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Appendix: Numerical Solution Method
We thank Rui Albuquerque, Kenza Benzima, Charles Engel, Tasos Karantounias, Tim Kehoe, Anton Korinek, Amartya Lahiri, Vincenzo Quadrini, and Jaume Ventura for useful comments and discussions. We also thank conference participants at the ITAM 2010 Summer Camp, MNB-CEPR Workshop on Financial Frictions, ECB Conference on “What Future for Financial Globalisation?”, The San Francisco Fed’s 2010 Pacific Basin Research Conference, and the Bank of Canada conference on “Financial Globalization and Financial Instability” and seminar participants at the Federal Reserve Board, Bank of England and the Philadelphia Fed.
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This is also related to the classic work on financial accelerators by Bernanke and Gertler (1989) and Kiyotaki and Moore (1997) and the more recent quantitative literature on this topic as in the work of Jermann and Quadrini (2009).
In a related paper Benigno et al. (2009) found that intervening during financial crisis by subsidizing nontradable goods leads to large welfare gains.
Galati and Moessner (2010) conduct an exhaustive survey of the growing literature in research and policy circles on macro-prudential policy.
They also examined the existence of deterministic cycles in a non-stochastic version of the model.
He provided analytical results for a canonical endowment economy model with a credit constraint where there is an exact equivalence between the two sets of allocations. In addition, he examined a model in which the exact equivalence of his first example does not hold, but still overborrowing is negligible.
We could also change to the standard setup, but in our calibration, θ = 0.14 and R = 1.028, and hence working capital loans would add 0.4 percent to the cost of labor implying that our findings would remain largely unchanged.
Notice that this effect vanishes when κ = 1, because when 100 percent of the value of land can be collateralized, the shadow value of relaxing the constraint by acquiring an extra unit of land equals the shadow value of relaxing it by reducing the debt by one unit.
We refer to the social planner’s equilibrium and constrained-efficient equilibrium interchangeably. Our focus is on second-best allocations, so when we refer to the social planner’s choices it should be understood that we mean the constrained social planner.
We could also allow the social planner to manipulate the borrowing ability state by state (i.e., by allowing the planner to alter
This implies that the social planner does not internalize the direct effects of choosing the contemporaneous labor allocation on contemporaneous wages. We have also investigated the possibility of having the planner internalize these effects but results are very similar. This occurs again because our calibrated interest rate and working capital requirement are very small.
See Bianchi (2009) for other decentralizations using capital and liquidity requirements and loan-to-value ratios.
αK represents the share of fixed assets in GDP, and not the standard share of capital income in GDP. There is little empirical evidence about the value of this parameter, with estimates that vary depending, for example, on whether we consider land used for residential or commercial purposes, or owned by government at different levels. We could also calibrate αK using the fact that, in a deterministic steady state where the collateral constraint does not bind, the value-of-land-GDP ratio is equal to αK/(R − 1), which would imply αK = 1.35(0.028) = 0.038. This yields very similar results as αK = 0.05.
The three crises correspond to the Great Depression, the Savings and Loans Crisis and the Great Recession (see Reinhart and Rogoff (2008)). While a century may be a short sample for estimating accurately the probability of a rare event in one country, Mendoza (2010) estimates a probability of about 3.6 percent for financial crises using a similar definition but applied to all emerging economies using data since 1980.
The choice of b’ becomes slightly higher for the social planner as b gets closer to the upper bound of the constrained region, because the deleveraging that occurs around this point is small enough for the probability of a binding credit constraint next period to be strictly positive. As a result, for given allocations, conditions (15) and (6) imply that μ is lower in the constrained-efficient allocations.
Measuring “ex ante” leverage as
Similarly, the fatter right-tail in the distribution of returns of the competitive equilibrium corresponds to periods with positive TFP shocks, which were preceded by low asset prices due to fire sales.
The literature on disasters typically uses Epstein-Zin preferences so as to be able to match the large observed equity premia. Here we use standard CRRA preferences with a risk aversion coefficient of 2, and as we show later, we can obtain larger risk premia than in the typical CRRA setup without credit frictions. Moreover, we obtain realistically large risk premia when the credit constraint binds.
Because the asset is in fixed supply, these allocations would be the same if we use instead an ad-hoc borrowing limit such that
The unconditional premium in the fixed price economy, at 0.86 percent, is not trivial, but note that it results from the fact that the constraint binds with very high probability, given the smaller incentives to accumulate precautionary savings. The risk premium in the unconstrained region of the fixed-price model is only 0.03 percent, v. 0.23 in our baseline model.
We could also define crises in the constrained-efficient equilibrium by using the value of the credit threshold obtained in the competitive equilibrium. However, with this criterion we would obtain an even lower probability of crises, because credit declines equal to at least one standard deviation of the first difference of credit in the decentralized equilibrium are zero-probability events in the constrained efficient equilibrium.
The sequence of TFP shocks is 0.9960, 0.9881, 0.9724, 0.9841, 0.9920 and the initial level of debt is 1.6 percent above the average.
The model overestimates the drop in credit relative to what we have observed so far in the U.S. crisis (which as of the third quarter of 2010 reached about 7 percent of GDP). One reason for this is that in the model, credit is in the form of one-period bonds, whereas in the data, loans have on average a much larger maturity. In addition, our model does not take into account the strong policy intervention that took place with the aim to prevent what would have been a larger credit crunch.
Two caveats on this point. First, at lower frequencies the correlation is positive. As Boz and Mendoza (2010) report, the household leverage ratio rose together with GDP, land prices and debt between 1997 and 2007. Second, the countercyclicality of leverage for the household sector differs sharply from the strong procyclicality of leverage in the financial sector (see Adrian and Shin (2010)).
If we reduce the debt tax we can obtain again average welfare gains, which again illustrates the interdependence of macroprudential policies.
To illustrate this point, we recomputed our model assuming that the borrowing constraint depends on the aggregate value of assets, as in their setup. Because assets do not have individual value as collateral, asset prices drop even more during crises, and this leads private agents to accumulate more precautionary savings, which results in crises having zero-probability in the long-run under both competitive and constrained efficient equilibria for our baseline calibration.
Notice that the probability of a crisis in the competitive equilibrium becomes 10 percent, more than three times larger than the target employed in the baseline calibration due to the reduction in the level of precautionary savings.
The increase in leisure mitigates the decrease in the stochastic discount factor but does not compensate for the fall in consumption
For the social planner’s allocations, we use the same algorithm operating on the planner’s optimality conditions.