Notes on Financial Frictions Under Asymmetric Information and Costly State Verification by Lawrence Christiano
Incorporating Financial Frictions into a Business Cycle Model General idea: Standard model assumes borrowers and lenders are the same people..no conflict of interest Financial friction models suppose borrowers and lenders are different people, with conflicting interests Financial frictions: features of the relationship between borrowers and lenders adopted to mitigate conflict of interest.
Discussion of Financial Frictions Simple model to illustrate the basic costly state verification (csv) model. Original analysis of Townsend (1978), Bernanke Gertler. Integrating the csv model into a full blown dsge model. Follows the lead of Bernanke, Gertler and Gilchrist (1999). Empirical analysis of Christiano, Motto and Rostagno (23,212).
Simple Model There are entrepreneurs with all different levels of wealth, N. Entrepreneur have different levels of wealth because they experienced different idiosyncratic shocks in the past. For each value of N, there are many entrepreneurs. In what follows, we will consider the interaction between entrepreneurs with a specific amount of N with competitive banks. Later, will consider the whole population of entrepreneurs, with every possible level of N.
Simple Model, cont d Each entrepreneur has access to a project with rate of return, Here, is a unit mean, idiosyncratic shock experienced by the individual entrepreneur after the project has been started, 1 R k df 1 The shock,, is privately observed by the entrepreneur. F is lognormal cumulative distribution function.
Banks, Households, Entrepreneurs ~ F, df 1 entrepreneur entrepreneur Households Bank entrepreneur entrepreneur entrepreneur Standard debt contract
Entrepreneur receives a contract from a bank, which specifies a rate of interest, Z, and a loan amount, B. If entrepreneur cannot make the interest payments, the bank pays a monitoring cost and takes everything. Total assets acquired by the entrepreneur: total assets net worth loans A N B Entrepreneur who experiences sufficiently bad luck,, loses everything.
Cutoff, gross rate of return experience by entrepreneur with luck, 1 R k interest and principle owed by the entrepreneur ZB 1 R k A ZB Z 1R k B N A N Z 1R k leverage L A 1 N A N Z 1R k total assets A L 1 L Cutoff higher with: higher leverage, L higher Z/1 R k
Expected return to entrepreneur from operating risky technology, over return from depositing net worth in bank: Expected payoff for entrepreneur 1R k A ZBdF N1R For lower values of, entrepreneur receives nothing limited liability. gain from depositing funds in bank ( opportunity cost of funds )
Rewriting entrepreneur s rate of return: 1 R k A ZBdF N1 R 1 R k A 1 R k AdF N1 R df 1 R k 1 R L Z 1R k L 1 Z L L 1R k Gets smaller with L Entrepreneur s return unbounded above Larger with L Risk neutral entrepreneur would always want to borrow an infinite amount (infinite leverage).
Rewriting entrepreneur s rate of return: 1 R k A ZBdF N1 R 1 R k A 1 R k AdF N1 R df 1 R k 1 R L Z 1R k L 1 L L Z 1R k Entrepreneur s return unbounded above Risk neutral entrepreneur would always want to borrow an infinite amount (infinite leverage).
Expected entrepreneurial return, over opportunity cost, N(1+R) 2 1.8 1.6 1.4 1.2 Equilibrium spread in numerical example developed in these notes. Entrepreneur would prefer to be a depositor for (leverage, interest rate spread), combinations that produce results below horizontal line. 1 2 4 6 8 1 12 14 leverage Interest rate spread, Z/(1+R), = 1.16, or.63 percent at annual rate Return spread, (1+R k )/(1+R), = 1.73, or 2.9 percent at annual rate.26
Expected entrepreneurial return, over opportunity cost, N(1+R) 2 1.8 1.6 High leverage always preferred eventually linearly increasing 1.4 1.2 Z/(1+R)=1.5, or 2 percent at annual rate 1 2 4 6 8 1 12 14 leverage Interest rate spread, Z/(1+R), = 1.16, or.63 percent at annual rate Return spread, (1+R k )/(1+R), = 1.73, or 2.9 percent at annual rate.26
If given a fixed interest rate, entrepreneur with risk neutral preferences would borrow an unbounded amount. In equilibrium, bank can t lend an infinite amount. This is why a loan contract must specify both an interest rate, Z, and a loan amount, B.
Simplified Representation of Entrepreneur Utility Utility: Where df 1 R k 1 R L 1 1 Rk 1 R L 1 F G Easy to show: G df 1 1 F, lim, lim lim G, lim G 1 Share of gross entrepreneurial earnings kept by entrepreneur
Banks Source of funds from households, at fixed rate, R Bank borrows B units of currency, lends proceeds to entrepreneurs. Provides entrepreneurs with standard debt contract, (Z,B)
Banks, cont d Monitoring cost for bankrupt entrepreneur with Bankruptcy cost parameter 1 R k A Bank zero profit condition fraction of entrepreneurs with 1 F quantity paid by each entrepreneur with ZB quantity recovered by bank from each bankrupt entrepreneur 1 df1 R k A amount owed to households by bank 1 RB
Zero profit condition: Banks, cont d 1 F ZB 1 df1 R k A 1 RB 1 F ZB 1 df1 R k A B 1 R The risk free interest rate here is equated to the average return on entrepreneurial projects. This is a source of inefficiency in the model. A benevolent planner would prefer that the market price savers correspond to the marginal return on projects (Christiano-Ikeda).
Banks, cont d Simplifying zero profit condition: 1 F ZB 1 df1 R k A 1 RB 1 F 1 R k A 1 df1 R k A 1 RB share of gross return, 1R k A, (net of monitoring costs) given to bank 1 F 1 df 1 R k A 1 RB 1 F 1 df Expressed naturally in terms of Expressed naturally in terms of,l,l 1 R B/N 1 R k A/N 1 R L 1 1 R k L
Bank zero profit condition, in (leverage, - bar) space 2 1.8 1.6 1.4 parameters: 1 Rk 1 R 1.73,. 21,.26 - bar 1.2 1.8.6.4.2 1.5 2 2.5 3 3.5 4 4.5 5 leverage Our value of 1 Rk, 29 basis points at an annual rate, is a little higher than the 2 basis point value adopted in 1 R BGG (1999, p. 1368); the value of is higher than the one adopted by BGG, but within the range,.2-.36 defended by Carlstrom and Fuerst (AER, 1997) as empirically relevant; the value of Varlog is nearly the same as the.28 value assumed by BGG (1999,p.1368).
Bank zero profit condition, in (leverage, - bar) space 2 1.8 Free entry of banks ensures zero profits 1.6 1.4 zero profit curve represents a menu of contracts,,l, that can be offered in equilibrium. - bar 1.2 1.8 Only the upward sloped portion of the curve is relevant, because entrepreneurs would never select a high value of if a lower one was available at the same leverage..6.4.2 1.5 2 2.5 3 3.5 4 4.5 5 leverage
Expressing Zero Profit Condition In Terms of New Notation share of entrepreneurial profits (net of monitoring costs) given to bank 1 F 1 df 1 R L 1 1 R k L G 1 R L 1 1 R k L L 1 1 1Rk G 1R
Equilibrium Contract Entrepreneur selects the contract is optimal, given the available menu of contracts. The solution to the entrepreneur problem is the that maximizes, over the relevant domain (i.e.,,1.13 in the example): profits, per unit of leverage, earned by entrepreneur, given leverage offered by bank, conditional on log df 1 R k 1 R 1 1Rk 1R 1 G higer drives share of profits to entrepreneur down (bad!) higher drives leverage up (good!) log 1 log 1 Rk 1 R log 1 1 Rk 1 R G
Entrepreneur Objective entrepreneurial utility 1.12 1.1 1.8 1.6 1.4 1.2 entrepreneurial objective as a function of bar entrepreneurial utility 1.9.8.7.6.5 entrepreneurial objective as a function of bar 1.4.25.3.35.4.45.5.55.6.65 bar.1.2.3.4.5.6.7.8.9 1 1.1 bar derivative of log entrepreneurial objective -.5 derivative of log entrepreneurial objective -.5-1 -.1 -.15-1.5-2 -2.5 relevant range of s -3 -.2-3.5 -.25-4 -4.5.25.3.35.4.45.5.55.6.65 bar.1.2.3.4.5.6.7.8.9 1 1.1 bar
Computing the Equilibrium Contract Solve first order optimality condition uniquely for the cutoff, : elasticity of entrepreneur s expected return w.r.t. 1 F 1 elasticity of leverage w.r.t. 1R k 1 F 1R F 1 1Rk G 1R Given the cutoff, solve for leverage: L 1 1 1Rk 1R G Given leverage and cutoff, solve for risk spread: risk spread Z 1R 1Rk 1R L L 1
Result Leverage, L, and entrepreneurial rate of interest, Z, not a function of net worth, N. Quantity of loans proportional to net worth: L A N N B N B L 1N 1 B N To compute L, Z/(1+R), must make assumptions about F and parameters. 1 R k 1 R,, F
Formulas Needed to do the Computations Need: G df, F Can get these from the pdf and the cdf of the standard normal distribution. These are available in most computational software, like MATLAB. Also, they have simple analytic representations.
Numerical Example Parameters: 1 R k 1 R Percent of average product of entrepreneurial Projects, absorbed by monitoring costs:.6% 1.73,. 26,.21 (Micro) equilibrium quantities: cutoff fraction of gross entrepreneurial earnings going to lender bankruptcy rate:.56% average among bankrupt entrepreneurs.5,.58, F.56, G.26, leverage L 2.2, interest rate spread ZR.62 (APR) 1.15, avg earnings of entrepreneur, divided by opportunity cost 1 1 Rk 1 R L 1.135 1 Note: on average, entrepreneur better off leveraging net worth and investing in project, rather than depositing net worth in bank.
Effect of Increase in Risk, Keep df 1 But, double standard deviation of Normal underlying F. Impact on log normal density of doubling standard deviation.9.8.7.6 density.5.4 Doubled standard deviation.3.2.1 Increasing standard deviation raises density in the tails..5 1 1.5 2 2.5
Jump in Risk replaced by 3 cutoff fraction of gross entrepreneurial earnings going to lender bankruptcy rate: 1.8% average among bankrupt entrepreneurs. 12,. 12, F.18, G.11, leverage L 1.1418, interest rate spread ZR 1.66 (APR) 1. 41, avg earnings of entrepreneur, per unit of net worth 1 1 Rk 1 R L 1. 8 1 Comparison with benchmark: cutoff fraction of gross entrepreneurial earnings going to lender bankruptcy rate:.56% average among bankrupt entrepreneurs.5,.58, F.56, G.26, leverage L 2.2, interest rate spread ZR.62 (APR) 1.15, avg earnings of entrepreneur, divided by opportunity cost 1 1 Rk 1 R L 1.135 1
1 Impact on standard debt contract of a 5% jump in risk spread, 4(Z/R-1).9.8.7.6.5 Risk spread = 4 Z, Leverage = (B+N)/N 1 R 1 Zero profit curve Risk spread=.635 Leverage = 1.95 Risk spread=.616 Leverage = 2.2 Entrepreneur Indifference curve.4 1.9 1.95 2 2.5 leverage, L = (B+N)/N
Incorporating CSV Financial Frictions into Neoclassical Model Bernanke, Gertler and Gilchrist, 1999 Handbook of Macroeconomics Chapter. Outline Broad overview of model. Details of entrepreneur Must take into account the heterogeneity of entrepreneurs by net worth, N. Describe relationship of entrepreneur to household Describe household and general equilibrium.
Standard, Neoclassical Model max c t,b t1,l t t subject to: t t uc t, uc t logc t c t B t1 K t1 1 K t w t l t r t K t 1 R t 1 B t l t 1 Optimization: u c t u c t1 r t1 1, u c t u c t1 1 R t l t 1 Firms and market clearing: c t I t K t l 1 t, B t1, w t 1 K 1 t, r t K t
Standard Model L Firms K Labor market C I Market for Physical Capital household K 1 K I
Standard Model with CSV L Firms K K, ~ F Labor market Observed by entrepreneur, but supplier of funds must pay monitoring cost to see it. Entrepreneurs household
Standard Model with CSV L Firms K K, ~ F Labor market C I Capital Producers Entrepreneurs K 1 K I household Entrepreneurs sell their K to capital producers
Standard Model with CSV L Firms K K, ~F, t Labor market C I Capital Producers Entrepreneurs household Entrepreneurial net worth now established. = value of capital + earnings from capital -repayment of bank loans
Standard Model with CSV Firms K K, ~F, t Labor market Capital Producers K Entrepreneurs household banks Entrepreneur receives standard debt contract.
Details About the Entrepreneur Begin after period t production, when entrepreneurial net worth is known, and they go to banks for loans. End at the point in t+1 when net worth in t+1 is determined.
Entrepreneur at end of t Let f t N denote the number (density, actually) of entrepreneurs with net worth, N, N. An entrepreneur with net worth, N, goes to the bank, receives a loan, B N t+1, and buys raw, physical capital: N K t1 N N B t1 After purchasing the capital, the entrepreneur experiences an idiosyncratic shock, ω, so that physical capital is transformed into effective capital as follows: K N t1, ~ F, t
Entrepreneur in t+1 The entrepreneur with net worth N in t rents N K t1 in a competitive capital rental market for rental rate, r t+1. After goods production, entrepreneur sells undepreciated capital, K N t1 1, at price unity. So, rate of return on capital for entrepreneur is: k 1 R t1 k, 1 R t1 Constant rate of return project, just like in micro example r t1 1
Entrepreneur in t+1, cnt d Period t+1 net worth of entrepreneur with net worth N in period t and who experiences shock ω: Gross rate of interest on loan N k max,1 R t1 K N t1 Z t1 B N t1
Entrepreneurial loan contract in t The banking system in period t is competitive. The zero profit condition must be satisfied: Leverage K N t1 Here: k 1 R t1 1 N k t1 G t1 1 1R t1 1R t K N t1 t1 Z t1 B N t1, Entrepreneurs treat the zero profit condition as a menu of contracts. They select the contract at time t that maximizes expected N given N. Entrepreneurial efficiency condition: 1 F t1 1 t1 k 1R t1 1R t 1 F t1 t1 F t1 k 1 1R t1 1R t t1 G t1
Aggregate Net Worth, Loans and Capital The total amount of net worth held by all entrepreneurs at the end of time t: N t1 Nf t NdN Total effective capital supplied by all entrepreneurs during t+1: N Kt1 f t NdFdN total effective capital of all entrepreneurs with net worth, N N Kt1 df f t NdN Must aggregate over all N types and all ω types total effective capital for all entrepreneurs N Kt1 f t NdN
Aggregate Net Worth, Loans and Capital The total amount of net worth held by all entrepreneurs at the end of time t: N t1 Nf t NdN Total effective capital supplied by all entrepreneurs during t+1: N Kt1 f t NdFdN total effective capital of all entrepreneurs with net worth, N N Kt1 df f t NdN This is just K N t+1 because total effective an entrepreneur capital for all entrepreneurs with net worth, N, N Kt1 observes f t NdNω after selecting the loan contract, and, hence, the quantity of capital purchased.
Aggregate Net Worth, Loans and Capital The total amount of net worth held by all entrepreneurs at the end of time t: N t1 Nf t NdN Total effective capital supplied by all entrepreneurs during t+1: N Kt1 f t NdFdN total effective capital of all entrepreneurs with net worth, N N Kt1 df f t NdN total effective capital for all entrepreneurs N Kt1 f t NdN
Aggregate capital: K t1 N Kt1 Aggregates, cont d Aggregate net worth in t+1: max,1 k Rt1 f t NdFdN N Kt1 f t NdN 1 1 1R k t1 t1 G t1 1R t 1 1 1R k t1 t1 G t1 1R t Nft NdN N t1 K N t1 Z t1 B N t1 df f t NdN k 1 t1 1 R t1 K t1
Relationship of Entrepreneur to Household Simplest assumption, which avoids a lot of complications, is the large family assumption. There are many identical households Each has a worker. Each has many entrepreneurs.enough so that average net worth in the representative family is always equal to average net worth in the economy as a whole. Entrepreneurs receive perfect consumption insurance from the household: Entrepreneurs and the worker all consume the same amount, C t.
Large Family Assumption household household worker mass of entrepreneurs worker mass of entrepreneurs Competitive market for capital services Competitive labor market
Entrepreneurs and Households, cnt d Aggregate net worth of all entrepreneurs, after accounting for all their income in t+1. k 1 t1 1 R t1 Then, a fraction, 1 ϒ, of each entrepreneur s net worth is transferred to the household as a lump sum. the household transfers resources, W e t+1, as a lump sum to each entrepreneur. So, net worth of all entrepreneurs at the end of t+1 is: k N t2 1 t1 1 R t1 K t1 e K t1 W t1 The entrepreneurs as a whole take this net worth to the bank in t+1, get loans and buy capital, and so on.
One day in life of entrepreneur Period t Entrepreneur supplies capital to capital rental market Period t+1 Entrepreneur pays off debt to bank, determines current net worth. Fraction, 1-γ, of each entrepreneurs net worth transferred to households. Each entrepreneur receives a lump-sum transfer, W e, from household. * End of period t: Using net worth, N t+1, and loans, entrepreneur purchases new, end-of-period stock of capital from capital goods producers. Entrepreneur observes idiosyncratic disturbance to its newly purchased capital. Entrepreneur sells undepreciated capital to capital producers Density of entrepreneurs having net worth, N: f t1 N Entrepreneurs go to *.
Other Equations of the Model Goods market clearing (resource constraint): goods bought by households to feed their entrepreneurs and workers ct investment goods bought by capital producers It goods bought for monitoring by banks t df1 Rt k K t Capital producers technology: K t1 1 K t I t production function Kt
Other Equations Household problem: max c t,b t1,l t t subject to: t t uc t, uc t logc t 1 1 t 1R t k K t W t e c t B t1 w t l t 1 R t 1 B t l t 1 transfers from entrepreneurs Optimization: u c t u c t1 1 R t, l t 1 Bond market clearing: borrowing by entrepreneurs K t1 N t1 lending by households Bt1
Result There are seven aggregate variables among the unknowns: c t,i t, t,r t k,k t, N t,r t There are seven dynamic equilibrium conditions that can be used to pin the down (see next slide). Solution does not require knowing f t N Reflects constant returns to scale in entrepreneurial projects and entrepreneurial objective. Otherwise, leverage and interest rate in standard debt contract would be a function of N. In that case, what entrepreneurs as a group do depends on the distribution of N. Constant return to scale assumptions massively simplify the analysis. Whether this entails significant distortions in conclusions is an interesting issue to explore.
In practice, monitoring costs small Summary of Equations Equilibrium conditions familiar from standard neoclassical model resource constraint: c t I t t df1 R t k K t K t household fonc: u c t u c t1 1 R t In neoclassical model Capital accumulation: K t1 1 K t I t these are the same. Rate of return on capital: 1 R k t K 1 t 1 Financial frictions introduce a wedge. Equilibrium conditions specific to financial frictions entrepreneur fonc: 1 F t1 1 t1 zero profit condition: K t 1 1 1R t k Aggregate Net Worth: 1Rk t1 1R t 1 1R k t1 1R t 1 F t1 t1 F t1 N t 1R t G t t 1 t1 G t1 N t1 1 t 1 R t k K t W t e Large family assumption has consequence that model focuses exclusively on the implications of financial frictions for distortions in the intertemporal margin, abstracting from all other implications, such as for distribution of income between entrepreneurs and others.
Financial Friction as Intertemporal Wedge Equilibrium conditions familiar from standard neoclassical model resource constraint: household fonc: Capital accumulation: Rate of return on capital: entrepreneur fonc: c t I t t df1 R t k K t K t u c t u k c t1 1 R t1 1 t1 K t1 1 K t I t 1 R t k K t 1 1 Equilibrium conditions specific to financial frictions 1 F t1 1 t1 wedge t1 1 1R t k 1R t1 zero profit condition: K t 1 1 1R t k Aggregate Net Worth: 1R k t1 1R t 1 1R k t1 1R t 1 F t1 t1 F t1 t1 G t1 1 1 F t1 t1 F t1 1 F t1 N t 1R t G t t 1 1 t1 N t1 1 t 1 R t k K t W t e t1 G t1 Looks just like neoclassical model with a particular tax on capital income.
Solving the Model Perturbation methods can be applied. First, require steady state of the model. Then, linearize equilibrium conditions about the steady state and solve. Parameterization: uc logc,.26,.21,.97, 1/3,.2, 1.3 1/4, w e.1 log t /.97log t 1 / t Cross sectional dispersion evolves according to a first order autoregressive process.
Solving the Model Endogenous variables in steady state:.5286, G.49, F.1,.5282, n 11.65 G.5271, c.77, k n 4.6, k c i c i n 2.14, k 8.62 c i 4 Z 1 R 1 1.11, 1 1 Rk n k.9991, wedge.114 Financial distortion: without financial frictions: K 1 1 1 1 42.4, c 2.64 with financial frictions: K 1 1 wedge 1 1 1 25., c 2.42 steady state welfare cost of the financial frictions: 8.9 percent of consumption (1(2.64-2.42)/2.42)
Finding the steady state Delete time subscripts from equations. Use household fonc to set 1 R 1/ Fix an initial guess, R k R Compute (and, hence, G, F and from entrepreneur fonc Compute K from rate of return on capital 1 expression: K 1 R k 1 Compute N from net worth equation. Adjust R k until zero profit condition is satisfied. Compute c from the resource constraint. 1.
1% jump Response to.1 jump in t, where log t /. 97 log t 1 / t model parameters:.26,. 21,. 97, 1/3,. 2, 1.3 1/4, 1. % deviation from ss GDP (c + i) -.1 -.2 -.3 -.4 -.5 5 1 15 % deviation from ss investment -2-4 -6-8 -1 5 1 15 % deviation from ss 2 1.5 1.5 consumption -.5 5 1 15 dev from ss (APR) risk free rate -.5-1 -1.5 5 1 15 interest rate spread return on capital net worth x 1-3 wedge, 1-(1+R)/(1+Rk(+1)) dev from ss (APR).5.4.3.2.1 dev from ss (APR).1.5 % deviation from ss 2 1.5 1.5 deviation from ss 4 3 2 1 5 1 15 5 1 15 5 1 15 5 1 15 bankruptcy rate credit 1 sigma capital stock deviation *1.4.3.2.1 5 1 15 deviation from ss -1-2 -3-4 5 1 15 % deviation from ss 9 8 7 6 5 1 15 % deviation from ss -.4 -.6 -.8-1 -1.2 5 1 15
1% jump Response to.1 jump in t, where log t /. 97 log t 1 / t model parameters:.26,. 21,. 97, 1/3,. 2, 1.3 1/4, 1. Suggests (but see later slide) that risk not likely to be important in business cycles % deviation from ss GDP (c + i) -.1 -.2 -.3 -.4 -.5 5 1 15 % deviation from ss investment -2-4 -6-8 -1 5 1 15 % deviation from ss 2 1.5 1.5 consumption -.5 5 1 15 dev from ss (APR) risk free rate -.5-1 -1.5 5 1 15 interest rate spread return on capital net worth x 1-3 wedge, 1-(1+R)/(1+Rk(+1)) dev from ss (APR).5.4.3.2.1 dev from ss (APR).1.5 % deviation from ss 2 1.5 1.5 Stock market deviation from ss 4 3 2 1 5 1 15 5 1 15 5 1 15 5 1 15 bankruptcy rate credit 1 sigma capital stock deviation *1.4.3.2.1 5 1 15 deviation from ss -1-2 -3-4 5 1 15 % deviation from ss 9 8 7 6 5 1 15 % deviation from ss -.4 -.6 -.8-1 -1.2 5 1 15
% deviation from ss -.1 -.2 -.3 -.4 -.5 GDP (c + i) 5 1 15 % deviation from ss -2-4 -6-8 -1 investment 5 1 15 % deviation from ss 2 1.5 1.5 1% jump Response to.1 jump in t, where log t /. 97 log t 1 / t model parameters:.26,. 21,. 97, 1/3,. 2, 1.3 1/4, 1. The mechanism by which the shock moves economic variables operates through the intertemporal wedge. consumption -.5 5 1 15 dev from ss (APR) -.5-1 -1.5 risk free rate 5 1 15 interest rate spread return on capital net worth x 1-3 wedge, 1-(1+R)/(1+Rk(+1)) dev from ss (APR).5.4.3.2.1 dev from ss (APR).1.5 % deviation from ss 2 1.5 1.5 Stock market deviation from ss 4 3 2 1 In steady state =.11 5 1 15 5 1 15 5 1 15 5 1 15 bankruptcy rate credit 1 sigma capital stock deviation *1.4.3.2.1 5 1 15 deviation from ss -1-2 -3-4 5 1 15 % deviation from ss 9 8 7 6 5 1 15 % deviation from ss -.4 -.6 -.8-1 -1.2 5 1 15