Macroeconomic Models with Financial Frictions Jesús Fernández-Villaverde University of Pennsylvania May 31, 2010 Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 1 / 69
Motivation I Traditional macro models embody a Modigliani-Miller environment. For example, in the standard RBC and NKE models (Smets-Wouters/CEE): 1 It does not even matter who owns the capital (household, firm, an investment fund). 2 Asset market is pretty boring: complete markets for households and firms. 3 Money is introduced in a rather ad hoc way through MIU or CIA. Therefore, there is little room to think about finance-related policy. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 2 / 69
Motivation II This was always a nagging worry of macroeconomists, who suspected they were missing important mechanisms. However, relatively little progress was made. After the Great Recession of 2007-2010, there seems to be little excuse not to analyze the interactions between the macroeconomy and the financial markets in much more detail. A lot of recent work. Unfortunately, our understanding is still limited. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 3 / 69
Goal Illustrate how we can incorporate financial frictions in standard macro models. Think about limitations of current approaches. Think about implications for policy: 1 Fiscal and monetary. 2 Systemic regulator. 3 Financial supervision. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 4 / 69
Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 5 / 69
An Analytic Framework How can we analyze formally some of these issues? We want a model: 1 With financial markets. 2 Where the Modigliani-Miller theorem does not hold. 3 Where we have mechanisms that resemble some of the channels emphasized by observers of the recent market turbulences. 4 Where we can perform quantitative analysis. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 6 / 69
Strategies Several approaches: 1 Models of constraints on borrowing: 1 Costly state verification. 2 Costly enforcement model. 2 Models of financial intermediation. 3 Models of leverage cycles. Because of time constraints, I will focus on models of constraints on borrowing. Why? 1 Historical reasons. 2 Foundation of more recent models. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 7 / 69
A Model with Costly-State Verification Tradition of financial accelerator of Bernanke, Gertler, and Gilchrist (1999), Carlstrom and Fuerst (1997), and Christiano, Motto, and Rostagno (2009). Elements: 1 Information asymmetries between lenders and borrowers costly state verification (Townsend, 1979). 2 Debt contracting in nominal terms: Fisher effect. 3 Changing spreads. We will calibrate the model to reproduce some of the basic observations of the U.S. economy. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 8 / 69
Flowchart of the Model Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 9 / 69
Households Representative household: E 0 t=0 β t e d t { ( )} mt u (c t, l t ) + υ log p t d t is an intertemporal preference shock with law of motion: d t = ρ d d t 1 + σ d ε d,t, ε d,t N (0, 1). Why representative household? Heterogeneity? Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 10 / 69
Asset Structure The household saves on three assets: 1 Money balances, m t. 2 Deposits at the financial intermediary, a t, that pay an uncontingent nominal gross interest rate R t. 3 Arrow-Debreu securities (net zero supply in equilibrium). Therefore, the household s budget constraint is: c t + a t p t + m t+1 p t = w t l t + R t 1 a t 1 p t + m t p t + T t + Ϝ t + tre t where: tre t = (1 γ e ) n t w e Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 11 / 69
Optimality Conditions The first-order conditions for the household are: Asset pricing kernel: e d t u 1 (t) = λ t { λ t = βe t R t λ t+1 Π t+1 u 2 (t) = u 1 (t) w t SDF t = E t β λ t+1 λ t and standard non-arbitrage conditions. } Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 12 / 69
The Final Good Producer Competitive final producer with technology ( 1 y t = 0 ) ε y ε 1 ε 1 ε it di. Thus, the input demand functions are: y it = ( pit p t ) ε y t i, Price level: ( 1 p t = 0 ) 1 pit 1 ε 1 ε di. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 13 / 69
Intermediate Goods Producers Continuum of intermediate goods producers with market power. Technology: where y it = e z t k α it 1 l 1 α it z t = ρ z z t 1 + σ z ε z,t, ε z,t N (0, 1) Cost minimization implies: ( ) 1 1 α ( ) 1 α wt 1 α rt mc t = α 1 α α e z t k t 1 l t = α 1 α w t r t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 14 / 69
Sticky Prices Calvo pricing: in each period, a fraction 1 θ of firms can change their prices while all other firms keep the previous price. Then, the relative reset price Π t = p t /p t satisfies: εg 1 t = (ε 1)g 2 t g 2 t gt 1 = λ t mc t y t + βθe t Πt+1g ε t+1 1 ( Π = λ t Πt y t + βθe t Πt+1 ε 1 ) t gt+1 2 Π t+1 Given Calvo pricing, the price index evolves as: 1 = θπ ε 1 t + (1 θ) Π 1 ε t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 15 / 69
Capital Good Producers I Capital is produced by a perfectly competitive capital good producer Why? It buys installed capital, x t, and adds new investment, i t, to generate new installed capital for the next period: ( [ ]) it x t+1 = x t + 1 S i t where S [1] = 0, S [1] = 0, and S [ ] > 0. i t 1 The period profits of the firm are: ( [ ]) ) it q t (x t + 1 S i t q t x t i t = q t (1 S i t 1 where q t is the relative price of capital. [ it i t 1 ]) i t i t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 16 / 69
Capital Good Producers II q t (1 S Discounted profits: ( [ ]) ) E 0 β t λ t it (q t 1 S i t i t t=0 λ 0 i t 1 Since this objective function does not depend on x t, we can make it equal to (1 δ) k t 1. First-order condition of this problem is: [ it i t 1 ] [ ] ) S it it λ t+1 + βe t q t+1 S i t 1 i t 1 λ t and the law of motion for capital is: ( k t = (1 δ) k t 1 + 1 S [ it i t 1 [ it+1 i t ]) i t ] ( ) 2 it+1 = 1 i t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 17 / 69
Entrepreneurs I Entrepreneurs use their (end-of-period) real wealth, n t, and a nominal bank loan b t, to purchase new installed capital k t : q t k t = n t + b t p t The purchased capital is shifted by a productivity shock ω t+1 : 1 Lognormally distributed with CDF F (ω) and 2 Parameters µ ω,t and σ ω,t 3 E t ω t+1 = 1 for all t. Therefore: E t ω t+1 = e µ ω,t+1 + 1 2 σ2 ω,t+1 = 1 µω,t+1 = 1 2 σ2 ω,t+1 This productivity shock is a stand-in for more complicated processes such as changes in demand or the stochastic quality of projects. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 18 / 69
Entrepreneurs II The standard deviation of this productivity shock evolves: log σ ω,t = (1 ρ σ ) log σ ω + ρ σ log σ ω,t 1 + η σ ε σ,t, ε σ,t N (0, 1). The shock t + 1 is revealed at the end of period t right before investment decisions are made. Then: log σ ω,t log σ ω = ρ σ (log σ ω,t 1 log σ ω ) + η σ ε σ,t σ ω,t = ρ σ σ ω,t 1 + η σ ε σ,t More general point: stochastic volatility. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 19 / 69
Entrepreneurs III The entrepreneur rents the capital to intermediate goods producers, who pay a rental price r t+1. Also, at the end of the period, the entrepreneur sells the undepreciated capital to the capital goods producer at price q t+1. Therefore, the average return of the entrepreneur per nominal unit invested in period t is: R k t+1 = p t+1 p t r t+1 + q t+1 (1 δ) q t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 20 / 69
Debt Contract Costly state verification framework. For every state with associated Rt+1 k, entrepreneurs have to either: 1 Pay a state-contingent gross nominal interest rate R l t+1 on the loan. 2 Or default. If the entrepreneur defaults, it gets nothing: the bank seizes its revenue, although a portion µ of that revenue is lost in bankruptcy. Hence, the entrepreneur will always pay if it ω t+1 ω t+1 where: R l t+1b t = ω t+1 R k t+1p t q t k t If ω t+1 < ω t+1, the entrepreneur defaults, the bank monitors the entrepreneur and gets (1 µ) of the entrepreneur s revenue. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 21 / 69
Zero Profit Condition The debt contract determines Rt+1 l to be the return such that banks satisfy its zero profit condition in all states of the world: ωt+1 +(1 µ) [1 F (ω t+1, σ ω,t+1 )] R l t+1b t }{{} Revenue if loan pays ωdf (ω, σ ω,t+1 ) Rt+1p k t q t k t } 0 {{ } Revenue if loan defaults = s t R t b t }{{} Cost of funds s t = 1 + e s+ s t that: is a spread caused by the cost of intermediation such s t = ρ s s t 1 + σ s ε s,t, ε s,t N (0, 1). For simplicity, intermediation costs are rebated to the households in a lump-sum fashion. External finance premium. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 22 / 69
Optimality of the Contract This debt contract is not necessarily optimal. However, it is a plausible representation for a number of nominal debt contracts that we observe in the data. Also, the nominal structure of the contract creates a Fisher effect through which changes in the price level have an impact on real investment decisions. Importance of working out the optimal contract. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 23 / 69
Characterizing the Contract I Define share of entrepreneurial earnings accrued to the bank: Γ (ω t+1, σ ω,t+1 ) = ω t+1 (1 F (ω t+1, σ ω,t+1 )) + G (ω t+1, σ ω,t+1 ) where: G (ω t+1, σ ω,t+1 ) = ωt+1 0 ωdf (ω, σ ω,t+1 ) Thus, we can rewrite the zero profit condition of the bank as: R k t+1 s t R t [Γ (ω t+1, σ ω,t+1 ) µg (ω t+1, σ ω,t+1 )] q t k t = b t p t which gives a schedule relating R k t+1 and ω t+1. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 24 / 69
Characterizing the Contract II Now, define the ratio of loan over wealth: ϱ t = b t /p t n t = q tk t n t n t = q tk t n t 1 and we get R k t+1 s t R t [Γ (ω t+1, σ ω,t+1 ) µg (ω t+1, σ ω,t+1 )] (1 + ϱ t ) = ϱ t that is, all the entrepreneurs, regardless of their level of wealth, will have the same leverage, ϱ t. A most convenient feature for aggregation. Balance sheet effects. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 25 / 69
Problem of the Entrepreneur Maximize its expected net worth given the zero-profit condition of the bank: max ϱ t,ω t+1 E t R k t+1 R t (1 Γ (ω t+1, σ ω,t+1 )) + [ ] R k η t+1 t s t R t [Γ (ω t+1, σ ω,t+1 ) µg (ω t+1, σ ω,t+1 )] ϱ t 1+ϱ t After a fair amount of algebra: E t R k t+1 R t (1 Γ (ω t+1, σ ω,t+1 )) = E t η t n t q t k t where the Lagrangian multiplier is: η t = s t Γ ω (ω t+1, σ ω,t+1 ) Γ ω (ω t+1, σ ω,t+1 ) µg ω (ω t+1, σ ω,t+1 ) This expression shows how changes in net wealth have an effect on the level of investment and output in the economy. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 26 / 69
Death and Resurrection At the end of each period, a fraction γ e of entrepreneurs survive to the next period and the rest die and their capital is fully taxed. They are replaced by a new cohort of entrepreneurs that enter with initial real net wealth w e (a transfer that also goes to surviving entrepreneurs). Therefore, the average net wealth n t is: n t = γ e 1 [ ] (1 µg (ω t, σ ω,t )) R k b t 1 t q t 1 k t 1 s t 1 R t 1 + w e Π t p t 1 The death process ensures that entrepreneurs do not accumulate enough wealth so as to make the financing problem irrelevant. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 27 / 69
The Financial Intermediary A representative competitive financial intermediary. We can think of it as a bank but it may include other financial firms. Intermediates between households and entrepreneurs. The bank: 1 Lends to entrepreneurs a nominal amount b t at rate R l t+1, 2 But recovers only an (uncontingent) rate R t because of default and the (stochastic) intermediation costs. 3 Thus, the bank pays interest R t to households. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 28 / 69
The Monetary Authority Problem Conventional Taylor rule: R t R = ( Rt 1 R ) γr ( ) γπ (1 γ Πt R ) ( ) y γy (1 γ R ) t exp (σ m m t ) Π y through open market operations that are financed through lump-sum transfers T t. The variable Π represents the target level of inflation (equal to inflation in the steady-state), y is the steady state level of output, and R = Π β the steady state nominal gross return of capital. The term ε mt is a random shock to monetary policy distributed according to N (0, 1). Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 29 / 69
Aggregation Using conventional arguments, we find expressions for aggregate demand and supply: where v t = 1 0 dispersion. y t = c t + i t + µg (ω t, σ ω,t ) (r t + q t (1 δ)) k t 1 y t = 1 v t e z t k α t 1l 1 α t ( pit p t ) ε di is the ineffi ciency created by price By the properties of Calvo pricing, v t evolves as: v t = θπ ε tv t 1 + (1 θ) Π ε t. We have steady state inflation Π. Hence, v t = 0 and monetary policy has an impact on the level and evolution of measured productivity. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 30 / 69
Equilibrium Conditions I The first-order conditions of the household: e d t u 1 (t) = λ t R t λ t = βe t {λ t+1 } Π t+1 u 2 (t) = u 1 (t) w t The first-order conditions of the intermediate firms: εg 1 t = (ε 1)g 2 t g 2 t gt 1 = λ t mc t y t + βθe t Πt+1g ε t+1 1 ( Π = λ t Πt y t + βθe t Πt+1 ε 1 ) t gt+1 2 mc t = w t Π t+1 k t 1 = α l t 1 α r t ( ) 1 1 α ( ) 1 α wt 1 α rt α 1 α α e z t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 31 / 69
Equilibrium Conditions II Price index evolves: Capital good producers: 1 = θπ ε 1 t q t (1 S [ it i t 1 +βe t λ t+1 λ t q t+1 S k t = (1 δ) k t 1 + + (1 θ) Π 1 ε t ] S [ it [ it+1 i t 1 ] it i t 1 i t 1 ) ] ( ) 2 it+1 = 1 i t i t ( [ ]) it 1 S i t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 32 / 69
Equilibrium Conditions III Entrepreneur problem: R k t+1 = Π t+1 r t+1 + q t+1 (1 δ) q t R k t+1 s t R t [Γ (ω t+1, σ ω,t+1 ) µg (ω t+1, σ ω,t+1 )] = q tk t n t q t k t Rt+1 k E t (1 Γ (ω t+1, σ ω,t+1 )) = R ( t 1 F (ω t+1, σ ω,t+1 ) E t s t 1 F (ω t+1, σ ω,t+1 ) µω t+1 F ω (ω t+1, σ ω,t+1 ) R l t+1b t = ω t+1 R k t+1p t q t k t q t k t = n t + b t p t ) nt n t = γ e 1 Π t [ (1 µg (ω t, σ ω,t )) R k t q t 1 k t 1 s t 1 R t 1 b t 1 p t 1 q t k t ] + w e Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 33 / 69
Equilibrium Conditions IV The government follows its Taylor rule: R t R = ( Rt 1 R Market clearing ) γr ( ) γπ (1 γ Πt R ) ( ) y γy (1 γ R ) t exp (σ m m t ) Π y y t = c t + i t + µg (ω t, σ ω,t ) (r t + q t (1 δ)) k t 1 y t = 1 v t e z t k α t 1l 1 α t v t = θπ ε tv t 1 + (1 θ) Π ε t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 34 / 69
Equilibrium Conditions V Stochastic processes: d t = ρ d d t 1 + σ d ε d,t z t = ρ z z t 1 + σ z ε z,t s t = 1 + e s+ s t s t = ρ s s t 1 + σ s ε s,t log σ ω,t = (1 ρ σ ) log σ ω + ρ σ log σ ω,t 1 + η σ ε σ,t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 35 / 69
Calibration Utility function: u (c t, l t ) = log c t ψ l t 1+ϑ 1 + ϑ ψ: households work one-third of their available time in the steady state and ϑ = 0.5, inverse of Frisch elasticity. Technology: Entrepreneur: α δ ε S [1] 0.33 0.023 8.577 14.477 µ σ ω w e s 0.15 2.528 n n k 2 25bp. For the Taylor rule, Π = 1.005, γ R = 0.95, γ Π = 1.5, and γ y = 0.1 are conventional values. For the stochastic processes, all the autoregressive are 0.95. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 36 / 69
Computation We can find the deterministic steady state. We linearize around this steady state. We solve using standard procedures. Alternatives: 1 Non-linear solutions. 2 Estimation using likelihood methods. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 37 / 69
Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 38 / 69
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Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 47 / 69
How Can We Use the Model? Christiano, Motto, and Rostagno (2003): Great depression. Christiano, Motto, and Rostagno (2008): Business cycle fluctuations. Fernández-Villaverde and Ohanian (2009): Spanish crisis of 2008-2010. Fernández-Villaverde (2010): fiscal policy. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 48 / 69
Figure: IRFs of Output to Different Fiscal Policy Shocks Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 49 / 69
A Model with Costly Enforcement We keep the basic structure as before, except the financial market friction is the cost of enforcing contracts. Structure: 1 Borrower may decide to renege on debt. 2 If that is the case, the lender can only recover the fraction (1 λ) of the gross return R k t+1 p tq t k t where: (1 λ) R k t+1 < R t and the borrower keeps the rest, λr k t+1 p tq t k t. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 50 / 69
Costly Enforcement Model II Value of project: V t = R k t+1p t q t k t R t (q t k t n t ) Incentive constraint: V t λr k t+1p t q t k t Since the constraint must be binding: Rt+1p k t q t k t R t (q t k t n t ) = λrt+1p k t q t k t 1 q t k t = n 1 (1 λ) R t+1 k t R t p t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 51 / 69
Costly Enforcement Model III Advantage: much easier to handle than costly state verification model. Disadvantage: no default in equilibrium, no spreads. When to use each of them? Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 52 / 69
A Model with Financial Intermediation Previous models have a very streamlined financial intermediation structure. Many of the events of the 2007-2010 recession were about breakdowns in intermediation. Kiyotaki and Gertler (2010)incorporate now a richer financial intermediation sector. In particular, we will deal with liquidity. To keep the presentation simple, I will get rid of nominal rigidities. Also, this will facilitate comparison with a neoclassical framework. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 53 / 69
Environment 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 island, aggregate output is: y t = A t k α t l 1 α t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 54 / 69
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: and: k t = (1 δ) k t + i t y t = c t + i t + g t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 55 / 69
Representative Household Preferences: ( ) E 0 β t log c t ψ l t 1+ϑ t=0 1 + ϑ 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) Macro-Finance May 31, 2010 56 / 69
Representative Household Household can save in: deposits in the banking sector: 1 Deposits at the financial intermediary, a t, that pay an uncontingent nominal gross interest rate R t. 2 Arrow-Debreu securities (net zero supply in equilibrium). The budget constraint is then: c t + a t = w t l t + R t 1 a t 1 + T t + Ϝ t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 57 / 69
Optimality Conditions The first-order conditions for the household are: Asset pricing kernel: 1 c t = λ t λ t = βe t λ t+1 R t ψl ϑ t c t = w t SDF t = E t β λ t+1 λ t and standard non-arbitrage conditions. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 58 / 69
Banks 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}. 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. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 59 / 69
Balance Sheet Net worth: n h t. Besides equity, banks raise funds in a national financial market: 1 Retail market: from the households, a 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) Macro-Finance May 31, 2010 60 / 69
Flow-of-Funds Balance sheet constraint: q h t s h t = n h t + a t + b h t Evolution of net worth: n h t = [ z t + (1 δ) q h t Objective function of bank: V t = E t ] s t 1 R t 1 a t 1 R bt 1 b t 1 (1 σ) σ i 1 β t λ t+i nt+i h i=0 λ t Value function: maximized objective function: ) V (st h, bt h, a t = max V t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 61 / 69
Financial Friction We need some financial friction to make the intermediation problem interesting. Diversion of funds to family: ( ) θ qt h st h ωbt h and close down. Then, the incentive constraint is: ) V (st h, bt h, a t θ ( ) qt h st h ωbt h The Lagrangian associated with the previous constraint is λ h t and: λ h t = π i λ i t + π n λ n t Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 62 / 69
Guess of Value Functions We guess that value function is linear in states: ) V (st h, bt h, a t = ν st st h ν bt bt h ν at a t Interpretation of coeffi cients. Substituting the balance sheet constraint, two equivalent expressions: ) ( ) V (st h, bt h, a t = ν st ν bt qt h st h (ν at ν bt ) a t + ν bt nt h ) = ν st st h ν bt bt h ν at (q t h st h nt h bt h Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 63 / 69
Optimality Conditions The FOC are: ( νst ) st h : qt h ν bt = λ h t θ (1 ω) ( ) a t : (ν bt ν at ) 1 + λ h t = θωλ h t ( ( )) λ h t : ν at nt h νst θ ν at qt h st h (θω (ν bt ν at )) bt h Interpretation: q h t 1 Marginal value of assets is higher than marginal cost of interbank borrowing if λ h t > 0 and ω < 1. 2 Marginal cost of interbank borrowing is higher than cost of deposits if λ h t > 0 and ω > 0. 3 Balance sheet effect: equity in bank must be suffi ciently high in relation with assets and interbank borrowing. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 64 / 69
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. ] s t 1 R t 1 a t 1 } Total net worth of new banks: {[ ] } nyt h = ξ z t + (1 δ) qt h s t 1 R t 1 a t 1 Aggregate balance sheet constraint: a t = h=i,n ( q h t s h t n h t ) Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 65 / 69
Equilibrium 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 = 1 (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) Macro-Finance May 31, 2010 66 / 69
Policy Experiments 1 Lending facilities: the central bank lends directly to banks that are balance sheet constrained. 2 Liquidity facilities: the central bank discounts loans. Banks can divert less funds from the central bank than from the regular interbank market. 3 Equity injections: the Treasury transfers wealth to banks. Sargent and Wallace (1983) Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 67 / 69
Andrew Wiles On Doing Research Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it s completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they re momentary, sometimes over a period of a day or two, they are the culmination of and couldn t exist without the many months of stumbling around in the dark that proceed them. Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 68 / 69
Conclusions Integrating macro with finance is one of the great challenges ahead of us. While there has been some progress, we are still exploring the first rooms of the mansion. We discussed today the introduction of financial frictions. Other key aspect is asset pricing. But this is a whole new ball game... Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 69 / 69