beta^{t+s} U{\left(C_{t+s},H_{t+s}\right)} , \notag\end{align}

subject to

\begin{align}C_t + S_t + B_t + E_t\left(f_t,f_{t-1}\right) = R_{t-1} S_{t-1} + R_{t-1}^B B_{t-1} +W_t H_t + \Pi_t\left(f_t\right) ,\notag\end{align}

where $R_t$ and $R_t^B$ are the interest earned on savings and bonds, respectively, $f_t$ is the end-of-period mass of firms in the economy and $\Pi_t$ are profits from the household-owned banks and payoffs from equity holdings. The household consumption-savings decision and portfolio allocation is characterized by

(16) \begin{align}& 1 = \mathbb{E}_t \left[ \Lambda_{t,t+1} \right] R_t , \end{align}

where $\Lambda_{t,t+1} = \beta \frac{U'\left(C_{t+1}\right)}{U_{t}'\left(C_t\right)} $ , and with $R_t^B = R_t$ . Labor supply is determined by

\begin{align}& W_t = - \frac{U'\left(H_t\right)}{U_{t}'\left(C_t\right)}. \notag\end{align}

The amount of equity purchased, $E_t$ , corresponds to the fixed costs paid for new entrants and is a claim on future profit streams of the new firms. The number of new entrants at t is the difference between the number of firms in t and the non-exiting firms in $t-1$ . It follows that expenditure on equity is given by

\begin{align}E_t = \left( f_t - \left(\eta + \left(1-\eta\right) \left(\lambda x_{t-1}^s + \left(1-\lambda\right)x_{t-1}^r \right) \right) f_{t-1} \right) k F . \notag\end{align}

Using the return on capital given in equation (15), the total profits earned by the firms per unit k given as the sum of the information rents received by risky-project firms and profits received by corporates can be written as

(17) \begin{align}& \pi_t = \left(1-\eta\right) \left(1-\lambda\right) p_{t}^r x_{t-1}^s \left(R_{t}^r - R_{t}^s \right) + \eta \left(R_t^s - R_{t-1}\right).\end{align}

Using these, the choice of the number of new firms to finance gives the first-order condition

(18) \begin{align}F = \mathbb{E}_t \left[ \Lambda_{t,t+1} \left( \left(\eta + \left(1-\eta\right) \left(\lambda x_{t}^s + \left(1-\lambda\right)x_{t}^r \right) \right) F + \pi_{t+1} \right) \right],\end{align}

which, using equations (12) and (13), implies the entry condition $V_t=F$ . That is, the households will fund new firms until the present value of future profits equals the cost of entry. We can also define the ex post gross rate of return to banks as

(19) \begin{align}R_t^L = r^* + \left( \lambda {x}_{t-1}^s \left(\tau^s_{t-1}-r^*\right) + \left(1-\lambda\right){x}_{t-1}^r \left( p_t^r \tau^r_{t-1} - r^*\right) \right)\frac{1}{\phi_{t-1}}.\end{align}

$ \phi_{t}\equiv \frac{S_{t}}{\left(1-\eta\right)f_{t} k} $ is the loan supply-demand ratio where $\left(1-\eta\right)f_{t} k$ is the capital sought by firms, and $S_t$ the household savings that the bank is intermediating. Free-entry in the banking sector then implies the zero arbitrage condition must hold:

(20) \begin{align}1 = \mathbb{E}_t \left[ \Lambda_{t,t+1} R_{t+1}^L \right]. \end{align}

Given that bank liabilities are risk-free deposits but assets are risky loans, it is possible for there to be ex post profits or losses in equilibrium. When there are profits, the household will receive a dividend, bailing out the banks when there are losses. Finally, it is assumed that the household utility function is in the form proposed in King et al. (Reference King, Plosser and Rebelo1988):

\begin{align}U \left(C_{t},H_{t}\right) = \frac{\left(C_t^{1-\chi} \left(1-H_t\right)^\chi\right)^{1-\sigma}}{1-\sigma} \notag.\end{align}

2.4 Market Clearing and Aggregation

Labor market clearing implies that total labor demanded by the three types of firm will equal the labor supplied by households, $H_t$ . An equal efficiency-capital-labor ratio follows from the perfect labor market and so, defining the aggregate efficiency capital as

(21) \begin{align}& \hat{K}_t \equiv \left[ \eta + \left(1-\eta\right) \left( \lambda x_{t-1}^s + \left(1-\lambda\right) x_{t-1}^r p_t^r \omega_t^r \right) \right] k f_{t-1}, \end{align}

we can write the aggregate labor demand equation as

\begin{align}W_t = \left(1-\alpha\right) z_t \left( \frac{\hat{K}_t}{H_t} \right)^{\alpha}. \notag\end{align}

We can likewise give aggregate output as $Y_t = z_t \hat{K}_{t}^{\alpha} H_t^{1-\alpha}$ , or rather, with aggregate productivity defined as a function of the ratio of efficiency-capital to total capital stock:

(22) \begin{align}A_t = z_t \left( \frac{\hat{K}_t}{K_{t-1}} \right)^{\alpha} , \end{align}

with the familiar looking aggregate production function

(23) \begin{align}Y_t = A_t {K}_{t-1}^{\alpha} H_t^{1-\alpha}\end{align}

that follows. Finally, we close the model with an aggregate resource constraint

(24) \begin{align}Y_t = C_t + I_t,\end{align}

where investment is the difference between the new capital stock, $K_t$ , and the sum of the depreciated returned capital and the undepreciated, unused capital

(25) \begin{align}I_t &= K_t - K_{t-1} + \delta \hat{K}_t - \left(1-\eta \right) \left(1-\lambda\right)x^r_{t-1} \left( p_t^r \omega^r_t - 1 \right)k f_{t-1}.\end{align}

3.1 Two Channels

To draw comparison with the RBC model, we can identify two channels by which financial disturbances affect real macroeconomic outcomes. The first is an “investment-wedge” channel, whereby the adverse selection affects the marginal efficiency of investment primarily through movements in the information rents. This inefficiency is measured by the spread between the savings rate and the return to capital which, using the average return on bank lending (19) and the firm lending rates (3)–(4), can be given by

(27) \begin{align}\Delta_t \equiv \mathbb{E}_t \left[\left(1-\lambda\right) \left( 1 - p_{t+1}^r\right) {x}_{t}^s R_{t+1}^s + \left( R_{t+1}^s - r^* \right) \left(\phi_t - \lambda x_t^s - \left(1-\lambda\right)x_t^r\right) \right] \frac{1}{\phi_t} . \end{align}

From this we can see that two factors contribute to this wedge: the information rents, measured by $ \left(1-\lambda\right) \left( 1 - p_{t+1}^r\right) {x}_{t}^s$ , and a capital-misallocation effect in the second term. This misallocation occurs when banks use their low-return technology, rationing credit to borrowers, as the average rate of return on lending must fall relative to the return on capital. Recall that $\phi_t$ is the loan supply-demand ratio, so if all household savings are intermediated to firms, it follows that the condition $\phi_t = \bar{x}_t = \lambda x_t^s + \left(1-\lambda\right)x_t^r$ holds and this effect disappears. The information rents increase in the expected default rate, and because banks can only reduce them by lowering $ x_t^s $ and rationing credit to firms with safe projects, one can see that if the default rate increases sufficiently, the contribution of the misallocation effect will rise.

The second channel is the efficiency wedge, whereby the credit friction generates movements in total factor productivity during the capital-misallocation regime. From equation (22), this can be written as

(28) \begin{align}A_t = z_t \left( \frac{\eta + \left(1-\eta\right) \left( \lambda x_{t-1}^s + \left(1-\lambda\right) x_{t-1}^r \right)}{ \eta + \left(1-\eta\right) \phi_{t-1} } \right)^{\alpha} \le z_t . \end{align}

If banks are intermediating all available funds, then, as before, $\phi_t = \bar{x}_t = \lambda x_t^s + \left(1-\lambda\right)x_t^r$ , and TFP just depends on exogenous technology $ z_t $ . When the adverse selection problem for the bank increases, due to increased risky-project firm default, for example, then banks restrict credit to firms by reducing $ x_t^s $ and $A_t$ falls. ^{Footnote 24}

4.1 Parametrization and Calibration

In addition to the parameters common to the real business cycle (RBC) literature, we are left with several parameters specific to the adverse selection economy. The size of firms is pinned down by the required capital, k; however, this has no effect on aggregate outcomes, and so we set $ k=1 $ without loss of generality. ^{Footnote 25} The share of corporate firms, $\eta$ , is set to 0.5 in line with the proportion of employment at establishments with greater than 500 employees. ^{Footnote 26} We calibrate $\lambda=0.775$ , $p=0.971$ , and $ F =0.149$ to target the proportion of risky bank loans, the mean firm entry rate, and the mean loan default rate. For the former, we target 24%, which is the average share of bank loans classified as “acceptable risk” over the interval 1997Q2–2017Q2. ^{Footnote 27} For the latter, we target a value of 2.8% per annum, taken from the average delinquency rate on commercial and industrial loans over the period 1987Q1–2017Q1. ^{Footnote 28} Finally, we target a mean annual firm entry rate of 12.5% in line with the average entry of US establishments over the period 1977–2014. ^{Footnote 29} We set $ r^* $ to 1 so the low-return asset is a storage technology. ^{Footnote 30}

These calibrations are listed in Table 1. For the remaining parameters, we closely follow the RBC literature. The capital share of output $\alpha = 0.3$ ; capital depreciates at $\delta = $ 2.3% per quarter; and the household discount factor $\beta =$ 0.99. The utility weight on leisure, $\chi = 0.64$ to target a steady-state labor supply $H=1/3$ , and the intertemporal elasticity of substitution, $\sigma = 2$ . These are all shown in Table 2. We calibrate the shock processes using a simulated method of moments approach; some further detail is given in the next section.

Table 1. Calibrations of adverse selection model parameters

Table 2. Parametrization of common real business cycle parameters.

4.2 Simulations

We compute a second-order pruned perturbation approximation to the model and impose the inequality constraints following the algorithm of Holden (Reference Holden2016). ^{Footnote 31} We draw comparison to the first-best economy, which is equivalent to the standard RBC model. ^{Footnote 32} To calibrate the persistence parameter, we estimate an autoregression of TFP with a linear trend, ^{Footnote 33} finding $ \rho_z = 0.978$ . The remaining parameters controlling the shock processes are calibrated to target second moments and cross-correlations. The standard deviation of the technology shock was calibrated to $ \sigma_a = 0.00619 $ , ^{Footnote 34} while the standard deviation and persistence of the risk shock were calibrated to $ \sigma_p = 0.00633 $ and $ \rho_p = 0.800 $ , respectively. ^{Footnote 35} We did initially include a shock to the relative value of risky projects, but this was calibrated to zero.

4.2.1 Unconditional moments

To gain some insight into the empirical performance of the model as compared to the financially efficient model, we report simulated and empirical moments in Table 3. The model does well at matching the observed skewness in output and investment despite not being targeted in the calibration. Including the risk shock reduces the procyclicality of consumption and leads to a negative correlation between the interest spread and output. Although a countercyclical response of consumption might seem to count against the model setup, the response is non-monotonic; for risk shocks large enough to cause financial crisis, because the mechanism maps to a decline in TFP, consumption can fall, as it would in the RBC model with a negative technology shock. The simulated moments reflect that the risk shock has no effect on the RBC model. Furthermore, although not targets in the calibration, the mean and standard deviation of the spread between the average rate of return on capital and the risk-free rate, $ \Delta_t $ , is 0.64 and 0.181 percentage points, respectively. This is close to 0.57 and 0.178 percentage points, which are the observed first and second moments of the spread between Moody’s BAA corporate bond and 10-year Treasury bond yields. ^{Footnote 36}

Table 3. Simulated and empirical moments. Data for Y, I, and C is HP-filtered US time series 1983Q2–2016Q2; investment wedge, $ \Delta $ , is the spread between Moody’s BAA-rated corporate bond yields and 10-Year Treasury Constant Maturity. Simulated time series of Y, I, and C are HP-filtered. Standard deviations are in percent for Y, I, and C and percentage points for $\Delta$ .

4.2.2 Impulse response functions

We now turn to the analysis of the propagation of the risk shock, which is an exogenous increase in the default rate of firms with risky projects, caused by a decline in the success probability, $p_t^r$ . ^{Footnote 37} The central result is that risk matters as a first-order issue. While the disturbance generates economic fluctuations in our model, the value of projects remain equal under symmetric information because $\omega_t^r = 1/p_t^r$ , leaving the first-best economy unaffected. Whereas without hidden information, the only important factor regarding firm finance is the expected discounted value, with adverse selection, the increased risk leads to higher information rents and so an increase in the investment wedge. Figure 3 shows impulse response functions to a 1 standard deviation risk shock, that is, an increase in the default rate of 0.63 percentage points. By widening the investment wedge, the increased default rate leads to a sharp 2% downturn in investment. Facing a lower interest rate, households substitute investment for consumption, dampening the overall fall in aggregate demand, which only shrinks by 0.2%. The share of risky loans increases as banks reduce funding to firms holding safe projects, allowing the banks to charge risky borrowers a higher repayment rate, $ \tau^r $ .

Figure 3. Impulse response functions to a 1 standard deviation (SD) transitory risk shock. Time is quarterly, and plots show percentage deviation from ergodic mean for Y, I, and C, and percentage point deviation for $ x^s $ , $ \Delta $ , and R.

Figure 4 shows expected impulse responses found by increasing the shock to reach the default threshold, $d_{t}^*$ . In this case, the probability of risky-project firm default increases by 3 percentage points, and, due to higher information rents, leads to banks rationing credit to firms with safe projects to charge firms with risky ones higher repayment rates. While the proportion of safe projects that are approved for finance, $ x^s $ , falls in both Figures 3 and 4, the former is a general equilibrium result caused by the fall in household saving being greater than the fall in firm numbers, whereas the latter is due, in part, to banks being unwilling to lend all available funds. This leads to a sharp decline in TFP and much sharper contractions in investment and output. Figure 4 also plots a version of the model with the TFP channel “switched off.” This allows us to assess the relative contribution from the endogenous variation in the investment wedge and TFP. For smaller shocks, as in Figure 3, the financial friction is affecting the real economy via the investment wedge, whereas for larger shocks, fluctuations can be mapped to both the investment wedge and TFP.

Figure 4. Impulse response functions to a transitory risk shock of 3% points comparing our model (black line) with a version with the TFP “switched off” (gray line). Time is quarterly, and plots show percentage deviation from ergodic mean for Y, I, C, and A, and percent point deviation for $ x^s $ and $ \Delta $ .

In this framework, the focus is on supply-side frictions. To model episodes such as the 2007–09 recession, it is necessary to add an exogenous demand-side disturbance. In Figure 5, we plot expected impulse response functions to a combination of the risk shock and a negative demand shock. For the latter, we employ an unexpected increase in $ \beta $ of 0.0015. ^{Footnote 38} The time preference shock occurs simultaneously with a 4.1% shock to the default of firms holding risky projects. ^{Footnote 39} The financial friction affects real outcomes via the same channels just discussed. Quantitatively, the crisis experiment can capture much of the observed movements in 2008. In the UK, for example, between 2008Q1 and 2009Q2, new loans to SMEs fell by 21%, real GDP fell by 6.13%, real investment by 21.8%, and real consumption by 5.89%. ^{Footnote 40} With the exception of consumption, the magnitudes of responses shown in Figure 5 are close to that in the data. Furthermore, the 3.3% decline in TFP closely matches that of the OECD measure of multifactor productivity for the UK over the same period, found to be 3.24%. ^{Footnote 41} Note that given the size of the contraction in investment, without the sharp fall in TFP, it would not be possible to generate the size of the decline in output. Other papers employ shortcuts to account for this issue, including exogenous TFP shocks (e.g. Christiano et al. (Reference Christiano, Eichenbaum and Trabandt2015)) and capital quality shocks (e.g. Gertler and Kiyotaki (Reference Gertler and Kiyotaki2010)).

Figure 5. Impulse response functions to a simultaneous time-preference shock and risk shock comparing our model (black line) with the RBC model (blue dashed). Time is quarterly, and plots show percentage deviation from ergodic mean for Y, I, C, and A, and percentage point deviation for $ x^s $ and $ \Delta $ .

4.3 Robustness

Some robustness checks of the parametrization were carried out on both the implied deterministic steady state and the model dynamics. The choice of parameters controlling preferences and production technology are standard; we focus on the novel parametrization, beginning with their impact on the steady-state equilibrium. Specifically, we test the parameter calibration by ignoring the target, choosing alternative values, but recalibrating the other parameters to hit the other calibration targets. Increasing the share of firms that have an observable state, $ \eta $ , dilutes the asymmetric information problem. The financial constraints in the banking sector are independent of $\eta $ , so the default threshold leading to credit tightening is unchanged. However, because the proportion of firms affected by adverse selection falls in $ \eta $ , the impact of credit crunches on aggregate outcomes weakens, and fluctuations in TFP are smaller. If we consider secular increases in $ \eta $ , ^{Footnote 42} holding other parameters constant, we find that, although having a smaller impact on the macroeconomy, credit crunches occur with higher frequency. Because new firms have an increased probability of being a corporate, and receiving surplus $ \mathbb{E}_t \left[ R_{t+1}^s-R_t \right]$ , firm entry goes up. The larger number of firms and, in particular, the larger proportion of observable-project corporates reduces the interest spread and the average return on capital. As highlighted in Proposition 2, a lower capital return shifts the default threshold down, so it takes a smaller rise in default to generate credit contractions.

The fixed cost of entry, F, is chosen to target the rate of firm entry. Increasing F will reduce firm entry and thus raise profits until the value of a new firm, $ V_t = F $ . Fewer firms will result in a higher return on capital and increased investment wedge. This would cause the default threshold, $ d^* $ , to shift down; however, with fewer firms seeking loans, the proportion of firms with safe projects that receive funds, $ x^s $ , increases, raising $ d^* $ . To hit the calibration target of the share of risky loans, $ \lambda $ is calibrated to a lower value so there are fewer safe projects in the economy. This moves $ d^* $ down again, reinforcing the effect of a higher return to capital and causing an overall increase in financial instability under higher entry costs. The combined effect, however, is fairly modest.

As would be expected given their role in the optimal contract, the calibrations of p and $\lambda$ do have a significant impact on both the stationary and dynamic equilibrium. If $\lambda$ is increased, the adverse selection problem weakens because, with fewer risky borrowers, the information rents are reduced. ^{Footnote 43} Furthermore, a lower $ \lambda $ or $ \bar{p} $ also imply great financial instability. ^{Footnote 44} Fewer safe projects imply higher information rents, shifting in the default threshold so credit contractions become more likely (see equation (26)). Likewise, a higher steady-state default rate would be closer to the threshold, $ d^* $ , so a smaller risk shock would be needed to reach it.

4.4 Instability and the Real Interest Rate

As stated in Proposition 2, the interest rate affects the likelihood of a credit contraction as the default threshold, $ d^* $ , above which the economy will be in the capital-misallocation regime, rises in the interest rate. Figure 6 plots the impulse response functions to a 1 standard deviation risk shock, as in Figure 3, but this time including a simulation with $ \beta = 0.993 $ , thus cutting $ \bar{R} $ by a bit more than 1% annualized. The reduced interest rate shifts $ d^* $ such that a 1 standard deviation shock is large enough to cause banks to restrict lending, leading to a sharp downturn.

Figure 6. Impulse response functions to a 1 SD transitory risk shock comparing baseline calibration (black line) with a low $ \bar{R} $ calibration (gray line). Time is quarterly, and plots show percentage deviation from ergodic mean for Y, I, and C, and percentage point deviation for $ x^s $ , $ \Delta $ , and R.

The result of financial instability with lower interest rates finds support in the data. Figure 7 plots 10-year rolling averages of the real interest rate and output volatility. ^{Footnote 45} There is a negative trend on the whole dataset; however, it is interesting to sort the data into three subsets. The red squares represent the middle episode, 1977–1987, which, by virtue of the rolling window, captures observations from 1972 and includes the impact of the 1973 oil crisis and heightened volatility in the 1970s and early 1980s. The green circles include data between 1988 and 2011, covering the Great Moderation, and the black diamonds represent observations between 1966 and 1976. The negative relationship between the real interest rate and volatility supports our results. Shifts in these curves are likely due to structural factors not in the model, such as the evolving size and nature of financial markets, but could also be partly explained by the share of small establishments, which has been in steady decline. ^{Footnote 46} A higher share of large firms would reduce the adverse selection and indicate a dampening of volatility and could, in part, lie behind the reduced volatility during the Great Moderation, off-setting the declining real interest rate.

Figure 7. Ten-year rolling average US real interest rates against 10-year rolling average US output volatility. 1966–1976 black diamonds; 1978–1987 red squares; and 1988–2011 green circles.

Figure 8. Average impulse response functions to a positive transitory shock to technology $z_t$ of 1% for our model (black line) and the RBC (blue dashed). Time is quarterly, and plots show percentage point deviation from ergodic mean for R and $ \Delta $ , and percent deviation for other variables.

Figure 9. Percent of reporting small and medium businesses that applied for loans but were denied credit. United Kingdom, 2011–2017. Four quarter moving average. Source: SME Finance Monitor, BDRC Continental.