The analysis based on our benchmark model with input complementarity in the previ-
ous section shows that complementarity-induced procyclical returns to scale can gener-
Figure 3.5: Frisch elasticity : Increase in productivity by 1 %
Note. Y-axis represents a percent deviation from steady state. Blue solid lines represent
the model with a normalized Translog with micro-consistent Frisch elasticity of 0.86. Green dashed lines represent the model with a Cobb-Douglas with micro-consistent Frisch elasticity of 0.86. Red dotted-dashed lines represent the model with a
Cobb-Douglas with macro-consistent Frisch elasticity of 3.31.
countercyclical markups. This strongly suggests that even if we allow realistic frictions
in the model, its reliance on either price or wage markup shocks will drastically decrease.
Because the main criticism by Chari, Kehoe, and McGrattan,2009of the New Keynesian
literature concerns its reliance on “dubiously structural shocks” (i.e., price markup shock,
wage markup shock, government spending shock, risk premium shock), it would be an
interesting exercise to re-do business cycle accounting after allowing complementarity-
induced procyclicality.
To this end, we extend the well-known medium-scale DSGE model by Smets and
Wouters,2007by using the normalized Translog production function with energy input.²⁶
A full description of the model can be found in the online Technical Appendix. In the fol-
lowing analysis, we compare two models : (i) the original Smets and Wouters,2007model
in which, as in the original model, we assume Cobb-Douglas technology and do not explic-
itly include energy²⁷(SW model); and (ii) the extended version of the Smets and Wouters,
2007 model with normalized Translog technology and energy input (HK model).²⁸ We perform variance decomposition and compare across two models. Here, we present the
main variance decomposition results. The estimation result of the structural parameters
can be found in Appendix.
Table3.4provides the 10-quarter (business cycle frequency) conditional variance de- compositions of the HK model and Table3.5shows that of the SW model. In SW model, 27% of the output fluctuation is explained by wage markup shock. In HK model, wage
markup shock only explains 12% of the output variation. Instead, the role of investment
shock and monetary policy shock increases from 15% to 28% and 3% to 9%, respectively. In-
vestment shock and monetary policy shock are classified as “structural” shocks in Chari,
Kehoe, and McGrattan, 2009. In general, the role of “suspicious shocks” drastically de-
creases and that of “structural shocks” increases in the HK model. Table3.6and3.7provide unconditional variance decomposition for the HK model and the SW model, respectively.
Again, one can easily verify that the role of wage markup shock drastically decreases and
the roles of investment shock and monetary policy shock increase.
aggregators throughout the analysis, following Cantore et al.,2015. The results are not sensitive to the choice of aggregator.
²⁷Even if we include energy input, variance decomposition results are similar to the model without energy as long as we use Cobb-Douglas technology.
²⁸For direct comparison between the two models, we shut down energy price shock and assumed it to be constant.
Table 3.4: Conditional Variance Decomposition: 10 Quarters, HK model
Output Cons. Invest. Hours
Prod. 28.97 17.79 12.32 4.06 Risk P. 3.93 11.03 0.48 6.05 Gov. 9.82 3.29 0.65 16.27 Invest. 28.00 9.15 73.90 25.84 Monetary 9.05 15.76 4.00 11.60 P. Mark 8.70 10.40 5.60 8.02 W. Mark 11.52 32.58 3.05 28.17
Table 3.5: Conditional Variance Decomposition: 10 Quarters, SW model
Output Cons. Invest. Hours
Prod. 32.87 16.33 14.55 3.44 Risk P. 3.08 7.54 0.24 4.76 Gov. 7.96 4.57 2.16 13.95 Invest. 14.64 0.77 65.62 13.37 Monetary 3.15 5.34 0.96 4.39 P. Mark 10.81 9.86 7.86 11.61 W. Mark 27.48 55.60 8.62 48.48
Note. Table3.4and3.5provide 10 quarter conditional variance decomposition for the HK model and the SW model, respectively. In the HK model, we assumed a normalized Translog production function with energy input. For direct comparison with the SW model, we shut down the energy price shock and assumed it to be constant. In the SW model, we assume a Cobb-Douglas production function without energy input. The only difference between our SW model and the original Smets and Wouters,2007model is that we used the Dixit-Stiglitz aggregator instead of the Kimball aggregator.
Finally, Figure 3.6 plots the energy data and smoothed energy input timeseries ob- tained from our estimated model.²⁹ As shown in the graph, our model closely mimics the true dynamics of energy input even though we did not explicitly target any moment
related to the energy market. This shows the external validity of our model.
²⁹Energy data refers to “Total Energy Consumed by the Industrial Sector” from the U.S. Energy Infor- mation Administration. Both energy data and model-derived smoothed energy input are HP filtered.
Table 3.6: Unconditional Variance Decomposition, HK model
Output Cons. Invest. Hours
Prod. 33.42 17.25 20.89 2.65 Risk P. 1.62 2.71 0.31 2.81 Gov. 6.32 7.18 2.44 14.05 Invest. 14.44 10.51 49.87 14.21 Monetary 4.40 4.73 3.04 5.95 P. Mark 5.66 4.69 5.47 5.26 W. Mark 34.13 52.92 17.97 55.06
Table 3.7: Unconditional Variance Decomposition, SW model
Output Cons. Invest. Hours
Prod. 27.05 10.79 20.25 1.48 Risk P. 1.02 1.39 0.15 1.67 Gov. 3.84 4.64 4.28 9.50 Invest. 5.33 2.51 41.82 5.83 Monetary 1.05 1.00 0.58 1.55 P. Mark 6.34 4.19 8.07 5.80 W. Mark 55.36 75.49 24.84 74.16
Note. Table3.6and3.7provide unconditional variance decomposition for the HK model and the SW model, respectively. In the HK model, we assumed a normalized Translog production function with energy input. For direct comparison with the SW model, we shut down energy price shock and assumed it to be constant. In the SW model, we assumed a Cobb-Douglas production function without energy input. The only difference between our SW model and the original Smets and Wouters,2007model is that we used the Dixit-Stiglitz aggregator instead of the Kimball aggregator.
3.7 Conclusion
In this paper, we studied the business cycle with normalized Translog production function
exhibiting complementarity-induced procyclical returns to scale. Through our empirical
study, we identified a complementarity between labor and energy that leads to procycli-
cal returns to scale, which is not compatible with the tightly parametrized production
functions commonly used in the literature such as Cobb-Douglas or CES. The normalized
Figure 3.6: Energy Input Dynamics : Data vs. Model
Note. Data refers to “Total Energy Consumed by the Industrial Sector” from the U.S.
Energy Information Administration. Model refers to smoothed energy input time series obtained from the estimated HK model. Both “Data” and “Model” are HP filtered.
cyclical returns to scale but is also consistent with the balanced growth path. We showed
that a simple calibrated business cycle model with the proposed production function gen-
erates strikingly data-consistent aggregate variable fluctuations due to demand shock
without relying on nominal rigidities or countercyclical markups. Our model also pro-
duces a stronger amplification effect than the model without complementarity. We then
showed that by incorporating our production function into the benchmark medium-scale
New Keynesian model (Smets and Wouters,2007), input complementarity leads to a dra-
matic decrease in the role of the “suspicious shocks” compared to that of the “structural
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