A Model of Financial Intermediation

Transcription

1 A Model of Financial Intermediation Jesús Fernández-Villaverde University of Pennsylvania December 25, 2012 Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

2 A Model with Financial Intermediation Previous models have a very streamlined financial intermediation structure. Many of the events of the recession were about breakdowns in intermediation. Kiyotaki and Gertler (2011) incorporate a richer financial intermediation sector. In particular, we will deal with liquidity. Different concepts of liquidity. Help us to think about unconventional monetary policy. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

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6 Representative Household Preferences: ( ) E 0 β t log (c t γc t 1 ) χ l t 1+ϑ t=0 1 + ϑ Household can save in: 1 Deposits at the financial intermediary, d t, that pay an uncontingent nominal gross interest rate R t. 2 Public debt, d gt, that pay an uncontingent nominal gross interest rate R t. 3 Arrow securities (net zero supply in equilibrium). The budget constraint is then: where d h,t = d t + d gt. c t + d h,t = w t l t + R t 1 d h,t 1 + T t + Ϝ t Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

7 Representative Household Continuum of members of measure one with perfect consumption insurance within the family. A fraction 1 f are workers and f are bankers. Workers work and send wages back to the family. Bankers run a bank that sends (non-negative) dividends back to the family. In each period, a fraction (1 σ) of bankers become workers and a fraction (1 σ) of workers become bankers. Why? f 1 f Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

8 Optimality Conditions The first-order conditions for the household are: 1 γ βe t = λ t c t γc t 1 c t+1 γc t λ t = βe t λ t+1 R t χl ϑ t = λ t w t Asset pricing kernel: SDF t = β λ t λ t 1 and standard non-arbitrage conditions. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

9 Technology Island model: continuum of islands of measure 1. In each island, there is a firm that produces the final good with capital (not mobile) and labor (mobile across islands) and a Cobb-Douglas production function. Then, by equating the capital-labor ratio across islands, aggregate output is: y t = A t k α t l 1 α t where A t is a random variable. Wages satisfy: w t = (1 α) y t l t Profits per unit of capital: z t = y t w t l t k t Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

10 Liquidity Risk Each period, investment opportunities arrive randomly to a fraction π i of islands. There is no opportunity in π n = 1 π i. Investment opportunities are i.i.d. across time and islands. Only firms in islands with investment opportunities can accumulate capital. Then: k t+1 = [ ψ t+1 π i ] (1 δ) k t + i t + ψt+1 π n (1 δ) k t = ψ t+1 [(1 δ) k t + i t ] where ψ t+1 is a shock to productivity of capital. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

11 Capital Good Producers Adjustment costs in investment: f ( it i t 1 with f (1) = f (1) = 0 and f ( i t i t 1 ) > 0. Relative price of capital: q i t. Capital good producers: max E 0 β t λ t t=0 λ 0 Optimality condition: ( ) qt i it = 1 + f + i t f i t 1 i t 1 ) ( ( ( )) ) qt i it 1 + f i t i t 1 ( it i t 1 ) E t β λ t+1 λ t ( it+1 i t ) 2 ( ) f it+1 i t Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

12 Aggregate Resource Constraint Government consumption g t. Then: ( ( )) it y t = c t f i t + g t i t 1 We will also have, later on, a wide set of policies, that will imply a relatively involved government budget constraint. I will skip details because it is mere accounting. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

13 Banks I Banks are born with a small initial transfer from the family. Initial equity is increased with retained earnings. Dividends are only distributed when the bank dies. Banks are attached to a particular island, which in this period may be h = {i, n}. Thus, ex post, they may not be able to lend wholesale market. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

14 Banks II Banks move across islands over time to equate expected rate of return: 1 Before moving, they sell their loans. 2 This allows us to forget about distributions: ratio of total financial intermediary net worth to total capital is the same in each island. Discussion: specificity in bank relational capital? Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

15 Balance Sheet Net worth: n h t. Besides equity, banks raise funds in a national financial market: 1 Retail market: from the households, d t at cost R t. Before investment shock is realized. 2 Wholesale market: from each other, b h t at cost R bt. After investment shock is realized. Then, bank lend to non-financial firms in their island st h. No enforcement problem (we can think about st h as equity). Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

16 Flow-of-Funds I Balance sheet constraint (where q h t is the price of a loan): q h t s h t = n h t + b h t + d t Evolution of net worth: n h t = [ z t + (1 δ) q h t Objective function of bank: V t = E t ] ψ t s h t 1 R t 1 d t 1 R bt 1 b h t 1 (1 σ) σ i 1 β i λ t+i nt+i h i=1 λ t Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

17 Flow-of-Funds II Value function: maximized objective function: = E t β t λ t+i λ t ( ) V t st h, bt h, d t = max V t { π h (1 σ) nt+1 h +σ max h dt+1 max s h t,bt h V ( t s h t+1, bt+1 h, d t+1) } Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

18 Financial Friction I We need some financial friction to make the intermediation problem interesting. Simply costly-enforcement mechanism. Diversion of funds to family: default, and close down. θ ( ) qt h st h ωbt h Interpretation of ω. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

19 Financial Friction II Three cases: 1 ω = 1 (frictionless interbank market). The interbank and loan rates are the same. They are bigger than the deposit rate if banks are constrained (only one aggregate constrain holds). 2 ω = 0 (symmetric frictions). The interbank and deposit rate are the same. The returns on loans if banks on non-investing islands are constrained. 3 ω (0, 1). The interbank rate lies between the return on loans and the deposit rate. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

20 Incentive Constraint Then, the incentive constraint (IC) is: ( ) ( ) V t st h, bt h, d t θ qt h st h ωbt h The Lagrangian associated with the IC is λ h t and: λ h t = π i λ i t + π n λ n t Problem: ( ) ) max V t st h, bt h, d t + λ h t (V t (st h, bt h, d t ( ) ) = max 1 + λ h t V t (st h, bt h, d t λ h t θ ( θ qt h st h ωbt h ) ( q h t s h t ωb h t )) Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

21 Guess of Value Functions We guess that value function is linear in states: ) V (st h, bt h, d t = ν st st h ν bt bt h ν dt d t Interpretation of coeffi cients: 1 ν st : marginal value of assets. 2 ν bt : marginal cost of interbank borrowing. 3 ν dt : marginal cost of deposits. Then: ( ) ) ( ) 1 + λ h t (ν st st h ν bt bt h ν dt d t λ h t θ qt h st h ωbt h Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

22 Optimality Conditions Remember that, from the balance sheet constraint: b h t = q h t s h t n h t d t The FOC are (note the bank takes nt h as given and use chain rule to take derivatives of bt h ): ( ) d t : 1 + λ h t (ν bt ν dt ) = λ h t θω ( ) ( ) st h : 1 + λ h ν st t qt h ν bt = λ h t θ (1 ω) ( ( )) λ h t : ν dt nt h νst θ ν dt qt h st h (θω (ν bt ν dt )) bt h q h t Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

23 Reading the Optimality Conditions Interpretation: 1 Marginal cost of interbank borrowing is higher than cost of deposits iff λ h t > 0 and ω > 0. 2 Marginal value of assets is higher than marginal cost of interbank borrowing if λ h t > 0 and ω < 1. 3 Balance sheet effect: equity in bank must be suffi ciently high in relation with assets and interbank borrowing. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

24 Case A: Frictionless Wholesale Financial Market I ω = 1. Arbitrage across asset markets: q b t = q l t = q t Marginal value of asset must be equal to the marginal cost of borrowing on the interbank market ν st = ν bt q t The incentive constraint is simply: q t s t b t = φ t n t where φ t = ν t θ µ t Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

25 Case A: Frictionless Wholesale Financial Market II With a bit of work (which I skip), and by matching coeffi cients µ t = βe t λ t+1 λ t Ω t+1 ( R k t+1 R t+1 ) ν t = βe t λ t+1 λ t Ω t+1 R t+1 where Ω t+1 = 1 σ + σ ( ν t+1 + φ t+1 µ t+1 ) R k t+1 = ψ t z t+1 + (1 δ) q t+1 q t Aggregating: q t s t = φ t n t In this economy, a crisis increases the excess returns for banks of all types. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

26 Case B: Symmetric Frictions I ω = 0. Deposits and interbank loans become perfect substitutes: ν t = ν bt Thus, in general, there will be differences in prices of assets across islands: q n t > q i t and µ i t > µn t 0 Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

27 Case B: Symmetric Frictions II The leverage ratio: q i ts i t n i t = φ i t = ν t θ µ i t qt n st n nt n φ n t = ν t θ µ n t ( q n t st n ) φ n t µ n t = 0 n n t With a bit of work (which I skip), and by matching coeffi cients λ ( ) t+1 µ t = βe t Ω h t+1 Rt+1 hh R t+1 λ t ν t = βe t λ t+1 λ t Ω h t+1r t+1 Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

28 Case B: Symmetric Frictions III where ( ) Ω h t+1 = 1 σ + σ ν t+1 + φ h t+1 µh t+1 R hh t+1 = ψ t z t+1 + (1 δ) q h q h t t+1 Note that know we need to index also by the type of the island in next period and integrate over it. Aggregating: qts i t i = φ i t ni t qt n st n φ n t nn t (qt n st n φ n t nn t ) µ n t = 0 Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

29 Case B: Symmetric Frictions IV In this economy, a crisis increases the excess returns for banks of all types. Also: E t λ t+1 λ t Ω h t+1r ih kt+1 > E t λ t+1 λ t Ω h t+1r nh kt+1 E t λ t+1 λ t Ω h t+1r bt+1 = E t λ t+1 λ t Ω h t+1r t+1 Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

30 Aggregation Total bank net worth: Total net worth of old banks: n h t = n h ot + n h yt n h ot = σπ h {[ z t + (1 δ) q h t where we have net out the interbank loans. Total net worth of new banks: n h yt = ξ Aggregate balance sheet constraint: [ z t + (1 δ) q h t d t = h=i,n ] ψ t s t 1 R t 1 d t 1 } ( q h t s h t n h t ] ψ t s t 1 ) Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

31 Market Clearing Market for securities: s i t = i t + (1 δ) π i k t s n t = (1 δ) π n k t Labor market Debt market: χl ϑ t = λ t w t d ht = d t + d g Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

32 Policy Experiments Unconventional monetary policy: 1 Lending facilities. 2 Liquidity facilities. 3 Equity injections. Classic discussion from Sargent and Wallace (1983): real bills doctrine. How do we decide between these different policies? Effect on government budget position. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

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35 Lending Facilities The central bank lends directly to banks that are constrained. Central bank is not constrained by its balance sheet (this is more subtle than it seems, but let us assume it for a moment). But additional cost τ of underwriting a loan (monitoring, politics...). The central bank does not subsidize loans......but, by increasing funds available, it has an impact on equilibrium prices and allocations. New equilibrium condition: q h t s h t = q h t ( ) spt h + sgt h Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

36 Liquidity Facilities Central bank discounts loans from the interbank lending market. Banks can divert less funds from the central bank than from the regular interbank market: θ (1 ω g ) with ω g > 0. Then: q h t s h t = n h t + b h t + m h t + d t Penalty rate for discount: difference in the default. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

37 Equity Injections Treasury transfers wealth to banks. Government takes direct ownership position. Then: q h t s h t = n h t + n g t + b h t + d t Government must pay a premium. Why? Problems of issuing equity for banks in a crisis. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43

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43 Issues Ahead More detailed structure of bank capital. Different wholesale markets. Heterogeneity. Non-linearities. Optimal policy. Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, / 43