Stationarization

The previous chapters derive all the equations necessary to solve for the steady-state and nonsteady-state equilibria of this model. However, because labor productivity is growing at rate \(g_y\) as can be seen in the firms’ production function (27) and the population is growing at rate \(\tilde{g}_{n,t}\) as defined in (5), the model is not stationary. Different endogenous variables of the model are growing at different rates. We have already specified three potential budget closure rules (54), (55), and (57) using some combination of government spending \(G_t\) and transfers \(TR_t\) that stationarize the debt-to-GDP ratio.

Table 1 lists the definitions of stationary versions of these endogenous variables. Variables with a ``\(\:\,\hat{}\,\:\)’’ signify stationary variables. The first column of variables are growing at the productivity growth rate \(g_y\). These variables are most closely associated with individual variables. The second column of variables are growing at the population growth rate \(\tilde{g}_{n,t}\). These variables are most closely associated with population values. The third column of variables are growing at both the productivity growth rate \(g_y\) and the population growth rate \(\tilde{g}_{n,t}\). These variables are most closely associated with aggregate variables. The last column shows that the interest rates \(r_t\), \(r_{p,t}\) and \(r_{gov,t}\), and household labor supply \(n_{j,s,t}\) are already stationary.

Table 1 Stationary variable definitions. Note: The interest rate \(r_t\) in firm first order condition is already stationary because \(Y_t\) and \(K_t\) grow at the same rate. Household labor supply \(n_{j,s,t}\in[0,\tilde{l}]\) is stationary.

Sources of growth

\(e^{g_y t}\)

\(\tilde{N}_t\)

\(e^{g_y t}\tilde{N}_t\)

Not growing

\(\hat{y}_{j,s,t}\equiv \frac{c_{j,s,t}}{e^{g_y t}}\)

\(\hat{\omega}_{s,t}\equiv\frac{\omega_{s,t}}{\tilde{N}_t}\)

\(\hat{Y}_t\equiv\frac{Y_t}{e^{g_y t}\tilde{N}_t}\)

\(n_{j,s,t}\)

\(\hat{b}_{j,s,t}\equiv \frac{b_{j,s,t}}{e^{g_y t}}\)

\(\hat{L}_t\equiv\frac{L_t}{\tilde{N}_t}\)

\(\hat{K}_t\equiv\frac{K_t}{e^{g_y t}\tilde{N}_t}\)

\(r_t\)

\(\hat{bq}_{t,s,j}\equiv \frac{bq_{t,s,j}}{e^{g_y t}}\)

\(\hat{BQ}_{j,t}\equiv\frac{BQ_{j,t}}{e^{g_y t}\tilde{N}_t}\)

\(r_{p,t}\)

\(\hat{c}_{j,s,t}\equiv \frac{y_{j,s,t}}{e^{g_y t}}\)

\(\hat{C}_t\equiv\frac{C_t}{e^{g_y t}\tilde{N}_t}\)

\(r_{gov,t}\)

\(\hat{tr}_{j,s,t}\equiv \frac{tr_{j,s,t}}{e^{g_y t}}\)

\(\hat{TR}_t\equiv\frac{TR_t}{e^{g_y t}\tilde{N}_t}\)

\(r_{K,t}\)

\(\hat{ubi}_{j,s,t}\equiv\frac{ubi_{j,s,t}}{e^{g_y t}}\)

\(\hat{UBI}_t\equiv\frac{UBI_t}{e^{g_y t}\tilde{N}_t}\)

\(\hat{T}_{j,s,t}\equiv \frac{T_{j,s,t}}{e^{g_y t}}\)

\(\hat{D}_t\equiv\frac{D_t}{e^{g_y t}\tilde{N}_t}\)

\(\hat{w}_t\equiv \frac{w_t}{e^{g_y t}}\)

\(\hat{K}_{g,t}\equiv\frac{K_{g,t}}{e^{g_y t}\tilde{N}_t}\)

The usual definition of equilibrium would be allocations and prices such that households optimize (18), (19), and (20), firms optimize (30) and (31), and markets clear (68), (72), (73), (76), and (77). In this chapter, we show how to stationarize each of these characterizing equations so that we can use our fixed point methods described in Sections Steady-state solution method and Stationary non-steady-state solution method of Chapter Equilibrium to solve for the equilibria in the steady-state and transition path equilibrium definitions.

Stationarized Household Equations

The stationary version of the household budget constraint (10) is found by dividing both sides of the equation by \(e^{g_y t}\). For the savings term \(b_{j,s+1,t+1}\), we must multiply and divide by \(e^{g_y(t+1)}\), which leaves an \(e^{g_y} = \frac{e^{g_y(t+1)}}{e^{g_y t}}\) in front of the stationarized variable.

(78)\[\begin{split} \hat{c}_{j,s,t} + e^{g_y}\hat{b}_{j,s+1,t+1} &= (1 + r_{p,t})\hat{b}_{j,s,t} + \hat{w}_t e_{j,s} n_{j,s,t} + \zeta_{j,s}\frac{\hat{BQ}_t}{\lambda_j\hat{\omega}_{s,t}} + \eta_{j,s,t}\frac{\hat{TR}_{t}}{\lambda_j\hat{\omega}_{s,t}} + \hat{ubi}_{j,s,t} - \hat{T}_{s,t} \\ &\quad\forall j,t\quad\text{and}\quad s\geq E+1 \quad\text{where}\quad \hat{b}_{j,E+1,t}=0\end{split}\]

Because total bequests \(BQ_t\) and total government transfers \(TR_t\) grow at both the labor productivity growth rate and the population growth rate, we have to multiply and divide each of those terms by the economically relevant population \(\tilde{N}_t\). This stationarizes total bequests \(\hat{BQ}_t\), total transfers \(\hat{TR}_t\), and the respective population level in the denominator \(\hat{\omega}_{s,t}\).

We stationarize the Euler equations for labor supply (18) by dividing both sides by \(e^{g_y(1-\sigma)}\). On the left-hand-side, \(e^{g_y}\) stationarizes the wage \(\hat{w}_t\) and \(e^{-\sigma g_y}\) goes inside the parentheses and stationarizes consumption \(\hat{c}_{j,s,t}\). On the right-and-side, the \(e^{g_y(1-\sigma)}\) terms cancel out.

(79)\[\begin{split} \hat{w}_t e_{j,s}\bigl(1 - \tau^{mtrx}_{s,t}\bigr)(\hat{c}_{j,s,t})^{-\sigma} = \chi^n_{s}\biggl(\frac{b}{\tilde{l}}\biggr)\biggl(\frac{n_{j,s,t}}{\tilde{l}}\biggr)^{\upsilon-1}\Biggl[1 - \biggl(\frac{n_{j,s,t}}{\tilde{l}}\biggr)^\upsilon\Biggr]^{\frac{1-\upsilon}{\upsilon}} \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall j,t, \quad\text{and}\quad E+1\leq s\leq E+S \\\end{split}\]

We stationarize the Euler equations for savings (19) and (20) by dividing both sides of the respective equations by \(e^{-\sigma g_y t}\). On the right-hand-side of the equation, we then need to multiply and divide both terms by \(e^{-\sigma g_y(t+1)}\), which leaves a multiplicative coefficient \(e^{-\sigma g_y}\).

(80)\[\begin{split} (\hat{c}_{j,s,t})^{-\sigma} = e^{-\sigma g_y}\biggl[\chi^b_j\rho_s(\hat{b}_{j,s+1,t+1})^{-\sigma} + \beta_j\bigl(1 - \rho_s\bigr)\Bigl(1 + r_{p,t+1}\bigl[1 - \tau^{mtry}_{s+1,t+1}\bigr]\Bigr)(\hat{c}_{j,s+1,t+1})^{-\sigma}\biggr] \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall j,t, \quad\text{and}\quad E+1\leq s\leq E+S-1 \\\end{split}\]
(81)\[ (\hat{c}_{j,E+S,t})^{-\sigma} = e^{-\sigma g_y}\chi^b_j(\hat{b}_{j,E+S+1,t+1})^{-\sigma} \quad\forall j,t\]

Stationarized Firms Equations

The nonstationary production function (27) can be stationarized by dividing both sides by \(e^{g_y t}\tilde{N}\). This stationarizes output \(\hat{Y}_t\) on the left-hand-side. Because the general CES production function is homogeneous of degree 1, \(F(xK,xK_g,xL) = xF(K,K_g,L)\), which means the right-hand-side of the production function is stationarized by dividing by \(e^{g_y t}\tilde{N}_t\).

(82)\[\begin{split} \begin{split} \hat{Y}_t &= F(\hat{K}_t, \hat{K}_{g,t}, \hat{L}_t) \\ &\equiv Z_t\biggl[(\gamma)^\frac{1}{\varepsilon}(\hat{K}_t)^\frac{\varepsilon-1}{\varepsilon} + (\gamma_{g})^\frac{1}{\varepsilon}(\hat{K}_{g,t})^\frac{\varepsilon-1}{\varepsilon} + (1-\gamma-\gamma_{g})^\frac{1}{\varepsilon}(\hat{L}_t)^\frac{\varepsilon-1}{\varepsilon}\biggr]^\frac{\varepsilon}{\varepsilon-1} \quad\forall t \end{split}\end{split}\]

Notice that the growth term multiplied by the labor input drops out in this stationarized version of the production function. We stationarize the nonstationary profit function (29) in the same way, by dividing both sides by \(e^{g_y t}\tilde{N}_t\).

(83)\[ \hat{PR}_t = (1 - \tau^{corp}_t)\Bigl[F(\hat{K}_t,\hat{K}_{g,t},\hat{L}_t) - \hat{w}_t \hat{L}_t\Bigr] - \bigl(r_t + \delta\bigr)\hat{K}_t + \tau^{corp}_t\delta^\tau_t \hat{K}_t \quad\forall t\]

The firms’ first order equation for labor demand (30) is stationarized by dividing both sides by \(e^{g_y t}\). This stationarizes the wage \(\hat{w}_t\) on the left-hand-side and cancels out the \(e^{g_y t}\) term in front of the right-hand-side. To complete the stationarization, we multiply and divide the \(\frac{Y_t}{e^{g_y t}L_t}\) term on the right-hand-side by \(\tilde{N}_t\).

(84)\[ \hat{w}_t = (Z_t)^\frac{\varepsilon-1}{\varepsilon}\left[(1-\gamma-\gamma_g)\frac{\hat{Y}_t}{\hat{L}_t}\right]^\frac{1}{\varepsilon} \quad\forall t\]

It can be seen from the firms’ first order equation for capital demand (31) that the interest rate is already stationary. If we multiply and divide the \(\frac{Y_t}{K_t}\) term on the right-hand-side by \(e^{g_y t}\tilde{N}_t\), those two aggregate variables become stationary. In other words, \(Y_t\) and \(K_t\) grow at the same rate and \(\frac{Y_t}{K_t} = \frac{\hat{Y}_t}{\hat{K}_t}\).

(85)\[\begin{split} r_t &= (1 - \tau^{corp}_t)(Z_t)^\frac{\varepsilon-1}{\varepsilon}\left[\gamma\frac{\hat{Y}_t}{\hat{K}_t}\right]^\frac{1}{\varepsilon} - \delta + \tau^{corp}_t\delta^\tau_t \quad\forall t \\ &= (1 - \tau^{corp}_t)(Z_t)^\frac{\varepsilon-1}{\varepsilon}\left[\gamma\frac{Y_t}{K_t}\right]^\frac{1}{\varepsilon} - \delta + \tau^{corp}_t\delta^\tau_t \quad\forall t\end{split}\]

Stationarized Government Equations

Each of the tax rate functions \(\tau^{etr}_{s,t}\), \(\tau^{mtrx}_{s,t}\), and \(\tau^{mtry}_{s,t}\) is stationary. The total tax liability function \(T_{s,t}\) is growing at the rate of labor productivity growth \(g_y\) This can be see by looking at the decomposition of the total tax liability function into the effective tax rate times total income (11). The effective tax rate function is stationary, and household income is growing at rate \(g_y\). So household total tax liability is stationarized by dividing both sides of the equation by \(e^{g_y t}\).

(86)\[\begin{split} \hat{T}_{s,t} &= \tau^{etr}_{s,t}(\hat{x}_{j,s,t}, \hat{y}_{j,s,t})\left(\hat{x}_{j,s,t} + \hat{y}_{j,s,t}\right) \qquad\qquad\qquad\qquad\forall t \quad\text{and}\quad E+1\leq s\leq E+S \\ &= \tau^{etr}_{s,t}(\hat{w}_t e_{j,s}n_{j,s,t}, r_{p,t}\hat{b}_{j,s,t})\left(\hat{w}_t e_{j,s}n_{j,s,t} + r_{p,t}\hat{b}_{j,s,t}\right) \quad\forall t \quad\text{and}\quad E+1\leq s\leq E+S\end{split}\]

We can stationarize the simple expressions for total government spending on public goods \(G_t\) in (48) and on household transfers \(TR_t\) in (39) by dividing both sides by \(e^{g_y t}\tilde{N}_t\),

(87)\[ \hat{G}_t = g_{g,t}\:\alpha_{g}\:\hat{Y}_t \quad\forall t\]
(88)\[ \hat{TR}_t = g_{tr,t}\:\alpha_{tr}\:\hat{Y}_t \quad\forall t\]

where the time varying multipliers \(g_{g,t}\) and \(g_{tr,t}\), respectively, are defined in (96) and (97) below. These multipliers \(g_{g,t}\) and \(g_{tr,t}\) do not have a ``\(\:\,\hat{}\,\:\)’’ on them because their specifications (54) and (55) that are functions of nonstationary variables are equivalent to (96) and (97) specified in stationary variables.

We can stationarize the expression for total government revenue \(Rev_t\) in (46) by dividing both sides of the equation by \(e^{g_y t}\tilde{N}_t\).

(89)\[ \hat{Rev}_t = \underbrace{\tau^{corp}_t\bigl[\hat{Y}_t - \hat{w}_t\hat{L}_t\bigr] - \tau^{corp}_t\delta^\tau_t \hat{K}_t}_{\text{corporate tax revenue}} + \underbrace{\sum_{s=E+1}^{E+S}\sum_{j=1}^J\lambda_j\hat{\omega}_{s,t}\tau^{etr}_{s,t}\left(\hat{x}_{j,s,t},\hat{y}_{j,s,t}\right)\bigl(\hat{x}_{j,s,t} + \hat{y}_{j,s,t}\bigr)}_{\text{household tax revenue}} \quad\forall t\]

Every term in the government budget constraint (47) is growing at both the productivity growth rate and the population growth rate, so we stationarize it by dividing both sides by \(e^{g_y t}\tilde{N}_t\). We also have to multiply and divide the next period debt term \(D_{t+1}\) by \(e^{g_y(t+1)}\tilde{N}_{t+1}\), leaving the term \(e^{g_y}(1 + \tilde{g}_{n,t+1})\).

(90)\[ e^{g_y}\left(1 + \tilde{g}_{n,t+1}\right)\hat{D}_{t+1} + \hat{Rev}_t = (1 + r_{gov,t})\hat{D}_t + \hat{G}_t + \hat{I}_{g,t} + \hat{TR}_t + \hat{UBI}_t \quad\forall t\]

The stationarized infrastructure investment spending rule \(I_{g,t}\) in (49), and the law of motion for the public capital stock \(K_{g,t}\) in (50) are given by:

(91)\[ \hat{I}_{g,t} = \alpha_{I,t} \hat{Y}_t \quad\forall t \quad\forall t\]
(92)\[ \hat{K}_{g,t+1} = \frac{(1 - \delta^{g})\hat{K}_{g,t} + \hat{I}_{g,t}}{e^{g_y}(1 + \tilde{g}_{n,t+1})} \quad\forall t\]

Stationary aggregate universal basic income expenditure \(\hat{UBI}_t\) is found by dividing (51) by \(e^{g_y t}\tilde{N}_t\).

(93)\[ \hat{UBI}_t = \sum_{s=E+1}^{E+S}\sum_{j=1}^J \lambda_j\hat{\omega}_{s,t} \hat{ubi}_{j,s,t} \quad\forall t\]

The expression for the interest rate on government debt \(r_{gov,t}\) in (52) is already stationary because every term on the right-hand-side is already stationary. The net return on capital, \(r_{K,t}\) is also stationary because the marginal products private and public capital are stationary. The expression for the return to household savings \(r_{p,t}\) in (67) is equivalent to its stationary representation because the same macroeconomic variables occur linearly in both the numerator and denominator.

(94)\[ r_{p,t} = \frac{r_{gov,t}D_{t} + r_{K,t}K_{t}}{D_{t} + K_{t}} = \frac{r_{gov,t}\hat{D}_{t} + r_{K,t}\hat{K}_{t}}{\hat{D}_{t} + \hat{K}_{t}} \quad\forall t\]

The long-run debt-to-GDP ratio condition is also the same in both the nonstationary version in (53) as well as the stationary version below because the endogenous side is a ratio of macroeconomic variables that are growing at the same rate.

(95)\[ \frac{D_t}{Y_t} = \frac{\hat{D}_t}{\hat{Y}_t} = \alpha_D \quad\text{for}\quad t\geq T\]

The three potential budget closure rules (54), (55), and (57) are the last government equations to stationarize. In each of the cases, we simply divide both sides by \(e^{g_y t}\tilde{N}_t\).

(96)\[\begin{split}\begin{split} &\hat{G}_t = g_{g,t}\:\alpha_{g}\: \hat{Y}_t \\ &\text{where}\quad g_{g,t} = \begin{cases} 1 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\:\:\text{if}\quad t < T_{G1} \\ \frac{e^{g_y}\left(1 + \tilde{g}_{n,t+1}\right)\left[\rho_{d}\alpha_{D}\hat{Y}_{t} + (1-\rho_{d})\hat{D}_{t}\right] - (1+r_{gov,t})\hat{D}_{t} - \hat{TR}_{t} - \hat{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\alpha_g \hat{Y}_t} \quad\text{if}\quad T_{G1}\leq t<T_{G2} \\ \frac{e^{g_y}\left(1 + \tilde{g}_{n,t+1}\right)\alpha_{D}\hat{Y}_{t} - (1+r_{gov,t})\hat{D}_{t} - \hat{TR}_{t} - \hat{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\alpha_g \hat{Y}_t} \qquad\qquad\quad\,\text{if}\quad t \geq T_{G2} \end{cases} \\ &\text{and}\quad g_{tr,t} = 1 \quad\forall t \end{split}\end{split}\]

or

(97)\[\begin{split}\begin{split} &\hat{TR}_t = g_{tr,t}\:\alpha_{tr}\: \hat{Y}_t \\ &\text{where}\quad g_{tr,t} = \begin{cases} 1 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{if}\quad t < T_{G1} \\ \frac{e^{g_y}\left(1 + \tilde{g}_{n,t+1}\right)\left[\rho_{d}\alpha_{D}\hat{Y}_{t} + (1-\rho_{d})\hat{D}_{t}\right] - (1+r_{gov,t})\hat{D}_{t} - \hat{G}_{t} - \hat{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\alpha_{tr} \hat{Y}_t} \quad\text{if}\quad T_{G1}\leq t<T_{G2} \\ \frac{e^{g_y}\left(1 + \tilde{g}_{n,t+1}\right)\alpha_{D}\hat{Y}_{t} - (1+r_{gov,t})\hat{D}_{t} - \hat{G}_{t} - \hat{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\alpha_{tr} \hat{Y}_t} \qquad\qquad\quad\,\text{if}\quad t \geq T_{G2} \end{cases} \\ &\text{and}\quad g_{g,t} = 1 \quad\forall t \end{split}\end{split}\]

or

(98)\[\begin{split}\begin{split} &\hat{G}_t + \hat{TR}_t = g_{trg,t}\left(\alpha_g + \alpha_{tr}\right)\hat{Y}_t \quad\Rightarrow\quad \hat{G}_t = g_{trg,t}\:\alpha_g\:\hat{Y}_t \quad\text{and}\quad \hat{TR}_t = g_{trg,t}\:\alpha_{tr}\:\hat{Y}_t \\ &\text{where}\quad g_{trg,t} = \begin{cases} 1 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\text{if}\quad t < T_{G1} \\ \frac{e^{g_y}\left(1 + \tilde{g}_{n,t+1}\right)\left[\rho_{d}\alpha_{D}\hat{Y}_{t} + (1-\rho_{d})\hat{D}_{t}\right] - (1+r_{gov,t})\hat{D}_{t} - \hat{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\left(\alpha_g + \alpha_{tr}\right)\hat{Y}_t} \quad\text{if}\quad T_{G1}\leq t<T_{G2} \\ \frac{e^{g_y}\left(1 + \tilde{g}_{n,t+1}\right)\alpha_{D}\hat{Y}_{t} - (1+r_{gov,t})\hat{D}_{t} - \hat{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\left(\alpha_g + \alpha_{tr}\right)\hat{Y}_t} \qquad\qquad\quad\,\text{if}\quad t \geq T_{G2} \end{cases} \end{split}\end{split}\]

Stationarized Market Clearing Equations

The labor market clearing equation (68) is stationarized by dividing both sides by \(\tilde{N}_t\).

(99)\[ \hat{L}_t = \sum_{s=E+1}^{E+S}\sum_{j=1}^{J} \hat{\omega}_{s,t}\lambda_j e_{j,s}n_{j,s,t} \quad \forall t\]

Total savings by domestic households \(B_t\) from (69) is stationarized by dividing both sides by \(e^{g_y t}\tilde{N}_t\). The \(\omega_{s,t-1}\) terms on the right-hand_side require multiplying and dividing by \(\tilde{N}_{t-1}\), which leads to the division of \(1 + \tilde{g}_{n,t}\).

(100)\[ \hat{B}_t \equiv \frac{1}{1 + \tilde{g}_{n,t}}\sum_{s=E+2}^{E+S+1}\sum_{j=1}^{J}\Bigl(\hat{\omega}_{s-1,t-1}\lambda_j b_{j,s,t} + i_s\hat{\omega}_{s,t-1}\lambda_j \hat{b}_{j,s,t}\Bigr) \quad \forall t\]

And the total domestic savings constraint (70) is stationarized by dividing both sides by \(e^{g_y t}\tilde{N}_t\).

(101)\[ \hat{K}^d_t + \hat{D}^d_t = \hat{B}_t \quad \forall t\]

The stationarized law of motion for foreign holdings of overnment debt (71) and the government debt market clearing condition (72), respectively, are solved for by dividing both sides by \(e^{g_y t}\tilde{N}_t\).

(102)\[ e^{g_y}\bigl[1 + \tilde{g}_{n,t+1}\bigr]\hat{D}^{f}_{t+1} = \hat{D}^{f}_{t} + \zeta_{D}\Bigl(e^{g_y}\bigl[1 + \tilde{g}_{n,t+1}\bigr]\hat{D}_{t+1} - \hat{D}_{t}\Bigr) \quad\forall t\]
(103)\[ \hat{D}_t = \hat{D}^d_t + \hat{D}^f_t \quad\forall t\]

The private capital market clearing equation (73) is stationarized by dividing both sides by \(e^{g_y t}\tilde{N}_t\), as is the expression for excess demand at the world interest rate (74) and the exogenous expression for foreign private capital flows (75).

(104)\[ \hat{K}_t = \hat{K}^d_t + \hat{K}^f_t \quad\forall t\]
(105)\[ \hat{ED}^{K,r^*}_t \equiv \hat{K}^{r^*}_t - \hat{K}^d_t \quad\forall t\]
(106)\[ \hat{K}^{f}_t = \zeta_{K}\hat{ED}^{K,r^*}_t \quad\forall t\]

We stationarize the goods market clearing (76) condition by dividing both sides by \(e^{g_y t}\tilde{N}_t\). On the right-hand-side, we must multiply and divide the \(K^d_{t+1}\) term and the \(D^f_{t+1}\) term, respectively, by \(e^{g_y(t+1)}\tilde{N}_{t+1}\) leaving the coefficient \(e^{g_y}(1+\tilde{g}_{n,t+1})\).

(107)\[\begin{split} \begin{split} \hat{Y}_t &= \hat{C}_t + \Bigl(e^{g_y}\bigl[1 + \tilde{g}_{n,t+1}\bigr]\hat{K}^d_{t+1} - \hat{K}^d_t\Bigr) + \delta\hat{K}_t + \hat{G}_t + \hat{I}_{g,t} + r_{p,t}\hat{K}^f_t ... \\ &\quad\quad - \Bigl(e^{g_y}\bigl[1 + \tilde{g}_{n,t+1}\bigr]\hat{D}^f_{t+1} - \hat{D}^f_t\Bigr) + r_{p,t}\hat{D}^f_t \quad\forall t \\ &\quad\text{where}\quad \hat{C}_t \equiv \sum_{s=E+1}^{E+S}\sum_{j=1}^{J}\hat{\omega}_{s,t}\lambda_j\hat{c}_{j,s,t} \end{split}\end{split}\]

We stationarize the law of motion for total bequests \(BQ_t\) in (77) by dividing both sides by \(e^{g_y t}\tilde{N}_t\). Because the population levels in the summation are from period \(t-1\), we must multiply and divide the summed term by \(\tilde{N}_{t-1}\) leaving the term in the denominator of \(1+\tilde{g}_{n,t}\).

(108)\[ \hat{BQ}_{t} = \left(\frac{1+r_{p,t}}{1 + \tilde{g}_{n,t}}\right)\left(\sum_{s=E+2}^{E+S+1}\sum_{j=1}^J\rho_{s-1}\lambda_j\hat{\omega}_{s-1,t-1}\hat{b}_{j,s,t}\right) \quad\forall t\]