DSGE Models with Financial Frictions Simon Gilchrist 1 1 Boston University and NBER September 2014
Overview OLG Model New Keynesian Model with Capital New Keynesian Model with Financial Accelerator
Introduction Bernanke and Gertler describe a general equilibrium model in which financial frictions cause fluctuations in output. The model is highly stylized and relies on an overlapping generations structure to obtain closed-form general equilibrium dynamics. Importantly, the contracting structure is developed from first principles rather than ad hoc assumptions regarding the relationship between borrower balance sheets and economic activity. It conveys the essence of the argument that financial market distortions create a powerful source of propagation via a financial accelerator mechanism.
OLG Model Overlapping generations framework with two types of agents households and entrepreneurs. Entrepreneurs borrow from financial intermediaries and face frictions in capital markets owing to a costly-state-verification problem as described above. Households lend to financial intermediaries and hence serve as the ultimate source of funds. In this sense, the financial intermediaries are a veil and can be ignored in the analysis.
Households work when young, consume, and save for consumption when old. Households can either lend to entrepreneurs or can save via a storage technology that yields gross return R. Assume that some fraction of households invest in the storage technology so that the household return on savings is R. Alternatively, one can view this as a simple open economy model.
Savings of young A young household chooses savings B t to solve: max ln C y t + β ln Co t+1 subject to C y t = W t B t, C o t+1 = RB t. Household optimality implies that household savings is linear in the current wage: B t = bw t where b denotes the savings rate.
Entrepreneurs Entrepreneurs work when young, earn a wage W t and save to purchase k t+1 units of capital. Capital goods are produced one-for-one from consumption goods so that the price of capital in terms of foregone consumption is unity (Q = 1). Entrepreneurs also pay a fixed cost C e which may be interpreted as a fixed amount of consumption when young.
Net Worth Entrepreneurs are risk neutral and, net of C e, only consume when old. Entrepreneurial net worth is therefore n t = W t C e. This net worth is used to finance capital expenditures k t+1.
Production Given capital k t+1, an individual entrepreneur has access to a constant returns to scale Cobb-Douglas production technology: y t+1 = l 1 α t+1 (θ t+1ωk t+1 ) α where θ t+1 denotes aggregate productivity and ω denotes an idiosyncratic shock to an individual firm s project return. Given k t+1 units of capital purchased at time t, entrepreneurs hire labor in t + 1 and earn the expost profits π t+1 which, given the form of production, is a linear function of capital π t+1 = ωr k t+1k t+1 Because ω can only be observed with cost µωr k t+1 k t+1, entrepreneurs face a CSV contracting problem as outlined above.
Aggregate Economy Let aggregate output be defined as Y t = y t dφ(ω) and define aggregate capital, K t+1, in an analogous manner. (This ignores bankruptcy costs as resource drain). Total labor is in fixed supply and normalized to unity so that W t = (1 α)y t Assume that shocks to the aggregate technology θ t are iid and, without loss of generality, normalize the mean of θ t such that the expected aggregate return satisfies E t R k t+1 = αk α 1 t+1 Thus, although individual entrepreneurs face constant returns to scale, the aggregate return on capital is a decreasing function of the aggregate stock of capital.
Benchmark Economy without Financial Frictions In the absence of frictions in financial markets, the expected return on capital is equated to the risk free rate. Consequently, the aggregate capital stock is determined by the user cost of capital: and is therefore constant. ( α 1/(1 α) K = R) Thus iid shocks to technology have no persistent effects on the economy.
Model with Financial Frictions Entrepreneurial net worth is equal to wages earned when young net of consumption: N t = (1 α)y t C e Aggregate capital next period is the sum of entrepreneurial net worth and household savings: where B t = b(1 α)y t. K t+1 = N t + B t Entrepreneurial leverage is therefore K t+1 N t = 1 + b(1 α)y t (1 α)y t C e and hence a decreasing function of current output.
Leverage With constant returns to scale, entrepreneurs leverage up to the point where the expected excess return on capital is equal to the premium on external funds. As a result, capital expenditures are determined by available net worth: E t R k t+1 R = s ( Kt+1 where s() is derived from the contracting problem defined above. If K < N t, entrepreneurs need not borrow to finance desired capital. In this case s = 1, there is no premium on external funds, and the capital stock is equal to the first best. N t )
Capital supply vs capital demand Assuming N t < K we have ( αkt+1 α 1 = Rs 1 + b(1 α)y ) t (1 α)y t C e where s() > 1. In this case, the premium on external funds is positive and aggregate capital is below first best.
Comments: The left-hand side of this equation can be interpreted as a capital demand equation. It is a downward sloping function of current capital. The right hand side can be interpreted as a capital supply equation, i.e. the price at which the market can finance a given level of capital depends on leverage at higher leverage, default risk is higher and hence the premium on external funds is higher. In this example household and entrepreneurial savings can be summarized by current output. As a result, the capital supply curve does not depend on the current capital choice K t+1.
Dynamics: With C e > 0, an increase in the current level of technology θ t leads to an increase in entrepreneurial net worth N t that is proportionately larger than the increase in household savings. As output rises, leverage falls. Given s () > 0, the premium on external finance falls. This is represented as an outward shift in the capital supply curve. Next period s capital will therefore increase. Furthermore, high capital tomorrow implies higher output tomorrow. As a result, following a positive shock to technology in period t, leverage and the premium on external funds will be persistently below steady-state, and capital will be persistently above steady-state.
Intuition In Bernanke-Gertler, iid shocks to technology are propagated through time via the financial accelerator mechanism. The essential ingredient necessary to obtain persistent procyclical movements in output is that entrepreneurial net worth increases more than household savings in response to an increase in current output. In the simple framework outlined above, this occurs because of the fixed entrepreneurial consumption requirement C e. More generally, any mechanism that makes net worth more procyclical than savings result in an amplification and propagation mechanism. Asset price movements are the most likely source of procyclicality in net worth.
Additional implications Any mechanism that transfers wealth from savers to borrowers will have expansionary effects on the economy. Bernanke and Gertler use this insight to argue that debt-deflation which transfers net worth from entrepreneurs to households can have persistent contractionary effects on economic activity.
Households choose i=0 Households: maximize ( E t β i 1 1 γ C1 γ t+i + a m 1 γ m subject to { C t+i, N t+i, B t+i P t+i, M t+i P t+i, K t+i } i=0 to ( Mt+i P t+i ) ) 1 γm 1 a n N 1+γn t+s 1 + γ n C t = W t N t + Π t + T R t + Z t K t + Q t (K t+1 (1 δ) K t ) P t + B t + M t 1 M ( ) t 1 Bt+1 P t P t P t 1 + i t P t where W t /P t denotes the real wage and Π t denotes profits received from firms owned by households, Q t denotes the price of capital and Z t denotes the rental rate on capital.
Household Optimality Conditions: The inter-temporal first-order conditions are: ( ) C γ Pt t = β(1 + i t )E t C γ t+1 P t+1 ) ( Mt + a m P t ( [Zt+1 + (1 δ) Q t+1 ] t = βe t C γ t = βe t ( Pt P t+1 C γ t+1 C γ The labor-leisure FOC is: Q t W t C γ t = a n N γn t P t ) γm C γ t+1 )
Labor-leisure: Household Optimality Conditions: W t C γ t = a n N γn t P t Consumption euler equation: { } C γ t = E t βr t+1 C γ t+1 where Money demand: R t+1 = Z t+1 + (1 δ) Q t+1 Q t ( M t = (a m ) 1 γm 1 1 ) 1 P t 1 + i t with R t+1 = (1 + i t ) Pt P t+1 γm C γ γm t
Final goods producers: Firms in the final goods sector producer a homogenous good, Y t, using intermediate goods, Y t (z) according to the CES production function ( 1 Y t = 0 ) ε Y t (z) ε 1 ε 1 ε dz where Y t (z) denotes intermediate good z and ε > 1 is the price elasticity of demand. The representative firm chooses inputs Y t (z) to solve ( 1 ) ε max Y t (z) ε 1 ε 1 E t ε dz P t subject to 0 E t = 1 0 P t (z)y t (z)dz
Lagrangean Problem: Substituting constraints and taking derivatives, the first order condition is ( ε 1 ) ε Y t (z) ε 1 ε 1 1 ε 1 ε dz Y t (z) 1 P ε t (z) = ε 1 ε P t 0 Rearranging we get the demand for the intermediate good ( ) Pt (z) ε Y t (z) = Y t Using this demand curve in conjunction with the definition of E t, it is straightforward to show that P t Y t = E t for ( 1 ) 1/(1 ε) P t = P t (z) 1 ε dz 0 where P t represents the minimum cost of achieving one unit of the final goods bundle. We interpret P t as the aggregate price index. P t
Intermediate goods producers: There is a continuum of monopolistically competitive firms owned by consumers, indexed by z [0, 1]. Each intermediate good firm operates a CRS production function and faces the demand curve for good z derived above. Nominal Rigidities: Calvo price setting. With probability 1 θ firms reset their price in any given period. Average price duration = 1 1 θ.
Cost Minimization Firm z chooses inputs N t (z) and K t (z) to minimize: subject to Cost minimization implies Marginal Cost: where C t = min W t P t N t (z) + Z t K t (z) Y t (z) = A t N t (z) α K t (z) 1 α W t = MC t α Y t (z) P t N t (z) MC t = Z t = MC t (1 α) Y t(z) K t (z) C t = MC t Y t (z) 1 α 1 α (1 α) α ( Wt P t ) 1 α Z α t A t
Flexible Prices Firm z chooses P t (z) to maximize subject to Π t (z) = P t(z) Y t (z) MC t Y t (z) P t ( ) Pt (z) ε Y t (z) = Y t The firm solves [ (Pt ) (z) 1 ε ( ) ] Pt (z) ε MC t max P t(z) P t P t P t Y t
Price vs Marginal Cost First order conditions imply that the firm sets its relative price as P t (z) P t = (1 + µ) MC t where (1 + µ) = ε ε 1 denotes a constant markup over real marginal cost.
Nominal Price Rigidities: Firm solves: ( [ ]) P max E t Λ t+s θ s t Y Pt t+s MC t+s Yt+s P s=0 t+s subject to ( ) P Yt+s ε = t Y t+s P t Write this as: ( [ ( ) P max E t Λ t+s θ s 1 ε t MC t+s P t(z) P t+s s=0 ( P t P t+s ) ε ] Y t+s )
Optimal reset price The first-order-conditions imply E t ( where s=0 ( P Λ t+s θ s [Pt (1 + µ) P t+s MC t+s ] t Y 1 + µ = ε ε 1 t+s P t+s ) ) = 0 Optimal reset price is a weighted average of expected future marginal costs: ( ( E t Pt s=0 Λ t+sθ s P )) P t+s MC t Yt+s t+s P t+s = (1 + µ) ( ( E t s=0 Λ P )) t Yt+s t+s P t+s
Log-linearization of P t Log-linearize this equation: ˆp t = (1 βθ) E t β s θ s [ˆp t+s + mc t+s ] which may be expressed as s=0 or equivalently: ˆp t = (1 βθ) mc t + βθe t ˆp t+1 ˆp t ˆp t = (1 βθ) ˆmc t + ˆπ t + βθe t (ˆp t+1 ˆp t+1 )
Price index: so P t = ( θp 1 ε t 1 P t P t 1 = Phillips Curve ) 1 ε 1/(1 ε) + (1 θ) Pt ( ( P θ + (1 θ) t P t 1 ) 1 ε ) 1/(1 ε) Log linearize this equation around a constant price level we have ˆπ t = (1 θ) (ˆp t ˆp t 1 ) Combine with optimal reset price to obtain: ˆπ t = κ ˆmc t + βe tˆπ t+1 where κ = (1 θ) (1 βθ) θ
Total input demand satisfies Aggregation: Aggregate output is N t = K t = 1 0 1 0 N t (z)dz K t (z)dz ( 1 Y t = A t (N t (z) α K t (z) 1 α) ) ε/(ε 1) (ε 1)/ε dz 0 = A t N α t K 1 α t ( 1 ( ) Nt (z) (ε 1)/ε ε/(ε 1) dz) 0 The term in brackets reflects the (second-order) loss in output owing to price dispersion. N t
Capital Production Capital accumulation is subject to adjustment costs: ( ) It K t+1 = (1 δ)k t + φ K t K t where φ(δ) = δ and φ (δ) = 1. Capital producers choose I t to max ( ) It Q t φ I t K t K t FOC implies Q t = 1 ( ) φ It K t
Monetary Policy Monetary policy satisfies a Taylor rule: 1 + i t = ( Pt P t 1 ) φπ ( ) φy Yt Y p t
Overview Bernanke, Gertler and Gilchrist develop a quantitative dynamic stochastic general equilibrium framework that embeds financial frictions in an otherwise workhorse dynamic New Keynesian economy. Their framework allows one to quantitatively assess the strength of the financial accelerator mechanism described by Bernanke-Gertler and Kiyotaki and Moore. It also provides a basis for a quantitative evaluation of alternative monetary policy rules.
Households: Households make consumption and labor supply decisions to maximize the present value of utility subject to standard constraints. Let R t denote the risk-free rate. Household optimality conditions imply a consumption Euler equation U (C t ) = E t R t+1 βu (C t+1 ) and combined with firm s hiring decisions the labor equation U (C t ) 1 µ t αy t /N t = v (N t ) where v (N) denotes the marginal disutility of labor.
Aggregate Capital Return Entrepreneurs in this economy purchase capital at period t, produce in period t + 1 and then resell their capital at market price Q t+1. The required return on capital is E t R k t+1 = E t [ (1/µt+1 ) Y t+1 /K t+1 + (1 δ)q t+1 Q t where denotes the marginal profitability of capital. The term µ t is the markup over marginal cost owing to monopolistic-competition features of the New Keynesian framework. In the absence of financial frictions, the return on household savings is equal to the return on capital: R t+1 = R k t+1 ]
Tobin s Q With adjustment costs to capital, Tobin s Q is an increasing function of the rate of investment: where φ () 0 1 Q t = ( ) φ It K t
Resource constraints Production:: Y t = A t N α t K 1 α t = C t + I t Capital Accumulation: ( ) It K t+1 = (1 δ)k t + φ K t K t
Log-linearization Consumption-euler equation: Return on capital: ĉ t = σe tˆr t+1 + E t ĉ t+1 Markup: ˆr k t = (1 ν)(ˆµ t + ŷ t ˆk t ) + ν ˆq t ˆq t 1 ˆµ t = ŷ t γĉ t (1 + γ n )ˆn t Tobin s Q: î t ˆk t = ηˆq t
Log-linearization Production and resource constraints: ŷ t = â t + αˆn t + (1 α)ˆk t ŷ t = c y ĉt + i y ît ˆk t+1 = (1 δ)ˆk t + δî t
Frictionless Model (RBC) With no nominal rigidities we impose: ˆµ t = 0 With no financial frictions we impose: ˆr t+1 = ˆr k t+1
Nominal Rigidities and Monetary Policy The New Keynesian features of the model imply a relationship between inflation and markups via the Phillips curve: π t = κˆµ t + βe t π t+1 Monetary policy specifies a Taylor type monetary policy rule where the nominal interest rate depends on inflation and output: ˆr n t = φ πˆπ t + φ y ŷ t The Fisher equation determines the relationship between real and nominal rates: ˆr t+1 = ˆr n t E tˆπ t+1
The Financial Sector: Entrepreneurs are long-lived but risk neutral. Face an exogenous failure rate and hence discount the future more than households. Financial frictions based on the costly-state-verification model of BGG.
Capital Return of Entrepreneur Continuum of risk-neutral entrepreneurs with exogenous birth and death rate γ. Entrepreneur with capital K it has access to a technology that transforms labor and capital services into wholesale output goods. Entrepreneur uses net worth N it to buy capital K i,t+1 at price Q t to be used in production at t + 1. She then resells the capital at price ω i,t+1 Q t+1. The realized (gross) return on capital: [ ] ω 1 it+1 Ri,t+1 k µ t+1 (1 α) Y i,t+1 K i,t+1 + Q t+1 (1 δ) = Q t
Default and leverage With constant returns to scale in production and monitoring technologies, the contracting problem implies E t R k t+1 R t = ρ( ω t ) Given a default barrier ω, capital expenditures are determined by available net worth: [ ( )] Γ(ωt ) µg(ω t )] QK i,t+1 = 1 + λ N i,t 1 Γ(ω t ) Summing across entrepreneurs: [ ( )] Γ(ωt ) µg(ω t )] QK t+1 = 1 + λ N t 1 Γ(ω t ) where N t = i N it and K t = i K it
Leverage Let N t+1 denote aggregate net worth. Because all entrepreneurs face the same aggregate return on capital, they face the same premium on external funds, hence summing across entrepreneurs implies that this equation holds in the aggregate. BGG obtain a relationship between the aggregate return on capital and the aggregate degree of entrepreneurial leverage: E t R K t+1 R t ( ) Qt K t+1 = s where the function s() is derived from the costly state verification contracting structure. N t
Entrepreneurial savings: Entrepreneurs discount the future more than households and are risk neutral implies that they defer consumption until they exit.
Aggregate net worth Aggregating across entrepreneurs then implies the aggregate net worth accumulation equation: N t = (1 γ)rt k Q t 1 K t ( ) (1 γ) (R t 1 (Q t 1 K t N t 1 ) + µg( ω t )Rt k Q t 1 K t ) + γw e t where γ is the death rate of entrepreneurs and γw e t represents a small exogenous transfer to new startups. The first term in this expression is the aggregate return on capital. The second term is the payment to bond holders (households) inclusive of compensation for default costs.
Aggregate net worth In equilibrium, the expected return on capital compensates for the risk free rate plus the loss in monitoring. Rewriting we obtain ( ) N t = (1 γ)r t 1 N t 1 +(1 γ) Rt k E t 1 Rt k Q t 1 K t +γwt e Thus ( surprise movements in the return on capital R k t E t 1 Rt k ) influence current net worth Nt. This is analogous to the Kiyotaki-Moore result. To fully describe model dynamics one must solve the full system using standard numerical solution methods. Nonetheless, one can get some idea of model dynamics from this last equation.
Log-linearization of financial sector External finance premium and net worth equation: s t = χ (n t q t k t + ε fd t n t = K ( ) K N rk t N 1 (s t 1 + r t 1 π t ) + θn t 1 + ε nw t Allow for credit-supply shocks: ε fd ε nw t t : disturbances to credit intermediation process : disturbances to asset values that serve as collateral
Intuition: Consider a positive shock to the economy that leads to an increase in the current value of assets in place Q t. For a given level of capital expenditures, a 1% increase in asset prices raises financing requirements Q t K t+1 by 1%. Net worth will also increase owing to the rise in asset prices. Around the steady-state we have that approximately d ln N t d ln Q t = K N > 1
Intuition: The percent change in net worth is roughly proportional to the steady-state degree of leverage. As a result, holding other things fixed, surprise movements in asset prices cause net worth to rise proportionately more than the increase in capital expenditures hence leverage falls and the external finance premium decreases. This causes a further increase in asset prices and further reductions in the premium on external funds and hence an expansion of investment spending.
The role of nominal price rigidities A positive shock that increases asset prices also leads to an increase in marginal cost which is only partially transmitted to the economy owing to nominal price rigidities. As a result, markups fall, and output expands. The reduction in markups and expansion of output puts further upward pressure on asset prices and thus strengthens the financial accelerator. The conduct of monetary policy also plays a key role: A relatively weak response of monetary policy to expected inflation reduces the effect of increased investment demand on real interest rates and causes a much larger increase in asset prices.
Quantitative results According to the calibration used in BGG, the financial accelerator leads to a 30% amplification of the output in response to increases in technology relative to a benchmark model without financial frictions. It also provides substantial amplification to shocks to monetary policy and other demand-side disturbances. A reallocation of wealth from households to entrepreneurs has a strong amplification mechanism: A reallocation that is equivalent to 1% of entrepreneurial net worth implies a 2% rise in actual net worth owing to the fact that asset prices rise as net worth expands. Such reallocation mechanisms allow one to consider the role of debt deflation as a source of business cycle dynamics. It also raises the possibility that the financial sector can serve as a source of economic volatility.