**6. Conclusions**

In this paper, the impact of total government, non-military and military spending shocks on the U.S. economy was assessed. We accounted for the established evidence that public spending shocks have changed substantially in the post-1980s. Therefore, we estimated our DSGE model with recent Bayesian techniques for two sample periods: 1954:3–1979:2 and 1983:1–2008:2. Our new Keynesian DSGE model featured limited asset market participation as a potential institutional explanation for different degrees of fiscal policy effectiveness. Therefore, our model allowed us to relate the differences in the transmission of public spending shocks to important financial changes in the U.S. economy.

Our results suggested that asset market participation increased noticeably in the post-1980s, in line with previous evidence in the economic literature. Moreover, we found that an exogenous increase in total governmen<sup>t</sup> spending led to a higher output, but decreased consumption. Our findings also indicated that, in the first sub-sample, an increase in non-military spending induced a crowding-in effect on consumption. On the contrary, positive shocks to military spending tended to depress private consumption. We also found that military spending shocks had a less positive effect on output than civilian spending shocks in both sub-samples. Finally, we assessed the role of monetary policy in the presence of different public spending shocks. Our findings suggested that a more aggressive monetary policy tended to lower private consumption and output.

Overall, our results indicated that the U.S. economy seemed to benefit from increases in non-military spending. On the other hand, the military Keynesianism hypothesis, which still has many supporters in the U.S., can be at least questionable. The policy implications that can be drawn from our analysis suggested that switching governmen<sup>t</sup> priorities in favour of supplying civilian goods and services, rather than financing federal defence spending, should foster the U.S. economy.

As future work, it will be intriguing to extend this work by considering a Markov switching rational expectation new-Keynesian model in order to analyse in more detail the change in volatility of fiscal spending shocks in the pre- and post-financial liberalisation periods.

**Author Contributions:** The work was equally divided between the two co-authors.

**Funding:** This work was supported by the Department of Political Science, University of Perugia.

**Acknowledgments:** We thank the editor, the three anonymous referees and Francesco Ravazzolo for their useful comments and suggestions that improved this work.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Maximization Problems of the Model**

From the asset holders utility maximization problem, we obtain the following FOCsfor *CA*,*<sup>t</sup>* and *LA*,*t*:

$$\lambda\_t = -\frac{L\_{A,t}^{\phi}}{\left(\mathbb{C}\_{A,t} L\_{A,t}^{\phi}\right)^{\sigma}} \frac{1}{P\_t} \tag{A1}$$

$$\frac{q\mathbb{C}\_{A,t}L\_{A,t}^{\varphi-1}}{\left(\mathbb{C}\_{A,t}L\_{A,t}^{\varphi}\right)^{\sigma}} = -\lambda\_t \left[\left(1-\tau\right)W\_t\right] \tag{A2}$$

Putting (A1) into (A2), we obtain the labour decision equation:

$$\frac{C\_{A,t}}{L\_{A,t}} = \frac{(1-\tau)}{\varrho} \frac{W\_t}{P\_t} \tag{A3}$$

The FOC for *BA*,*t*+<sup>1</sup> is:

$$
\lambda\_t \frac{1}{R\_t} = \lambda\_{t+1} \beta \tag{A4}
$$

Putting (A1) into (A5), we obtain the Euler equation:

$$\frac{1}{R\_t} = \beta \left(\frac{\mathbb{C}\_{A,t}}{\mathbb{C}\_{A,t+1}}\right)^{\sigma} \left(\frac{L\_{A,t+1}}{L\_{A,t}}\right)^{\rho(1-\sigma)} \frac{P\_t}{P\_{t+1}} \tag{A5}$$

Thus:

$$\mathcal{R}\_t^{-1} = \beta E\_t \left[ \Lambda\_{t, t+1} \right]$$

where:

$$
\Lambda\_{t,t+s} = \beta^s \left(\frac{\mathbb{C}\_{A,t}}{\mathbb{C}\_{A,t+s}}\right)^{\sigma} \left(\frac{L\_{A,t+s}}{L\_{A,s}}\right)^{q(1-\sigma)} \frac{P\_t}{P\_{t+s}} \tag{A6}
$$

This is the stochastic discount factor.

From the non-asset holders utility maximization problem, we obtain the following FOCs for *C N*,*t* and *LN*,*t*:

$$
\lambda\_t = \frac{L\_{N,t}^{\varphi}}{\left(C\_{N,t}L\_{N,t}^{\varphi}\right)^{\sigma}} \frac{1}{P\_t} \tag{A7}
$$

$$\frac{q\mathbb{C}\_{N,t}L\_{N,t}^{q-1}}{\left(\mathbb{C}\_{N,t}L\_{N,t}^{q}\right)^{\sigma}} = \lambda\_t \left[\left(1-\tau\right)\mathcal{W}\_t\right] \tag{A8}$$

Putting (A8) into (A9) gives the labour decision equation:

$$\frac{C\_{N,t}}{L\_{N,t}} = \frac{(1-\tau)}{\varrho} \frac{W\_t}{P\_t} \tag{A9}$$

Given the following production function for intermediate goods:

$$N\_t\left(i\right) = N\_t\left(i\right) - F\_\prime \tag{A10}$$

we can write real profits as:

$$\mathcal{O}\_{t}\left(\dot{i}\right) \equiv \left[\frac{P\_{t}\left(\dot{i}\right)}{P\_{t}}\right] \mathcal{Y}\_{t}\left(\dot{i}\right) - \left[\frac{\mathcal{W}\_{t}}{P\_{t}}\right] \mathcal{N}\_{t}\left(\dot{i}\right) \tag{A11}$$

A firm *i* sets *P* (*i*) in order to solve the following problem:

$$\begin{aligned} \max\_{\left\{P\_{t}^{\star}(i)\right\}} & E\_{t} \sum\_{s=0}^{\infty} a^{s} \Lambda\_{t,t+s} \left[P\_{t}^{\star}\left(i\right) Y\_{t,t+s}\left(i\right) - \mathcal{W}\_{t+s} Y\_{t,t+s}\left(i\right)\right] \\ \text{s.t.} & \left. Y\_{t}\left(i\right) = \left(\frac{P\_{t}^{\star}\left(i\right)}{P\_{t}}\right)^{-x} Y\_{t} \end{aligned}$$

that is:

$$\max\_{\left\{P\_{t}^{\star}(i)\right\}} E\_{t} \sum\_{s=0}^{\infty} a^{s} \Lambda\_{t, t+s} \left[ P\_{t}^{\star} \left(i\right) \left(\frac{P\_{t}^{\star}\left(i\right)}{P\_{t}}\right)^{-\varepsilon} Y\_{t} - \mathcal{W}\_{t} \left(\frac{P\_{t}^{\star}\left(i\right)}{P\_{t}}\right)^{-\varepsilon} Y\_{t} \right]$$

The FOC is given by:

$$E\_t \sum\_{s=0}^{\infty} a^s \Lambda\_{t, t+s} \left[ P\_t^\* \left( i \right) - \frac{\varepsilon}{\varepsilon - 1} W\_{t+s} \right] = 0 \tag{A12}$$

#### **Appendix B. Steady States**

The Euler equation in the steady state gives:

$$R = \frac{1}{\beta} \tag{A13}$$

In the steady state, from the FOC of the price setting in the intermediate goods firm's problem, we have for the real wage:

$$\frac{W}{P} = \frac{\varepsilon - 1}{\varepsilon} \tag{A14}$$

we can rewrite (A14) as:

$$\frac{\mathcal{W}}{P} = \frac{Y}{N} \frac{1 + F\_Y}{1 + \mu} \tag{A15}$$

The ratio of profits to output is given by:

$$O\_Y \equiv \frac{\mu - F\_Y}{1 + \mu} \tag{A16}$$

We assume, in the steady state, that:

$$N\_N = N\_A = N \tag{A17}$$

Because of preference homogeneity, we need to ensure that steady-state consumption shares are also equal across groups. This can be seen comparing the two labour decision equations evaluated in the steady state:

$$\frac{\mathbf{C}\_A}{L} = \frac{1-\tau}{\varrho} \frac{\mathcal{W}}{P} = \frac{\mathbf{C}\_N}{L} \tag{A18}$$

implying:

$$\mathbb{C}\_{A} = \mathbb{C}\_{N} = \mathbb{C} \tag{A19}$$

The steady-state coefficients needed for our log-linear approximation above are fully determined as:

$$(1 - \tau) \frac{W}{P} \frac{N}{Y} = (1 - \tau) \frac{1 + F\_Y}{1 + \mu} \tag{A20}$$

$$\frac{C\_N}{Y} = (1 - \tau) \frac{1 + F\_Y}{1 + \mu} - T\_Y \tag{A21}$$

$$T\_Y = G\_Y - \mathfrak{r} \tag{A22}$$

$$\frac{C\_A}{Y} = (1 - \pi) \frac{1}{1 - \lambda} \left( 1 - \lambda \frac{1 + F\_Y}{1 + \mu} \right) - T\_Y \tag{A23}$$

We thus achieve equalization of steady-state consumption shares by making an assumption on technology. Specifically, we ensure that asset income in the steady state is zero. This requires assuming that the fixed cost of production is characterised by:

$$F\_Y = \mu \tag{A24}$$

Substituting in (A22) gives:

$$\frac{C\_A}{Y} = \frac{C\_N}{Y} = 1 - \tau - T\_Y = 1 - G\_Y \tag{A25}$$

We want to find hours in steady state. Given the equalization of hours and consumption between the two groups and normalizing *P* = 1, the intratemporal optimality condition implies:

$$\left(\left(1-\tau\right)\,\mathrm{WN}-T=\frac{\left(1-\tau\right)}{\varrho}\,\mathrm{W}\left(1-N\right)\tag{A26}$$

dividing by *Y* and using (A20) and the expression for the fixed cost, we obtain the following expression for the steady-state hours:

$$\frac{N}{1-N} = \frac{1}{q} \frac{1-\tau}{1-G\_Y} \tag{A27}$$

Given *τ* and *GY*, we chose the steady state *N* to match average hours worked. From (A27), this implies a unique value for *ϕ*.

#### **Appendix C. The Log-Linearized Model**

Below, we show the log-linearized equations of our model around the non-stochastic steady state. We denote by small letters the log deviation of a variable from its steady-state value, while for any variable *Xt*, *X* stands for its steady-state value and *XY* its steady-state share in output, *X*/*Y*.

The log-linearized Euler equation for asset-holders is given by:

$$\varepsilon\_{A,t} = E\_t \varepsilon\_{A,t+1} - \frac{1}{\sigma} \left( r\_t - E\_t \pi\_{t+1} \right) + \left( \frac{1}{\sigma} - 1 \right) \left( 1 + \frac{T\_Y}{1 - G\_Y} \right) \left( E\_t n\_{A,t+1} - n\_{A,t} \right) \tag{A28}$$

The log-linearization of the labour decision equation for asset holders is given by:

$$\frac{N}{1-N}n\_{A,t} = w\_t - c\_{A,t} \tag{A29}$$

The log-linearized labour decision equation for non-asset holders is equal to:

$$\frac{N}{1-N}\mathbf{n}\_{N,t} = \mathbf{w}\_t - \mathbf{c}\_{N,t} \tag{A30}$$

The consumption for non-asset holders is obtained log-linearizing their budget constraint and is given by:

$$(1 - G\_Y)c\_{N,t} = (1 - \tau) \left( w\_t + n\_{N,t} \right) - T\_Y t\_t \tag{A31}$$

From the last two relations, we obtain a reduced-form labour supply for non-asset holders:

$$m\_{N,t} = \frac{\varrho}{1+\varrho} \left[ \frac{-T\_Y}{1-G\_Y+T\_Y} \right] (w\_l - t\_l) \tag{A32}$$

The log-linearized expression for aggregate hours is given by:

$$n\_t = \lambda n\_{N,t} + \left(1 - \lambda\right) n\_{A,t} \tag{A33}$$

The log-linearized expression for aggregate consumption is given by:

$$
\mathbf{c}\_{l} = \lambda \mathbf{c}\_{N,t} + (1 - \lambda) \mathbf{c}\_{A,t} \tag{A34}
$$

The log-linearized aggregate production function is given by:

$$\left(\;y\_t = (1 + F\_Y)\;\right)\_{tt} \tag{A35}$$

We note that the share of the fixed cost *F* in the steady-state output governs the degree of increasing returns to scale. The log-linearized new-Keynesian Phillips curve is given by:

$$
\pi\_t = \beta E\_t \pi\_{t+1} + \frac{\left(1 - a\right)\left(1 - a\beta\right)}{a} w\_t \tag{A36}
$$

In both models of aggregate governmen<sup>t</sup> spending and disaggregated non-military and military components, the log-linearization of the budget constraint around a steady state with zero debt and a balanced primary budget gives the following expression:

$$
\beta b\_{t+1} = b\_t + G\_{\Upsilon \mathcal{G}\_t} - T\_T t\_t - \tau y\_t \tag{A37}
$$

Moreover, in the model with disaggregated non-military and military spending, we have that:

$$
\mathcal{g}\_t G\_Y = \mathcal{N} M\_Y nm\_t + M\_Y m\_t \tag{A38}
$$

The log-linearized structural primary deficit is given by:

$$d\_{s\mathcal{I}} = G\_Y g\_t - T\_Y t\_l \tag{A39}$$

Finally, the log-linearized goods market clearing can be written as:

$$
\mathfrak{g}\_l = \mathfrak{g}\_l \mathbf{G}\_Y + \mathfrak{c}\_l \left( 1 - \mathbf{G}\_Y \right) \tag{A40}
$$

#### **Appendix D. Diagnostic Tests**

*Appendix D.1. Prior and Posterior Distributions*

> S1

(1954:Q3–1979:Q2)

**Figure A1.** Total governmen<sup>t</sup> spending model. Notes: In the above graphs, the grey lines represent the prior distributions, whereas the black lines correspond to the posterior distributions.

42

**Figure A2.** Non-military and military spending model. Notes: In the above graphs, the grey lines represent the prior distributions, whereas the black lines correspond to the posterior distributions.

*Appendix D.2. Monte Carlo Markov Chain Univariate Diagnostics*

**Figure A3.** Total governmen<sup>t</sup> spending model: S1 (1954:Q3–1979:Q2). Notes: In the above graphs, the blue lines represent the 80% interval range based on the pooled draws from all sequences, whereas the red lines indicate the mean interval based on the draws of the individual sequences. The first column shows the convergence diagnostics for the 80% interval. The second and the third column with labels denote an estimate of the same statistics for the second and third central moments.

**Figure A4.** Total governmen<sup>t</sup> spending model, S2 (1983:Q1–2008:Q2). Notes: In the above graphs, the blue lines represent the 80% interval range based on the pooled draws from all sequences, whereas the red lines indicate the mean interval based on the draws of the individual sequences. The first column shows the convergence diagnostics for the 80% interval. The second and the third column with labels denote an estimate of the same statistics for the second and third central moments.

**Figure A5.** Non-military and military spending model: S1 (1954:Q3–1979:Q2). Notes: In the above graphs, the blue lines represent the 80% interval range based on the pooled draws from all sequences, whereas the red lines indicate the mean interval based on the draws of the individual sequences. The first column shows the convergence diagnostics for the 80% interval. The second and the third column with labels denote an estimate of the same statistics for the second and third central moments.

**Figure A6.** Non-military and military spending model: S2 (1983:Q1–2008:Q2). Notes: In the above graphs, the blue lines represent the 80% interval range based on the pooled draws from all sequences, whereas the red lines indicate the mean interval based on the draws of the individual sequences. The first column shows the convergence diagnostics for the 80% interval. The second and the third column with labels denote an estimate of the same statistics for the second and third central moments.

#### *Appendix D.3. Multivariate Convergence Diagnostics*

S1 (1954:Q3–1979:Q2)

**Figure A7.** Total governmen<sup>t</sup> spending model. Notes: In the above graphs, the diagnostics is based on the range of the posterior likelihood function.

S1

**Figure A8.** Non-military and military spending model. Notes: In the above graphs, the diagnostics is based on the range of the posterior likelihood function.

#### *Appendix D.4. Smoothed Shocks*

S1

**Figure A9.** Total governmen<sup>t</sup> spending model. Notes: In the above graphs, the black line represents the estimate of the smoothed structural shocks.

S1

**Figure A10.** Non-military and military spending model. Notes: In the above graphs, the black line represents the estimate of the smoothed structural shocks.

S1 (1954:Q3–1979:Q2)

**Figure A11.** Total governmen<sup>t</sup> spending model. Notes: In the above graphs, the dotted black lines indicate the observed data. The red lines indicate the estimates of the smoothed variables.

**Figure A12.** Non-military and military spending model. Notes: In the above graphs, the dotted black lines indicate the observed data. The red lines indicate the estimates of the smoothed variables.

#### *Appendix D.6. Parameters' Identification*

**Figure A13.** Total governmen<sup>t</sup> spending model. Notes: In the above graphs, blue bars indicate the identification strength of the parameters based on their prior means, whereas orange bars denote the identification strength of the parameters based on their standard deviations.

S1 (1954:Q3–1979:Q2)

**Figure A14.** Non-military and military spending model. Notes: In the above graphs, blue bars indicate the identification strength of the parameters based on their prior means, whereas orange bars denote the identification strength of the parameters based on their standard deviations.

#### **Appendix E. Estimated Impulse Response Functions**

**Figure A15.** Total governmen<sup>t</sup> spending shock. Notes: The above graphs show the responses of the key variables together with their 95% confidence intervals.

**Figure A16.** Non-military spending shock. Notes: The above graphs show the responses of the key variables together with their 95% confidence intervals.

**Figure A17.** Military spending shock. Notes: The above graphs show the responses of the key variables together with their 95% confidence intervals.

#### **Appendix F. Benchmark Model vs. DSGE-VARs**

**Table A1.** Comparison between the benchmark model and DSGE-VARs: model with total governmen<sup>t</sup> spending.


Notes: As in Bekiros and Paccagnini (2014), the DSGE-VARs are estimated with different numbers of lags (from 1–4). The tightness parameter is set equal to 0.5.



Notes: As in Bekiros and Paccagnini (2014), the DSGE-VARs are estimated with different numbers of lags (from 1–4). The tightness parameter is set equal to 0.5.
