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Interpreting Structural Shocks and Assessing Their Historical Importance

  • Luis Herrera und Jesús Vázquez EMAIL logo
Veröffentlicht/Copyright: 7. April 2022
The B.E. Journal of Macroeconomics
Aus der Zeitschrift The B.E. Journal of Macroeconomics Band 23 Heft 1

Abstract

We revisit three major US recessions through the lens of a standard medium-scale DSGE model (Smets, F., and R. Wouters. 2007. “Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach.” The American Economic Review 97: 586–606) augmented with financial frictions. We first estimate the DSGE model using a Bayesian approach for three alternative periods, each containing a major US recession: the Great Depression, the Stagflation and the Great Recession. Then, we assess the stability of structural parameters, and analyze what frictions were particularly important and what shocks were the main drivers of aggregate fluctuations in each historical period. This exercise can be understood as a test of the standard New-Keynesian DSGE model with financial accelerator in closed economies. We find that the estimated DSGE model is able to provide a sound explanation of all three recessions by closely relating both estimated structural shocks and frictions with well known economic events.

Keywords: medium-scale DSGE model; financial frictions; structural shocks; major recessions
JEL Classification: E30; E44
Corresponding author: Jesús Vázquez, Universidad del País Vasco (UPV/EHU), Bilbao, Spain, E-mail:

Funding source: Ministerio de Economía y Competitividad (FEDER/EU)

Award Identifier / Grant number: ECO2016-78749P

Funding source: Ministerio de Ciencia, Innovación y Universidades

Award Identifier / Grant number: FPU17/06331

Funding source: Eusko Jaurlaritza

Award Identifier / Grant number: IT-1336-19

Funding source: Ministerio de Ciencia e Innovación

Award Identifier / Grant number: PID2020-118698GB-I00

Acknowledgement

We are indebted to the Associate Editor, Evi Pappa and two anonymous referees for their detailed comments and suggestions. We also thank Luca Pensieroso and Rigas Oikonomou for their helpful comments on a preliminary version of this paper.

  1. Research funding: This research was supported by the Spanish Ministry of Science and Innovation under grant number PID2020-118698GB-I00, the Spanish Ministry of Economy and Competition under grant number ECO2016-78749P (FEDER/EU), and the Basque Government under grant number IT-1336-19. Moreover, the first author acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities under scholarship grant FPU17/06331.

  2. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

  3. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

Appendix A

We consider a DSGE model that resembles the Smets and Wouters (2007) model, extended with the financial accelerator of Bernanke, Gertler, and Gilchrist (1999). This appendix briefly describes the model and the log-linearized equations that characterize the general equilibrium. All variables in the log-linearized equations are expressed in log-deviations from their steady states except those measured in percentage terms, which are expressed in simple deviations from their respective steady-state values.

A.1 Households

The representative household i decides consumption, hours worked, and holds riskless assets, aiming at maximizing a non-separable utility function in consumption and labor which also includes external habit formation.

E t ∑ k = 0 ∞ β k ln ( C t + k ( i ) − h C t + k − 1 ) − L t + k ( i ) 1 + σ l 1 + σ l ,

where C t and L t are consumption and hours worked, β is the discount factor, h denotes the habit persistence parameter, and σ l is the labor elasticity.

Household savings are allocated to deposit liabilities in banks and holdings of government bonds. These riskless assets, B, are perfect substitutes and pay the same interest rate, R n . Households also obtain dividends, D, from intermediate firms, capital goods producers, and labor unions. Hence, the budget constraint is as follows:

C t + k ( i ) + B t + k ( i ) e ϵ t b R t + k n P t + k − T t + k = W t + k ( i ) L t + k ( i ) P t + k + B t − 1 + k ( i ) P t + k + D t + k P t + k ,

where R t n is the nominal interest rate, P t is the aggregate price level, and the risk premium shock, ϵ t b , is an exogenous premium on the bond yields, which is assumed to follow an AR(1) process.[33] T are lump-sum taxes and W is the nominal wage.

From the optimality conditions of the household maximization problem the following Euler equation is obtained:

(1) c t = h 1 + h c t − 1 + ( 1 − h 1 + h ) E t c t + 1 − 1 − h ( 1 + h ) R t n − E t π t + 1 + ϵ t b .

A.2 Labor Union

As in Smets and Wouters (2007), households supply homogeneous labor to intermediate labor unions that differentiate labor services. These intermediate labor unions then set wages to sell labor services to a labor packer who aggregates the differentiated labor and resells it to intermediate goods firms. Aggregation of labor services follows

L t = ∫ 0 1 L t ( i ) 1 1 + ϵ t w d i 1 + ϵ t w ,

where 1 + ϵ t w is the desired markup of wages over the household’s marginal rate of substitution, which is assumed to follow a stochastic process around its steady-state value. Labor packers maximize profits in a perfectly competitive market

max L t ( i )   W t L t − ∫ 0 1 W t ( i ) L t ( i ) ,

where L t is subject to the labor aggregation function, W t is the aggregate wage that intermediate firms pay for labor services, and W t (i) is the wage that labor packers pay for the differentiated labor. This optimization problem gives rise to the following labor demand function

L t ( i ) = W t ( i ) W t − 1 + ϵ t w ϵ t w L t .

Both labor demand function and labor services aggregation function imply the wage aggregation function

(2) W t = ∫ 0 1 W t ( i ) − 1 ϵ t w d i − ϵ t w .

Following Calvo’s lottery scheme, it is assumed that labor unions can only adjust nominal wages with probability: 1 − ξ w . The fraction of labor unions ξ w that cannot adjust wages are assumed to follow the indexation rule, W t + 1 ( i ) = W t ( i ) P t P t − 1 γ w , where γ w is the wage indexation parameter to past inflation. Hence, the labor unions choose an optimal W to maximize,

(3) E t ∑ k = 0 ∞ β k ξ w k Λ t + k W t ( i ) L t + k ( i ) − ϵ t + k b L t + k ( i ) 1 + σ l 1 + σ l ,

subject to the labor demand and the indexation rule. β k Λ t + k = β λ t + k λ t denotes the stochastic discount factor between t and t + k, and λ t is the marginal utility of consumption for households at time t.

Hence, this setup gives rise to the following log-linearized equation for real wages, w t :

(4) w t = β 1 + β ( E t w t + 1 + E t π t + 1 ) + ( 1 − β 1 + β ) w t − 1 − 1 + β γ w 1 + β E t π t + γ w 1 + β π t − 1 − 1 1 + β ( 1 − β ξ w ) ( 1 − ξ w ) ( 1 + ( 1 + λ w ) σ l / λ w ) ξ w × w t − σ l l t − 1 1 − h ( c t − h c t − 1 ) + ϵ t w ,

where the term in brackets on the right-hand side of the equation defines the wedge between the real wage and the marginal rate of substitution between labor and consumption.

A.3 Final Good Firm

Competitive final good producers buy intermediate goods and assemble them to finally sell homogeneous goods to households. The intermediate goods aggregation follows

Y t = ∫ 0 1 Y t ( i ) 1 1 + ϵ t p d i 1 + ϵ t p ,

where Y t is the homogeneous good, Y t (i) is the heterogeneous good supplied by firm i, and 1 + ϵ t p is the desired markup of prices over the marginal costs of firms, which is assumed to follow a stochastic process around its steady-state value. Final good firms maximize profits in a perfectly competitive market

max Y t ( i )   P t Y t − ∫ 0 1 P t ( i ) Y t ( i ) ,

where Y t is subject to the goods aggregation function, P t (i) is the price for differentiated goods, and P t is the aggregate price index. The optimality condition of this maximization problem results in the following demand function for goods

(5) Y t ( i ) = P t ( i ) P t − 1 + ϵ t p ϵ t p Y t .

Hence, the goods demand function and the intermediate goods aggregator imply the following price aggregator

(6) P t = ∫ 0 1 P t ( i ) − 1 ϵ t p d i − ϵ t p .

A.4 Intermediate Goods Firms

As in the labor market, it is assumed that intermediate goods firms can only adjust prices with probability: 1 − ξ p . Those firms which cannot adjust prices in period t simply reset their prices according to the indexation rule: P t + 1 ( i ) = P t ( i ) P t P t − 1 γ p , where γ p represents the degree of price indexation to past inflation. Firms able to decide their optimal prices P t * at time t choose them by maximizing current and future expected profits. Denoting the marginal costs and the inflation rate by MC t and π t , respectively, the price setting optimization problem faced by intermediate goods firms is

(7) E t ∑ k = 0 ∞ β k ξ p k Λ t + k P t P t + k P t * ( i ) ∏ l = 1 k π t + l − 1 γ p − M C t + k   Y t + k ( i ) ,

subject to the price indexation rule and the demand function for goods.

Hence, this setup gives rise to the following so-called New-Keynesian Phillips curve:

(8) π t = β 1 + β γ p E t π t + 1 + γ p 1 + β γ p π t − 1 + 1 1 + β γ p ( 1 − β ξ p ) ( 1 − ξ p ) ξ p α r t k + ( 1 − α ) w t − ϵ t a + ϵ t p ,

where the term in brackets on the right-hand side of the equation is the firms’ marginal cost.

In addition to setting prices, intermediate goods firms decide on the output of goods. They choose the amount of production inputs by maximizing the flow of discounted profits

(9) E t β Λ t + 1 Y t + 1 ( i ) − r t + 1 k K t + 1 s ( i ) − W t + 1 P t + 1 L t + 1 ( i ) ,

where r t + 1 k is the rental rate of capital, and K t + 1 s ( i ) denote capital services. The production function is assumed to follow a Cobb–Douglas technology:

(10) Y t = ϵ t a K t s ( i ) α L t ( i ) 1 − α − ϕ p ,

where ϕ p is the share of fixed costs implied in production, and ϵ t a is a disturbance capturing TFP shocks. The log-linearized equilibrium conditions of this maximization problem are:

y t = ϕ p α k t − 1 s + ϕ p ( 1 − α ) l t + ϕ p ϵ t a
l t = k t − 1 s + r t k − w t

A.5 Capital-Good Producers

Capital-good producers build capital goods and sell them to entrepreneurs at price Q t . They turn out capital goods by combining investment goods purchased from final good producers and installed capital rented from entrepreneurs.[34] We also consider that capital-good producers face quadratic adjustment costs, S(I t /It−1). We assume that S(I t /It−1) is a strictly increasing, twice differentiable function. Then, the optimization problem of the capital-good producers is:

Max I t E t ∑ k = 0 ∞ β k Λ t + k Q t + k I t + k ϵ t + k i − I t + k − Q t + k I t + k ϵ t + k i S I t + k I t + k − 1 ,

where the disturbance ϵ t i follows an AR(1) process that represents the investment-specific technology shock. The log-linearization of the first order condition of this optimization problem gives rise to the following investment equation:

(11) i t = 1 1 + β i t − 1 + β 1 + β E t i t + 1 + 1 φ ( 1 + β ) q t + ϵ t i ,

where φ captures the steady-state elasticity of investment adjustment cost function, S(I t /It−1). Notice that a higher elasticity reduces the sensitivity of investment to the value of the existing capital stock, q t .

The capital stock evolves according to:

K t = ( 1 − τ ) K t − 1 ϵ t k + 1 − S I t I t − 1 I t .

In log-linearized form:

(12) k t + 1 = ( 1 − τ ) k t + τ i t + τ ϵ t i ,

where τ is the depreciation rate of capital.

A.6 Entrepreneurs and Banks

As mentioned above, some households are assumed to be entrepreneurs. Since entrepreneurs face a constant survival probability the proportion of such households is also constant. Entrepreneurs use their own and external funds to finance the acquisition of capital, which is rented to goods producers. Once capital is acquired they observe the realization of an idiosyncratic shock (ω) and decide the degree of capital utilization (U t ) facing an adjustment cost.[35] Hence, the amount of capital that the entrepreneurs rent to firms is K t + 1 s = U t + 1 K t . At the end of the period, they sell back the undepreciated capital to the capital-good producers at a price Qt+1. Formally, entrepreneurs solve the following optimization problem:

max U t + 1 r t + 1 k U t + 1 − a ( U t + 1 ) ω K t .

The (aggregate) log-linearized optimality conditions derived from this optimization problem are as follows:

(13) k t + 1 s = u t + 1 + k t ,
(14) u t = ( ( 1 − ψ ) / ψ ) r t k ,

where Eq. (13) describes the capital used in production, and Eq. (14) determines the degree of capital utilization as a function of the rental rate of capital. The parameter ψ is a positive function of the elasticity of the capital utilization adjustment cost, which is normalized to take a value between zero and one. Thus, the higher the value of ψ, the higher the cost of adjustment faced by that entrepreneurs.

Moreover, the average (across entrepreneurs) rate of return of capital utilized in production should be consistent with the following (non-arbitrage) equilibrium equation:

E t R t + 1 k = E t r t + 1 k U t + 1 − a ( U t + 1 ) + Q t + 1 ( 1 − τ ) Q t ,

where the real expected interest rate on external funds is equal to the expected marginal return of capital—otherwise, entrepreneurs would not be behaving rationally in their decision on capital utilization. Log-linearizing the previous equation gives the corresponding arbitrage-free condition for the value of capital:

(15) q t = r ̄ k R ̄ k E t r t + 1 k − E t R t + 1 k + 1 − τ R ̄ k E t q t + 1 ,

where R ̄ k and r ̄ k are the steady-state values of capital return and the rental rate, respectively.

The equilibrium condition that describes the cost of external funding is obtained from the optimal-debt contract problem, which implies the maximization of entrepreneurs’ utility and the zero-profit condition associated with the assumption of perfectly competitive banks:[36]

E t R t + 1 k = E t s N t + 1 Q t K t + 1 e ϵ t b R t ,

where s N t + 1 Q t K t + 1 is a function that represents the elasticity of external funding rates to the leverage ratio. The log-linearization of this expression results in:

(16) E t R t + 1 k = − ϵ E t [ n t + 1 − q t − k t + 1 ] + R t n − E t π t + 1 + ϵ t b ,

where the term in brackets is the wedge between the (log of the) net worth of the entrepreneurs (nt+1) and the (log of the) gross value of capital (q t + kt+1). This difference represents the proportion of projects that the entrepreneur is not able to self-finance and, by the same token, the external funding required. The parameter ϵ is the elasticity of the external finance premium to this entrepreneurial financial wealth. Therefore, a higher value of this parameter would make the interest rate spread more sensitive to the leverage ratio. Moreover, this spread is also characterized by an exogenous process, ϵ t b , that captures the fluctuations of this risk premium beyond those generated by the financial frictions considered in the model. Notice that this shock also shows up in the consumption-Euler Eq. (1). Therefore, it can also be viewed as a shock to the supply of credit that exogenously changes the interest rate spread.

Entrepreneurial net worth accumulation is given by the profits of the surviving entrepreneurs:

(17) n t + 1 = γ R t k Q t − 1 K t − E t − 1 R t k Q t − 1 K t − N t + W t e ϵ t n w .

where γ is the survival rate of entrepreneurs, and W t e is the transfer to all the entrepreneurs who are in business in period t. This profit is determined by the difference between the revenue from holding capital and the cost of external finance. The log-linearized equation is given by

(18) n t + 1 = γ R ̄ k K ̄ N ̄ R t k − E t − 1 R t k + E t − 1 R t k + n t + ϵ t n w ,

where K ̄ N ̄ is the steady-state capital net worth ratio, and ϵ t n w is an AR(1) shock capturing exogenous fluctuations in entrepreneurs’ net worth.[37] This equation shows that the variations in entrepreneurs’ net worth are mostly explained by unexpected changes in the real return. Therefore, one of the main sources of volatility in this framework is unexpected changes in asset prices.

A.7 Market Clearing Condition

The standard market clearing condition is augmented by considering the cost of capital utilization and bankruptcy:

(19) y t = c t + i t + c t k u t i l + c t b a n k r u p t .

Nevertheless, these additional terms are negligible under a reasonable parameterization (De Graeve 2008). Moreover, we consider a measurement error on y t .[38]

A.8 Central Bank

Finally, we close the model with a Taylor rule where the short-term nominal interest rate set by the central banker reacts to inflation, changes in inflation, output gap, and output gap growth. The output gap is defined as the ratio between output determined in the economy featuring price-wage rigidity and the one determined in a fully-flexible economy:

(20) R t n R n = R t − 1 n R n ρ π t π r π Y t Y t f r y 1 − ρ π t / π π t − 1 / π r Δ π Y t / Y t f Y t − 1 / Y t − 1 f r △ y e x p η t R .

In log-linearized form:

(21) R t n = ρ R t − 1 n + ( 1 − ρ ) r π π t + r y y t − y t f + r △ π ( π t − π t − 1 ) + r △ y ( y t − y t f − y t − 1 − y t − 1 f ) + η t R ,

where η t R is an i.i.d. process.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/10.1515/bejm-2021-0244).

Received: 2021-06-07
Revised: 2022-02-23
Accepted: 2022-03-10
Published Online: 2022-04-07

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Advances
  3. A New General Equilibrium Welfare Measure, with Application to Labor Income Taxes
  4. Labor Share Dynamics and Factor Complementarity
  5. Effect of Monetary Policy on Government Spending Multiplier
  6. News-Driven Housing Booms: Spain Versus Germany
  7. Sovereign Debt Crisis, Fiscal Consolidation, and Active Central Bankers in a Monetary Union
  8. Housing Taxation and Economic Growth: Analysis of a Balanced-Growth Model with Residential Capital
  9. Intergenerational Linkages, Uncertain Lifetime and Educational and Health Expenditures
  10. Contributions
  11. Tolerance of Informality and Occupational Choices in a Large Informal Sector Economy
  12. Uncertainty Shocks, Innovation, and Productivity
  13. Asymmetric Effects of Private Debt on Income Growth
  14. Interpreting Structural Shocks and Assessing Their Historical Importance
  15. Charge-offs, Defaults and the Financial Accelerator
  16. Filtering Persistent and Asymmetric Cycles
  17. Population Aging and Convergence of Household Credit
  18. Robustly Optimal Monetary Policy in a Behavioral Environment
  19. Forward Guidance Effectiveness in a New Keynesian Model with Housing Frictions
  20. The Welfare Effects of Social Insurance Reform in the Presence of Intergenerational Transfers
https://doi.org/10.1515/bejm-2021-0244
Schlagwörter für diesen Artikel
medium-scale DSGE model; financial frictions; structural shocks; major recessions

Artikel in diesem Heft

  1. Frontmatter
  2. Advances
  3. A New General Equilibrium Welfare Measure, with Application to Labor Income Taxes
  4. Labor Share Dynamics and Factor Complementarity
  5. Effect of Monetary Policy on Government Spending Multiplier
  6. News-Driven Housing Booms: Spain Versus Germany
  7. Sovereign Debt Crisis, Fiscal Consolidation, and Active Central Bankers in a Monetary Union
  8. Housing Taxation and Economic Growth: Analysis of a Balanced-Growth Model with Residential Capital
  9. Intergenerational Linkages, Uncertain Lifetime and Educational and Health Expenditures
  10. Contributions
  11. Tolerance of Informality and Occupational Choices in a Large Informal Sector Economy
  12. Uncertainty Shocks, Innovation, and Productivity
  13. Asymmetric Effects of Private Debt on Income Growth
  14. Interpreting Structural Shocks and Assessing Their Historical Importance
  15. Charge-offs, Defaults and the Financial Accelerator
  16. Filtering Persistent and Asymmetric Cycles
  17. Population Aging and Convergence of Household Credit
  18. Robustly Optimal Monetary Policy in a Behavioral Environment
  19. Forward Guidance Effectiveness in a New Keynesian Model with Housing Frictions
  20. The Welfare Effects of Social Insurance Reform in the Presence of Intergenerational Transfers
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