Government Consumption

Abstract

Empirically, government expenditures represent a large share of total demand and significantly affect output, employment, and welfare. In the introductory Sect. 4.2 of this chapter, we document some selected empirical facts of government consumption. In particular, we find that government consumption is procyclical, and after an unexpected increase in consumption, output, employment, and (to a smaller extent) private consumption all increase.

Appendices

Appendix 4.1: Reverse Shooting

To compute the transition dynamics in the Numerical Example of Sect. 4.3, we first have to set the length of the transition period. When is the transition complete? We will assume that the dynamics are complete when the state variables are reasonably close to the new steady-state values. For practical reasons, we will stop searching for a better length of the transition if the divergence between the value of the state variable K _t in the last period of the transition from the steady-state value is less than 0.01% or if the value of the divergence is small, e.g., less than 10⁻⁵ in absolute value. The latter number is used when the state variable is very small and 0.01% of the state variable would be close to the accuracy of the solution algorithm (of the non-linear equation solver).

As we will discover in the remainder of the book, the transition in the neoclassical growth model occurs relatively fast compared to that in the other benchmark model that we consider in this book, the overlapping generations (OLG) model. Often, a transition length of fewer than 50 periods is sufficient, while we need to consider more than 100 periods in the OLG model in later chapters. We will choose 40 periods for the present case.

In essence, we have to solve for the dynamics of the three endogenous variables K _t, $C^p_t$, and L _t for t = 1, …, 40 given $K_0=\bar K$, $L_0=\bar L$ and $C^p_0=\bar C^p$ for the initial values and $L_{41}=\tilde L$, $K_{41}=\tilde K$, and $C^p_{41}=\tilde C^p$ for the final values in case 1 and $L_{41}=\bar L$, $K_{41}=\bar K$, and $C^p_{41}=\bar C^p$ in case 2. More formally, we have to solve a two-point boundary value problem.

Let us first consider case 1 with a permanent change in government consumption $G_t=\tilde G$ for t = 1, …. For the solution, we will make use of the first-order conditions, which we restate for your convenience:

$$\displaystyle \begin{aligned} w_t & = \frac{\kappa}{\phi}\left(\frac{C_t}{1-L_t}\right)^{\frac{1}{\rho}} \left(\varXi_t \right)^{1-\frac{1}{1-1/\rho_c}} \left( C^p_t\right)^{\frac{1}{\rho_c}}, \end{aligned} $$

(4.75a)

$$\displaystyle \begin{aligned} \beta (1+r_{t+1}-\delta) &= \left(\frac{X_{t+1}}{X_t}\right)^{1-\frac{1-\sigma}{1-1/\rho_c}} \left(\frac{C_{t+1}}{C_t}\right)^{\frac{1}{\rho}} \left(\frac{\varXi_{t+1}}{\varXi_t}\right)^{1-\frac{1}{1-1/\rho_c}} \left(\frac{C^p_{t+1}}{C^p_t}\right)^{\frac{1}{\rho_c}}, \end{aligned} $$

(4.75b)

$$\displaystyle \begin{aligned} K_{t+1}&=(1-\delta) K_t + K_t^\alpha L_t^{1-\alpha}-G_t-C^p_t, \end{aligned} $$

(4.75c)

where effective consumption C _t and the real interest rate r _t are given by (4.21) and (4.9b), and X _t is defined as follows:

$$\displaystyle \begin{aligned} X_t\equiv C_t^{1-\frac{1}{\rho}}+\kappa (1-L_t)^{1-\frac{1}{\rho}}.\end{aligned} $$

(4.76)

In the first period of the transition at t = 1, the government unexpectedly increases public consumption G _t. As a consequence, the household can only adjust its behavior in period t = 1, not before. The beginning-of-period capital stock K ₁, therefore, is predetermined by the decision of the household in period t = 0. The household can only choose consumption $C^p_1$, labor L ₁, and the next-period capital stock K ₂. As a consequence, we have to solve for the 119 unknowns $\{K_t\}_{t=2}^{40}$, $\{L_t\}_{t=1}^{40}$, and $\{C^p_t\}_{t=1}^{40}$. We require that K ₄₀ diverges by less than 0.01% from $\tilde K=1.2946$.

The algorithm that we will apply is called reverse shooting. We will provide a guess of K ₄₀ that is close to the new steady-state value $K_{41}=\tilde K$. We can compute the values L ₄₀ and $C^p_{40}$ with the help of the first-order conditions (4.75a) and (4.75c) for period t = 40 and using the steady-state values $K_{41}=\tilde K$, $L_{41}=\tilde L$, and $C^p_{41}=\tilde C^p$ for the values of the endogenous variables in period t + 1 = 41. The problem is to solve a system of two non-linear equations in two unknowns, L ₄₀ and $C^p_{40}$. As the initial guess that we have to provide to the non-linear equation solver routine used in the Gauss program Ch4_subs_private_pub_dyn.g, we simply take the new steady-state values $\tilde L$ and $\tilde C^p$, respectively.

With the help of the values $\{K_{40},L_{40},C^p_{40}\}$, we can compute the values $\{K_{39},L_{39},C^p_{39}\}$ using the three non-linear equations (4.75a), (4.75b), and (4.75c). More generally, we can compute $\{K_{t},L_{t},C^p_{t}\}$ with the help of $\{K_{t+1},L_{t+1},C^p_{t+1}\}$ in the same way for t = 39, 38, …, 1 providing $\{K_{t+1},L_{t+1},C^p_{t+1}\}$ as an initial value to the non-linear equation solver. In the final iteration, we have $\{K_1,L_1,C^p_1\}$. If K ₁ is close to its value in the initial steady state $\bar K$ we are done. Otherwise, we have to adjust our guess of K ₄₀ and retry.

How do we choose an initial guess K ₄₀? This is not a trivial task. First, we know that K ₁ is smaller than K ₄₁ from our steady-state computation in the Gauss program Ch4_subs_private_pub.g. Since we know from the standard continuous-time Ramsey model that the transition path is saddle-point stable and that the speed of convergence declines during the transition, we would expect K ₄₀ to lie very close to K ₄₁.⁶⁴ For this reason, we perform a grid search of the optimal start value K ₄₀ in the interval $[\bar K+0.9 (\tilde K-\bar K),\tilde K]$. We choose an equispaced grid of nk = 1, 000 points. For the lower values in this grid, the capital stock falls below zero in fewer than 40 periods during the recursive iteration over $\{K_t,L_t,C^p_t\}$, and we subsequently have to exclude these values. As it turns out, the optimal point (that produces a value of K ₁ closest to $\bar K$) is very close to $\tilde K$, and again, we use a much tighter grid, $[\bar K+0.999 (\tilde K-\bar K),\tilde K]$, and find K ₄₀ = 1.2945418 as our initial guess. Given this value, we simply find the solution to the non-linear equation $f(K_1)=K_1-\bar K=0$.

How does our algorithm perform? We compute a value of K ₁ that diverges by less than 10⁻⁸ from the old steady-state value $\bar K=1.2882145$. Therefore, we have very accurately computed the transition dynamics. Let us conclude this section with some qualifying remarks:

• 1. If we had chosen a higher number for the transition periods than 40, we might have failed to compute the transition dynamics. Why? Let us consider the final values of K ₃₉ and K ₄₀. We have computed the values of K ₃₉ = 1.2945418 and K ₄₀ = 1.2945502 for an accuracy of 10⁻⁸. The values are very close to one another, and the transition is basically complete after 30–35 periods. If we had chosen an even higher number, say 100, we would not have been able to provide an initial guess for K ₁₀₀ that is different from $K_{101}=\tilde K$ given machine accuracy. If we, however, use K ₁₀₀ = K ₁₀₁, our algorithm will just compute the steady-state values for L ₁₀₀ and $C^p_{100}$ in the first step and for all other values $\{K_t,L_t,C^p_t\}$ for t = 99, …, 1. If we, however, consider a smaller value, e.g., K ₁₀₀ = 1.2946334 instead of $\tilde K=1.2946335$, we will obtain a negative value for K _t in our recursive iteration for t > 1 and be unable to find a solution.

• 2. Is it possible to solve for the dynamics if we select K ₂ as a starting value and iterate forward (so-called forward shooting)? Yes. The reason is as follows: Given K ₁ and K ₂, we can solve for $C^p_1$ and L ₁ using (4.75a) and (4.75c). In the next step, we seek to solve for $\{K_3,C^p_2,L_2\}$. Therefore, we use (4.75a) and (4.75c) for period t = 2 and (4.75b) for period t = 1. The solution $\{K_3,C^p_2,L_2\}$ is used in the next step. We iterate forward until we have found the solution for $\{K_{41},C^p_{40},L_{40}\}$ and compare K ₄₁ to the new steady-state value $\tilde K$. If we are close, we are done. Otherwise, we have to specify a new guess for $\tilde K$.⁶⁵

  In Chap. 6, you will encounter a problem (an OLG model with pay-as-you-go pensions) where reverse shooting is possible, while forward shooting is not. In practice, reverse shooting is often used even if forward shooting is possible. The reason is because it is often easier to provide an initial guess for the capital stock in the last period rather than the first period of the transition because the speed of convergence declines and you have to search in a smaller neighborhood around the final steady state than in the case for a guess of the capital stock in the initial period. If you use forward shooting, you have to search in a larger neighborhood of the initial steady state. This is particularly cumbersome if your state variable is not a single variable but is multi-dimensional.

• 3. In the case of a temporary government shock, we know that the new steady-state values are equal to the old steady-state values. In this case, however, the capital stock K _t approaches K ₄₁ from above. Therefore, we have to provide a guess for K ₄₀ that is larger than $K_{41}=\bar K$. The rest of the computational procedure is completely equivalent to the case of a permanent increase in government consumption.

Appendix 4.2: Frisch Labor Supply Elasticity for Cobb-Douglas Utility

The Frisch labor supply elasticity or intertemporal labor supply elasticity η _L,w is defined as the percentage change in the labor supply in response to a 1% increase in the wage given a constant marginal utility of consumption u _C:

$$\displaystyle \begin{aligned} \eta_{L,w} \equiv \frac{d L}{dw}|{}_{u_C = \mbox{const}} \frac{w}{L}. \end{aligned} $$

(4.77)

Let utility u(C, L) be a function of consumption C and labor supply L. Ignoring taxes, contributions, and pensions, the first-order condition of the household with respect to its labor supply is given by:

$$\displaystyle \begin{aligned} -u_L(C,L) = w u_C(C,L). \end{aligned} $$

(4.78)

According to (4.78), the disutility from working another time unit is equal to the utility from the additional consumption that can be afforded by the additional wage income from working an additional time unit. The total differential of (4.78) for a constant u _C(C, L) is given by

$$\displaystyle \begin{aligned}- u_{LL} dL - u_{CL} dC = u_C dw.\end{aligned} $$

(4.79)

Furthermore, du _C = 0 implies

$$\displaystyle \begin{aligned} u_{CC} dC + u_{CL} dL =0,\end{aligned}$$

or

$$\displaystyle \begin{aligned} dC = -\frac{u_{CL}}{u_{CC}} dL.\end{aligned}$$

Inserting the last equation into (4.79), we obtain

$$\displaystyle \begin{aligned} \eta_{L,w} \equiv \frac{d L}{dw}|{}_{u_C = \mbox{const}} \frac{w}{L} = \frac{u_C}{\frac{u_{CL}^2}{u_{CC}}-u_{LL}} \frac{w}{L}, \end{aligned} $$

(4.80)

and for the Cobb-Douglas utility function with $u(C,L)=\frac {\left ( C^{\iota } (1-L)^{1-\iota }\right )^{1-\sigma }-1}{1-\sigma }$, the Frisch labor supply elasticity is

$$\displaystyle \begin{aligned} \eta_{L,w}= \frac{1-\iota (1-\sigma)}{\sigma} \frac{1-L}{L}.\end{aligned}$$

Appendix 4.3: Microfoundations of Calvo Price Setting

Let us consider a wholesale firm j with the relative price P _t+s(j)∕P _t+s.⁶⁶ In period t, the firm received the signal to choose its optimal relative price p _At = P _At∕P _t and, since then, has not received a signal to do so again up to period t + s. Between period t and t + s, the price of good j, P _t(j), increases with the lagged inflation rate π _t, π _t+1, …, π _t+s−1 in each period t + 1, t + 2, …, t + s, while the aggregate price level P _t increases with the inflation rates π _t+1, π _t+2, …, π _t+s:

$$\displaystyle \begin{aligned}\frac{P_{t+s}(j)}{P_{t+s}}=\frac{\pi_{t+s-1}\cdots\pi_t}{\pi_{t+s}\cdots\pi_{t+1}}p_{At}=\frac{\pi_t}{\pi_{t+s}}p_{At}.\end{aligned}$$

Accordingly, the firm will choose p _At in period t to maximize discounted dividends (after inserting the demand function (4.43) into dividends (4.46)):

$$\displaystyle \begin{aligned} \mathbb{E}_t \ \sum_{s=0}^\infty (\beta\varphi_y)^s \frac{\varLambda_{t+s}}{\varLambda_t} \left[ \left(\frac{\pi_t}{\pi_{t+s}}p_{At}\right)^{1-\epsilon_y}Y_{t+s}-g_{t+s}\left(\frac{\pi_t}{\pi_{t+s}}p_{At}\right)^{-\epsilon_y}Y_{t+s}\right],\end{aligned}$$

where (φ _y)^s denotes the probability that the firm cannot adjust its price for s consecutive periods.

Differentiating this equation with respect to p _At results in the first-order condition:

$$\displaystyle \begin{aligned}0= \mathbb{E}_t \sum_{s=0}^\infty (\beta\varphi_y)^s\frac{\varLambda_{t+s}}{\varLambda_t} \left[ (1-\epsilon_y)\left(\frac{\pi_t}{\pi_{t+s}}\right)^{1-\epsilon_y}Y_{t+s}p_{At}^{-\epsilon_y}+ \epsilon_y g_{t+s}\left(\frac{\pi_t}{\pi_{t+s}}\right)^{-\epsilon_y}Y_{t+s}p_{At}^{-\epsilon_y-1} \right],\end{aligned}$$

where we used the theorem that the derivative operator can be interchanged with the expectational operator and the sum operator.

The first-order equation can be re-written as

$$\displaystyle \begin{aligned} p_{At} &= \frac{\epsilon_y}{\epsilon_y-1}\frac{\varGamma_{1t}}{\varGamma_{2t}}, \end{aligned} $$

(4.81a)

$$\displaystyle \begin{aligned} \varGamma_{1t} &= \mathbb{E}_t \sum_{s=0}^\infty (\beta\varphi_y)^s \left(\frac{\pi_t}{\pi_{t+s}}\right)^{-\epsilon_y}g_{t+s}\varLambda_{t+s}Y_{t+s}= g_t \varLambda_t Y_t + (\beta\varphi_y) \mathbb{E}_t \left(\frac{\pi_t}{\pi_{t+1}}\right)^{-\epsilon_y}\varGamma_{1t+1}, \end{aligned} $$

(4.81b)

$$\displaystyle \begin{aligned} \varGamma_{2t} &= \mathbb{E}_t \sum_{s=0}^\infty (\beta\varphi_y)^s \left(\frac{\pi_t}{\pi_{t+s}}\right)^{1-\epsilon_y}\varLambda_{t+s}Y_{t+s}= \varLambda_t Y_t + (\beta\varphi_y) \mathbb{E}_t \left(\frac{\pi_t}{\pi_{t+1}}\right)^{1-\epsilon_y} \varGamma_{2t+1}. \end{aligned} $$

(4.81c)

Γ _1t and Γ _2t are simply auxiliary variables whose behaviors are described by (stochastic) first-order difference equations. Therefore, they are easily amenable to the solution with the linearization methods described in Appendix 2.3.

The price index (4.45) implies

$$\displaystyle \begin{aligned}P_t^{1-\epsilon_y}=(1-\varphi_y)P_{At}^{1-\epsilon_y}+\varphi_y P_{Nt}^{1-\epsilon_y}=(1-\varphi_y)P_{At}^{1-\epsilon_y}+\varphi_y (\pi_{t-1}P_{t-1})^{1-\epsilon_y}.\end{aligned}$$

The second equality follows from the updating rule (4.47) and the fact that the non-optimizers are a random sample of optimizers and non-optimizers. Dividing by P _t on both sides yields:

$$\displaystyle \begin{aligned} 1 = (1-\varphi_y)p_{At}^{1-\epsilon_y} + \varphi_y (\pi_{t-1}/\pi_t)^{1-\epsilon_y}. \end{aligned} $$

(4.81d)

Finally, consider the definition of $\tilde Y_t$ given in (4.50):

$$\displaystyle \begin{aligned}\tilde Y_t = \int_{0}^1 Y_{t}(j)\; dj = \int_0^1 \left(\frac{P_{t}(j)}{P_t}\right)^{-\epsilon_y}Y_t\; dj=\left(\frac{\tilde P_t}{P_t}\right)^{-\epsilon_y}Y_t,\end{aligned}$$

with the definitions

$$\displaystyle \begin{aligned} \tilde P_t^{-\epsilon_t}\equiv \int_0^1 P_{t}(j)^{-\epsilon_y}\; dj,\end{aligned}$$

and

$$\displaystyle \begin{aligned} s_t^y \equiv \left(\frac{\tilde P_t}{P_t}\right)^{-\epsilon_y}.\end{aligned}$$

Therefore,

$$\displaystyle \begin{aligned} \tilde Y_t&= s_t^y Y_t. \end{aligned} $$

(4.81e)

Using the same reasoning for $\tilde P_t$ as for the price index P _t above results in the following first-order difference equation for the dispersion of individual prices P _t(j) in the wholesale sector:

$$\displaystyle \begin{aligned} s_t^y = (1-\varphi_y)p_{At}^{-\epsilon_y} + \varphi_y (\pi_{t-1}/\pi_t)^{-\epsilon_y}s_{t-1}^y. \end{aligned} $$

(4.81f)

Appendix 4.4: Microfoundations of Wage Setting

Consider the real wage W _t(h)∕P _t of a household member h who has set his wage optimally in period t to $\tilde w_t=W_{At}/P_t$ and who has not been able to do so again until period s. Between period t and t + s, the nominal wage of the household member h increases with the lagged inflation rate π _t, π _t+1, …, π _t+s−1 in each period t + 1, t + 2, …, t + s, while the aggregate price level P _t increases with the inflation rates π _t+1, π _t+2, …, π _t+s:

$$\displaystyle \begin{aligned}\frac{W_{Nt+s}}{P_{t+s}}=\frac{\prod_{i=1}^s \pi_{t+i-1}W_{At}}{\prod_{i=1}^s \pi_{t+i}P_t}=\frac{\pi_t}{\pi_{t+s}}\tilde w_t.\end{aligned}$$

The demand for his type of labor service follows from (4.58)

$$\displaystyle \begin{aligned}L_{t+s}(h)=\left(\frac{(\pi_{t}/\pi_{t+s})\tilde w_t}{w_{t+s}}\right)^{-\epsilon_w} L_{t+s},\end{aligned} $$

where w _t+s = W _t+s∕P _t+s denotes the real wage prevailing in period t + s. Accordingly, the Lagrangian for the optimal real wage is represented by:

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} \mathscr{L} &=& \mathbb{E}_t \sum_{s=0}^\infty (\beta\varphi_w)^s&\Bigg\{ \frac{(C_{t+s}(h)-\chi \bar C_{t+s})^{1-\sigma}-1}{1-\sigma}\\ &&& - \frac{\nu_0}{1+\frac{1}{\nu_1}}\left[\left(\frac{(\pi_t/\pi_{t+s})\tilde w_{t}}{w_{t+s}}\right)^{-\epsilon_w} L_{t+s}\right]^{1+\frac{1}{\nu_1}}\\ &&&+ \gamma^M_0 \frac{\left(\frac{M_{t+1}(h)}{P_t}\right)^{1-\gamma^M_1}-1}{1-\gamma^M_1}\\ &&&+\varLambda_{ht+s}\left[ \frac{\pi_t}{\pi_{t+s}}\tilde w_t\left(\frac{(\pi_t/\pi_{t+s})\tilde w_{t}}{w_{t+s}}\right)^{-\epsilon_w} L_{t+s} + RMT \right]\Bigg\},\vspace{-2pt}\end{array}\end{aligned} $$

where (φ _w)^s denotes the probability that the household cannot adjust its wage optimally for s periods and RMT _t is a placeholder for the remaining terms of the household budget constraint (4.61).

The first-order condition with respect to $\tilde w_t$ is represented by

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} 0 &=\mathbb{E}_t \sum_{s=0}^\infty (\beta\varphi_w)^s\Bigg\{\epsilon_w \nu_0 \tilde w_t^{-\epsilon_w(1+\frac{1}{\nu_1})-1}\left(\frac{(\pi_t/\pi_{t+s})}{w_{t+s}}\right)^{-\epsilon_{w}(1+\frac{1}{\nu_1})} L_{t+s}^{1+\frac{1}{\nu_1}}\\ &\quad \quad +(1-\epsilon_w)\varLambda_{ht+s}\tilde w_t^{-\epsilon_w}w_{t+s}^{\epsilon_w}\left(\frac{\pi_t}{\pi_{t+s}}\right)^{1-\epsilon_w} L_{t+s}\Bigg\}. \vspace{-2pt}\end{array}\end{aligned} $$

Using Λ _ht+s = Λ _t+s, this equation can be arranged to read as

$$\displaystyle \begin{aligned} \tilde w_t &=\frac{\epsilon_w}{\epsilon_w-1}\frac{\varDelta_{1t}}{\varDelta_{2t}}, \end{aligned} $$

(4.82a)

where

$$\displaystyle \begin{aligned} \notag\varDelta_{1t} &=\nu_0 \mathbb{E}_t \sum_{s=0}^\infty (\beta\varphi_w)^s \left(\frac{\pi_t \tilde w_t}{\pi_{t+s}w_{t+s}}\right)^{-\epsilon_w(1+\frac{1}{\nu_1})} L_{t+s}^{1+\frac{1}{\nu_1}}, \end{aligned} $$

$$\displaystyle \begin{aligned} &=\nu_0\left(\frac{\tilde w_t}{w_t}\right)^{-\epsilon_w(1+\frac{1}{\nu_1})}L_t^{1+\frac{1}{\nu_1}} + (\beta\varphi_w)\mathbb{E}_t \left(\frac{\pi_t\tilde w_t}{ \pi_{t+1}\tilde w_{t+1}}\right)^{-\epsilon_w(1+\frac{1}{\nu_1})}\varDelta_{1t+1}, \end{aligned} $$

(4.82b)

$$\displaystyle \begin{aligned} \notag\varDelta_{2t} &=\mathbb{E}_t \sum_{s=0}^\infty (\beta\varphi_w)^s \varLambda_{t+s}\left(\frac{\tilde w_t}{w_{t+s}}\right)^{-\epsilon_w}\left(\frac{\pi_t}{\pi_{t+s}}\right)^{1-\epsilon_w} L_{t+s}, \end{aligned} $$

$$\displaystyle \begin{aligned} &=\varLambda_t \left(\frac{\tilde w_t}{w_t}\right)^{-\epsilon_w}L_t + (\beta\varphi_w)\mathbb{E}_t \left(\frac{\tilde w_t}{\tilde w_{t+1}}\right)^{-\epsilon_w}\left(\frac{ \pi_t}{\pi_{t+1}}\right)^{1-\epsilon_w}\varDelta_{2t+1}. \end{aligned} $$

(4.82c)

Again, Δ _1t and Δ _2t are simply auxiliary variables whose behaviors are described by (stochastic) first-order difference equations. Therefore, they are easily amenable to the solution with the linearization methods described in Appendix 2.3.

The wage index (4.59) implies

$$\displaystyle \begin{aligned}W_t^{1-\epsilon_w}=(1-\varphi_w)W_{At}^{1-\epsilon_w}+\varphi_w (\pi_{t-1}W_{t- 1})^{1-\epsilon_w}\end{aligned}$$

and thus, the real wage equals

$$\displaystyle \begin{aligned} w_t^{1-\epsilon_w} &= (1-\varphi_w)\tilde w_t^{1-\epsilon_w}+\varphi_w\left(\frac{\pi_{t-1}}{\pi_t}w_{t-1}\right)^{1-\epsilon_w}. \end{aligned} $$

(4.82d)

Finally, consider the index

$$\displaystyle \begin{aligned}\tilde L_t^{1+\frac{1}{\nu_1}} = \int_0^1 L_{t}(h)^{1+\frac{1}{\nu_1}}dh,\end{aligned}$$

in the families of current-period utility functions. Using labor demand function (4.58), this index can be re-written as

$$\displaystyle \begin{aligned}\tilde L_t^{1+\frac{1}{\nu_1}}=L_t^{1+\frac{1}{\nu_1}}\int_0^1 \left(\frac{W_{t}(h)}{W_t}\right)^{-\epsilon_w(1+\frac{1}{\nu_1})}dh.\end{aligned}$$

Therefore,

$$\displaystyle \begin{aligned}\tilde L_t &= s^w_t L_t.\end{aligned} $$

(4.82e)

implies the definition of the wage dispersion measure (4.63).

Next, we need to derive the dynamics of the wage dispersion measure $s^w_t$. For this reason, consider

$$\displaystyle \begin{aligned} \begin{aligned} \bar W_t^{-\epsilon_w(1+\frac{1}{\nu_1})}&=\int_0^1 W_{t}(h)^{-\epsilon_w(1+\frac{1}{\nu_1})}dh\\ &=(1-\varphi_w)(W_{At})^{-\epsilon_w(1+\frac{1}{\nu_1})}+\varphi_w (\pi_{t-1}W_{Nt-1})^{-\epsilon_w(1+\frac{1}{\nu_1})}. \end{aligned} \end{aligned}$$

Accordingly, wage dispersion $s_t^w$ is described by the following equation:

$$\displaystyle \begin{aligned}(s^w_t)^{1+\frac{1}{\nu_1}}=\left(\frac{\bar W_t}{W_t}\right)^{-\epsilon_w(1+\frac{1}{\nu_1})}=\left(\frac{\bar W_t/P_t}{W_t/P_t}\right)^{-\epsilon_w(1+\frac{1}{\nu_1})} =\left(\frac{\bar w_t}{w_t}\right)^{-\epsilon_w(1+\frac{1}{\nu_1})}.\end{aligned}$$

Using the same line of argument employed to derive (4.81f) yields the dynamic equation for the measure of wage dispersion $s^w_t$:

$$\displaystyle \begin{aligned} (s_t^w)^{1+\frac{1}{\nu_1}} = (1-\varphi_w)\left(\frac{\tilde w_{t}}{w_t}\right)^{-\epsilon_w(1+\frac{1}{\nu_1})} + \varphi_w\left(\frac{\pi_{t-1}w_{t-1}}{\pi_t w_t}\right)^{-\epsilon_w(1+\frac{1}{\nu_1})}(s^w_{t-1})^{1+\frac{1}{\nu_1}}.\end{aligned} $$

(4.82f)

Appendix 4.5: Monetary Policy Analysis in the New Keynesian Model

Figure 4.25 illustrates the impulse response functions to an interest rate shock of one percentage point in the benchmark New Keynesian model with ϕ = 1.0. In accordance with empirical observations, restrictive monetary policy decreases all private demand, investment and private consumption; moreover, output and labor decline. As expected, inflation also decreases under a restrictive monetary policy.

Fig. 4.25

Impulse responses to an interest rate shock

The reasons are as follows: Following an increase in the nominal interest rate Q _t, prices only adjust slowly so that also the real interest rate in the economy increases. As a consequence, firms reduce their investment, and Tobin’s q decreases. In addition, households postpone consumption to later periods in accordance with their Euler condition (4.66c). Therefore, demand decreases, and prices fall. Since wholesale producers are slow to adjust their prices, the mark-up increases.

Appendix 4.6: Data Sources

The time series on government expenditures that we use in this chapter are attached as separate Excel files to my Matlab/Gauss programs.

• Government Expenditures General government total expenditure as percent of GDP (National currency). The data for Fig. 4.1 are retrieved from the IMF’s World Economic Outlook database (Accessed on 15 February 2018).

  http://www.imf.org/external/pubs/ft/weo/2017/02/weodata/index.aspx.

  The German data for the years 1980–1990 are retrieved from the Deutsche Bundesbank (Accessed on 15 February 2018).

  http://www.bundesbank.de/Navigation/DE/Statistiken/Zeitreihen_ Datenbanken/zeitreihen_datenbank.html.

  The US data for the years 1980–2000 and the data for Fig. 4.2 are retrieved from ‘National Accounts at a Glance’: 6. General Government (Accessed on 15 February 2018).

  https://stats.oecd.org/Index.aspx?DataSetCode=NAAG#.

• Pensions Government, Public expenditure, Pension expenditure from OECD Factbook 2015–2016, which can be obtained via the Internet at the web site (Accessed on 15 February 2018)

  http://www.oecd-ilibrary.org/economics/oecd-factbook-2015-2016/public-and-private-expenditure-on-pensions-as-a-percentage-of-gdp-2011_factbook-2015-graph171-en.

• Health Expenditures are retrieved from the OECD database ‘OECD iLibrary’ (Accessed on 15 February 2018).

  http://stats.oecd.org/BrandedView.aspx?oecd_bv_id=health-data-en&doi=data-00349-en.

• Education Expenditures are retrieved from the OECD publication ‘Education at a Glance: 2014’, Table B2.3 – Expenditure on educational institutions as a percentage of GDP, by source of fund and level of education (2011) (Accessed on 15 February 2018).

  http://www.oecd-ilibrary.org/education/education-at-a-glance-2014/ expenditure-on-educational-institutions-as-a-percentage-of-gdp-by-source-of- fund-and-level-of-education-2011_eag-2014-table124-en.

• Defense Expenditures are retrieved from the OECD publication ‘National Accounts at a Glance: 2015’, General government, Table 24.1. General government expenditure by function, ‘Defence’ and ‘Public order and safety’, Percentage of GDP, 2012 (Accessed on 15 February 2018). http://www.oecd-ilibrary.org/economics/national-accounts-at-a-glance-2015/total-general-government-expenditure-by-function_na_glance-2015-table35-en.

• Government Consumption The series on government consumption is constructed with the help of the data from FRED provided by the Federal Reserve Bank of St. Louis (Accessed on 28 October 2015). In particular, I used the series GCEC1 Real Government Consumption Expenditures & Gross Investment, Billions of Chained 2009 Dollars, Quarterly, Seasonally Adjusted Annual Rate, and subtracted real government investments. I computed real government investment from the series A782RC1Q027SBEA Gross Government Investment, divided by the implicit price deflator (2009=100) of GDP (GDPDEF).

Problems

4.1

Solve the transition dynamics for the Numerical Problem in Sect. 4.3 by forward shooting as described in Appendix 4.1.

4.2

Derive the Frisch labor supply elasticity (4.39) of the Cobb-Douglas utility function (4.28).

4.3

In applied work, researchers often select model parameters that are not empirically observable or only estimated with a high degree of uncertainty by optimizing the behavior of the RBC model. Using the model in Sect. 4.4, use a grid search over ϕ ∈ [0, 1.5] and ρ ^C ∈ [0.3, 1.3] to find the minimum distance (i.e., the minimum of the squared deviations) of the theoretical second moments (as implied by the model) from the empirical second moments (as presented in Table 4.1).

4.4

Use the preferences

$$\displaystyle \begin{aligned} u(C^p,L)=\frac{(C^p)^{1-\sigma} (1-L)^{1+\vartheta}}{1-\sigma},\;\; \sigma>1, \vartheta>0,\end{aligned}$$

and recompute the RBC model with stochastic government. Set σ = 2.0, and calibrate 𝜗 such that steady-state labor supply is equal to L = 0.3. Does private consumption increase after an increase in government consumption?

4.5

Derive (4.72) using the individual budget constraint, the government budget, and the firms’ first-order equations. Apply Euler’s theorem according to which the aggregate output is equal to the sum of all factor payments for a constant-returns-to-scale technology under perfect competition.

4.6

Derive the equilibrium dynamics (4.73 ) for the New Keynesian model in Sect. 4.5. Assume, however, that firms in the wholesale sector who cannot optimally choose their price adjust their nominal price P _NT according to the average inflation rate:

$$\displaystyle \begin{aligned} P_{Nt}= \pi P_{Nt-1}.\end{aligned}$$