Appendix I. The Lending Problem
The lending problem results from the following maximization of the entrepreneurs’ payoff:
for all states in t + 1, where Bt = Qtkt+1—netnt is the domestic loan to the entrepreneurs with net worth netnt.
The first order conditions with respect to kt+1 and
where ρ(·) is a risk premium defined as:
To express in real terms we can define payrt = paynt/Pt and (qrt = Qt/Pt and re-write (A.4) as
where 1 + πt+1 = Pt+1/Pt. Similarly, defining entrepreneurs’ debt and net worth in real terms (bt = Bt/Pt and netrt = netnt/Pt), we can have the following expressions for the recovery rate, net worth evolution and entrepreneurs consumption:
Appendix II. The Problem of Liquidity Intermediaries
As noted in the main text, the optimization’s problem of the liquidity intermediaries is:
The Lagragian of the problem is:
where Ft and are Gt the lagrange multipliers in constraint (B.1) and (B.2), respectively. These lagrange multipliers can be interpreted as the marginal cost of providing liquidation services from final goods and excess reserves. Thus, the first order conditions of the liquidity intermediaries’ problem are:
The (real) fire sales price can be expressed as the marginal present discounted value of the defaulted capital, which can be obtained through:
where mglqt is the marginal cost of liquidation services per unit of defaulted capital, sdt,t+1 is the stochastic discount factor and
Using the Lagrange multipliers of constraints (B.1) and (B.2) we obtain an expression for the marginal cost of liquidation services, mglqt = v(ft + gt). Taking derivative with respect to kD,t in both sides of (B.8) we obtain an expression for fire sales price, which is net of the cost of liquidation and takes account that a fraction of the defaulted capital is becoming productive each period:
Appendix III. Price Rigidities, the Phillips Curve, and Aggregation of Final Goods Demand
There is one final good produced using the intermediate composite goods:
where ∊ is the elasticity of substitution across the composite intermediate goods and dat is total domestic demand. The final good market is perfectly competitive and the demand for each intermediate composite good i is given by
where Pi,t is the price of the intermediate composite good i. The aggregate price level of domestic demand is then:
Each intermediate composite producer has the same technology:
where yi,t and yf,i,t are, respectively, the amount of domestic and foreign goods used by the intermediate composite producer i. The cost minimization implies
And the marginal cost (expressed in real terms) is
which is the same for all intermediate goods producers, because they face the same prices of domestic and foreign goods and their technology is constant return to scale. For the same reason, we can obtain:
where we have used the fact that total demand for domestic goods is composed by the demand of intermediate composite producers and exports, xt. The intermediate composite good producers set prices following Calvo’s (1983) mechanism of price adjustment. In each period, a fraction 1—Φp of the producers can change optimally their prices. All other producers can only index their prices to past inflation with a weight χP. Thus, the problem for the intermediate composite producers i is the following:
The first order condition of this problem is:
Defining the following expressions in recursive manner:
the optimal condition for price Pi,t (C.8) can be written as:
Using Calvo’s pricing mechanism; we can express the price level aggregation as:
Finally, the relationship between domestic demand and supply of final goods is given by
where dispt ≥ 1 is the inefficiency attributed to price dispersion and yS,t is the aggregate supply of the composite goods, defined as:
Again using the properties of Calvo’s pricing mechanism, this price dispersion term evolves as:
Appendix IV. Complete Set of Equilibrium Conditions
The equilibrium for the model economy, given macroeconomic policy rules for
Appendix V. Extension with Foreign Funding for Lending Intermediaries and Entrepreneurs
In contrast to other studies, and to boost balance sheet effects, we can allow for the possibility that the financial intermediaries use foreign funds to finance entrepreneurs besides the interbank market. For simplicity, we assume the amount of external funds available in each period for the lending intermediaries is constant in foreign currency
Thus, the loan contract now solves the following maximization problem
for all states in t + 1
Thus, the equilibrium condition for the loan contract and entrepreneurs’ variables are stated as follows.
- Arbitrage condition for the loans to entrepreneurs:
- Definition of the recovery rate of financial intermediaries’ loans:
- Definition of the risk premium:
- Budget constraints of entrepreneurs:
- Break-even condition for financial intermediaries:
- Real net worth of entrepreneurs:
Again, all entrepreneurs receive a lump-sum transfers, τE, bit now they also make an interest payment for the constant amount of foreign debt
Appendix VI. Financial Dollarization of the Entrepreneurs’ Loans
Another modification of the baseline model can be the possibility that the loan contract to the entrepreneurs is set in or indexed to the foreign currency (dollars). Under this situation, we define ψ ∈ [0,1] as the fraction of the loan set in domestic currency and 1—ψ as the fraction of the loan in foreign currency. For the exercise reported we consider ψ = 0.5.
Thus, the loan contract now solves the following problem:
for all states in t + 1,
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Changes in average reserve requirements in Colombia underestimate the actual impact because they don’t capture changes in marginal rates and remuneration that increase the effectiveness of these measures (Vargas et al, 2010).
The importance of shocks to world interest rates for emerging market business cycles has been emphasized in Neumeyer and Perri (2005).
Real resources are needed to conduct due diligence, assess future cash-flows of failed capital, and return to productive use. “Excess reserves” are the financial or liquid resources needed to buy that capital or distressed assets. As noted by Gorton and Huang (2004) there are many notions of “liquidity” and they mostly refer to situations where not all assets can be used to buy all other assets at a point in time. This amounts to a “liquidity-in-advance” constraint, as summarized in the technology below.
The use of reduced-form technologies to produce financial services is common in monetary policy models (for instance, Chari, Christiano, and Eichenbaum (1995), Edwards and Vegh (1997), Goodfriend and McCallum (2007), Christiano, Motto and Rostagno (2010) and Curdia and Woodford (2010)).
Edwards and Vegh (1997) demonstrate the desirability of using a countercyclical reserve requirement in the context of a fixed-exchange-rate regime; however, they assume that the reserve requirement moves directly with foreign interest rates rather than with domestic financial conditions.
The model is calibrated to resemble a prototypical emerging market economy such as the ones in Figure 1.
This is akin to “leaning-against-the-wind,” although the expression could be applied more broadly to responses to asset prices and other indicators of financial conditions.
Bianchi (2010) demonstrates that, for a very generic bank balance sheet, capital and reserve requirements have similar effects (see also Benigno, 2012). Agénor et al (2013) study interactions between interest rate rules and a Basel III-type countercyclical capital regulatory rule in the management of housing demand shocks.
Gray (2010) noted that reserves are used to smooth settlement of transactions and respond to unexpected deposit withdrawals. When reserves are kept for prudential purposes, they could be held not just with vault cash and deposits at the central bank, but also liquid treasury securities.
The four countries in Figure 1, hold on average 12.4 percent of total assets in government securities, but there is wide variation across countries; for Brazil and Turkey the figure is around 20-24 percent, while for Colombia and Peru is around 2-4 percent (averages for the period 1998-2012). It is worth noting that this holding of government securities has been reduced in Brazil since 2006, reaching around 13 percent of total asset in 2012.
In the case of Peru, the sum of government securities and cash held by commercial banks in the period 2003–2012 was equivalent to 25 percent of deposits, while the effective reserve requirement was about 23.5 percent. In the same period, excess reserves were 0.35 percent of deposits.
This higher default rate could be rationalized as the “distress” rate obtained in estimates from credit default swap rates (see Hull, Predescu, and White, 2004)
This calibration implies the following for steady-state rateS: Rk = 38.5% > RL = 15.8% > RIB = 4.5% > RD = 4%