A Model with Costly-State Verification

Transcription

1 A Model with Costly-State Verification Jesús Fernández-Villaverde University of Pennsylvania December 19, 2012 Jesús Fernández-Villaverde (PENN) Costly-State December 19, / 47

2 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 basic observations of the U.S. economy. Jesús Fernández-Villaverde (PENN) Costly-State December 19, / 47

3 Flowchart of the Model Jesús Fernández-Villaverde (PENN) Costly-State December 19, / 47

4 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) Costly-State December 19, / 47

5 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 securities (net zero supply in equilibrium). Therefore, the household s budget constraint is: where: 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 tre t = (1 γ e ) n t w e + T t + Ϝ t + tre t Jesús Fernández-Villaverde (PENN) Costly-State December 19, / 47

6 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) Costly-State December 19, / 47

7 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) Costly-State December 19, / 47

8 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) Costly-State December 19, / 47

9 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) Costly-State December 19, / 47

10 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. Alternative: i t 1 1 Adjustment cost in capital. 2 Time to build. Jesús Fernández-Villaverde (PENN) Costly-State December 19, / 47

11 Capital Good Producers II Technology illiquidity. Importance of irreversibilities? 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) Costly-State December 19, / 47

12 Capital Good Producers III q t (1 S Discounted profits: E 0 β t λ t t=0 λ 0 ( [ ]) ) it (q t 1 S i t i t 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 + [ it+1 i t ( [ ]) it 1 S i t i t 1 ] ( ) 2 it+1 = 1 i t Jesús Fernández-Villaverde (PENN) Costly-State December 19, / 47

13 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 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) Costly-State December 19, / 47

14 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) Costly-State December 19, / 47

15 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) Costly-State December 19, / 47

16 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) Costly-State December 19, / 47

17 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 is a spread caused by the cost of intermediation such that: 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) Costly-State December 19, / 47

18 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) Costly-State December 19, / 47

19 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) Costly-State December 19, / 47

20 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) Costly-State December 19, / 47

21 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) Costly-State December 19, / 47

22 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) Costly-State December 19, / 47

23 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) Costly-State December 19, / 47

24 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) Costly-State December 19, / 47

25 Aggregation Using conventional arguments, we find expressions for aggregate demand and supply: where v t = 1 0 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 dispersion. 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) Costly-State December 19, / 47

26 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 Jesús Fernández-Villaverde (PENN) Costly-State December 19, / 47

27 Equilibrium Conditions II 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 = Π t+1 k t 1 = α w t 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) Costly-State December 19, / 47

28 Equilibrium Conditions III 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 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) Costly-State December 19, / 47

29 Equilibrium Conditions IV 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) Costly-State December 19, / 47

30 Equilibrium Conditions V The government follows its Taylor rule: R t R = Market clearing ( Rt 1 R ) 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) Costly-State December 19, / 47

31 Equilibrium Conditions VI 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) Costly-State December 19, / 47

32 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] µ σ ω w e s 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 Jesús Fernández-Villaverde (PENN) Costly-State December 19, / 47

33 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) Costly-State December 19, / 47

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46 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 Fernández-Villaverde (2010): fiscal policy. Jesús Fernández-Villaverde (PENN) Costly-State December 19, / 47

47 Figure: IRFs of Output to Different Fiscal Policy Shocks Jesús Fernández-Villaverde (PENN) Costly-State December 19, / 47