Back Matter

## Appendices

### APPENDIX 1

#### Population Growth

The population size at time 0 is assumed to equal N, with N(1 — ψ) OLG households and Nψ LIQ households. The size of a new cohort born at time t is given by $N{n}^{t}\left(1-\frac{\theta }{n}\right)$, so that by time t + k this cohort will be of size $N{n}^{t}\left(1-\frac{\theta }{n}\right){\theta }^{k}$. When we sum over all cohorts at time t we obtain

This means that the overall population grows at the rate n. When we normalize real quantities, we divide by the level of technology Tt and by population, but for the latter we divide by nt only, meaning real figures are not in per capita terms but rather in absolute terms adjusted for population growth.

### APPENDIX 2

#### Optimality Conditions for OLG Households

We have the following Lagrangian representation of the optimization problem of OLG households:32

where Λa,t is the marginal utility to the generation of age a at time t of an extra unit of domestic currency. Define the marginal utility of an extra unit of consumption goods output as

$\begin{array}{ccc}{\lambda }_{a,t}={\Lambda }_{a,t}{P}_{t},& \phantom{\rule{7.0em}{0ex}}& \left(286\right)\end{array}$

and let

$\begin{array}{ccc}{u}_{a,t}^{OLG}={\left({c}_{a,t}^{OLG}\right)}^{{\eta }^{OLG}}{\left({S}_{t}^{L}-{\ell }_{a,t}^{OLG}\right)}^{1-{\eta }^{OLG}}.& \phantom{\rule{7.0em}{0ex}}& \left(287\right)\end{array}$

Then we have the following first-order conditions for consumption and labor supply

$\begin{array}{ccc}\frac{{\eta }^{OLG}{\left({u}_{a,t}^{OLG}\right)}^{1-\gamma }}{{c}_{a,t}^{OLG}}={\lambda }_{a,t}\left({p}_{t}^{R}+{p}_{t}^{C}{\tau }_{c,t}\right),& \phantom{\rule{7.0em}{0ex}}& \left(288\right)\end{array}$
$\begin{array}{ccc}\frac{\left(1-{\eta }^{OLG}\right){\left({u}_{a,t}^{OLG}\right)}^{1-\gamma }}{{S}_{t}^{L}-{\ell }_{a,t}^{OLG}}={\lambda }_{a,t}{w}_{t}{\Phi }_{a,t}\left(1-{\tau }_{L,t}\right),& \phantom{\rule{7.0em}{0ex}}& \left(289\right)\end{array}$

which can be combined to yield

We can aggregate this as

and normalize it as

In this aggregation we have made use of the following assumptions about labor productivity:

Equation (293) is our specification of the profile of labor productivity over the lifetime. Equation (294) is the assumption that average labor productivity equals one. Equations (293) and (294), for a given productivity decline parameter χ, imply the initial productivity level κ in (295). Equation (296) is the definition of effective aggregate labor supply.

Next we have the first-order conditions for domestic and foreign bonds ${B}_{a,t}^{N}/{B}_{a,t}^{T}$33 and Fa,t:

Together these yield the uncovered interest parity condition

To write the marginal utility of consumption λa,t in terms of quantities that can be aggregated, specifically in terms of consumption, we use (287) and (290) in (288) to get

We use (300) in (297) to obtain the generation specific consumption Euler equations

$\begin{array}{ccc}{E}_{t}{c}_{a+1,t+1}^{OLG}={E}_{t}{j}_{t}{c}_{a,t}^{OLG},\text{ }{where}& \phantom{\rule{7.0em}{0ex}}& \left(301\right)\end{array}$

### APPENDIX 3

#### Consumption and Wealth

The key equation for OLG households is the one relating current consumption to current wealth. We start deriving this by reproducing the budget constraint:

We now derive an expression that decomposes human wealth into labor and dividend income. First, we note that after-tax wage income can be decomposed as follows:

$\begin{array}{ccc}{W}_{t}{\Phi }_{a,t}{\ell }_{a,t}^{OLG}\left(1-{\tau }_{L,t}\right)={W}_{t}{\Phi }_{a,t}\left(1-{\tau }_{L,t}\right){S}_{t}^{L}-{W}_{t}{\Phi }_{a,t}\left(1-{\tau }_{L,t}\right)\left({S}_{t}^{L}-{\ell }_{a,t}^{OLG}\right).& \phantom{\rule{7.0em}{0ex}}& \left(304\right)\end{array}$

The first expression on the right-hand side of (304) is the labor component of income, which equals the marginal value of the household’s entire endowment (one unit) of time. The second expression in (304), by (290), can be rewritten as

$\begin{array}{ccc}{W}_{t}{\Phi }_{a,t}\left(1-{\tau }_{L,t}\right)\left({S}_{t}^{L}-{\ell }_{a,t}^{OLG}\right)=\frac{1-{\eta }^{OLG}}{{\eta }^{OLG}}{P}_{t}{c}_{a,t}^{OLG}\left({p}_{t}^{R}+{p}_{t}^{C}{\tau }_{c,t}\right),& \phantom{\rule{7.0em}{0ex}}& \left(305\right)\end{array}$

which can be combined with the consumption expression in (303) to obtain, on the left-hand side of (303), ${P}_{t}{c}_{a,t}^{OLG}\left({p}_{t}^{R}+{p}_{t}^{C}{\tau }_{c,t}\right)/{\eta }^{OLG}$. The second component of income is dividend, remuneration for bankruptcy monitoring, and net transfer income net of redistribution to LIQ agents, the expression for which can be simplified by noting that in equilibrium all firms in a given sector pay equal dividends, so that we can drop the firm specific index and write $\underset{0}{\overset{1}{\int }}{D}_{a,t}^{j}\left(i\right)di={D}_{a,t}^{j}$. We also assume that per capita dividends, remuneration payments for bankruptcy monitoring, and net transfers received by each OLG agent are identical. Finally, we incorporate the assumption that a share of dividend and net transfer income is redistributed to LIQ agents:

These assumptions imply

$\begin{array}{l}\begin{array}{ccc}\sum _{j=N,TDCIMXFK\mathit{\text{EP}}}1\underset{0}{\overset{1}{\int }}{D}_{a,t}^{j}\left(i\right)di+{P}_{t}rb{r}_{a,t}-{P}_{t}{\tau }_{{T}_{a,t}}^{OLG}=& \phantom{\rule{7.0em}{0ex}}& \left(307\right)\end{array}\\ \frac{\left({\Sigma }_{j=N,T,D,C,I,M,X,F,K,EP}{D}_{t}^{j}+rb{r}_{t}\right)\left(1-\iota \right)}{N{n}^{t}\left(1-\psi \right)}+\frac{{\stackrel{ˇ}{c}}_{t}^{OLG}}{{\stackrel{ˇ}{C}}_{t}}\frac{\left({D}_{t}^{R}+{P}_{t}{{\gamma }}_{t}-{P}_{t}{\tau }_{t}^{ls}\right)}{N{n}^{t}\left(1-\psi \right)}+\frac{{\stackrel{ˇ}{\ell }}_{t}^{OLG}}{{\stackrel{ˇ}{L}}_{t}}\frac{{D}_{t}^{U}}{N{n}^{t}\left(1-\psi \right)}.\end{array}$

The preceding arguments imply that total nominal wage and dividend income of households of age a in period t is given by

$\begin{array}{l}\begin{array}{ccc}In{c}_{a,t}={W}_{t}{\Phi }_{a,t}\left(1-{\tau }_{L,t}\right){S}_{t}^{L}& \phantom{\rule{7.0em}{0ex}}& \left(308\right)\end{array}\\ +\frac{\left({\Sigma }_{j=N,T,D,C,I,M,X,F,K,EP}{D}_{t}^{j}+rb{r}_{t}\right)\left(1-\iota \right)}{N{n}^{t}\left(1-\psi \right)}+\frac{{\stackrel{ˇ}{c}}_{t}^{OLG}}{{\stackrel{ˇ}{C}}_{t}}\frac{\left({D}_{t}^{R}+{P}_{t}{{\gamma }}_{t}-{P}_{t}{\tau }_{t}^{ls}\right)}{N{n}^{t}\left(1-\psi \right)}+\frac{{\stackrel{ˇ}{\ell }}_{t}^{OLG}}{{\stackrel{ˇ}{L}}_{t}}\frac{{D}_{t}^{U}}{N{n}^{t}\left(1-\psi \right)}.\end{array}$

We now rewrite the household budget constraint as follows:

$\begin{array}{c}\begin{array}{ccc}{P}_{t}{c}_{a,t}^{OLG}\frac{\left({p}_{t}^{R}+{p}_{t}^{C}{\tau }_{c,t}\right)}{{\eta }^{OLG}}+{B}_{a,t}+{B}_{a,t}^{N}+{B}_{a,t}^{T}+{ℰ}_{t}{F}_{a,t}& \phantom{\rule{7.0em}{0ex}}& \left(309\right)\end{array}\\ =In{c}_{a,t}+\frac{1}{\theta }\left[{i}_{t-1}{B}_{a-1,t-1}+\frac{{i}_{t-1}}{\left(1+{\xi }_{t-1}^{b}\right)}\left({B}_{a-1,t-1}^{N}+{B}_{a-1,t-1}^{T}\right)+{i}_{t-1}\left(\stackrel{̃}{N}\right){ℰ}_{t}{F}_{a-1,t-1}\left(1+{\xi }_{t-1}^{f}\right)\right].\end{array}$

We proceed to derive a condition relating current consumption to lifetime wealth through successive forward substitutions of (309). In doing so we use the arbitrage condition (298) to cancel terms relating to foreign bonds. After the first substitution we obtain

$\begin{array}{c}\hfill \begin{array}{ccc}\frac{\theta \left(1+{\xi }_{t}^{b}\right)}{{i}_{t}}{E}_{t}\left\{{B}_{a+1,t+1}+{B}_{a+1,t+1}^{N}+{B}_{a+1,t+1}^{T}+{ℰ}_{t+1}{F}_{a+1,t+1}\right\}& \phantom{\rule{7.0em}{0ex}}& \left(310\right)\end{array}\hfill \\ +{P}_{t}{c}_{a,t}^{OLG}\frac{\left({p}_{t}^{R}+{p}_{t}^{C}{\tau }_{c,t}\right)}{{\eta }^{OLG}}+\frac{\theta \left(1+{\xi }_{t}^{b}\right)}{{i}_{t}}{E}_{t}\left\{{P}_{t+1}{c}_{a+1,t+1}^{OLG}\frac{\left({p}_{t+1}^{R}+{p}_{t+1}^{C}{\tau }_{c,t+1}\right)}{{\eta }^{OLG}}\right\}=\\ In{c}_{a,t}+\frac{\theta \left(1+{\xi }_{t}^{b}\right)}{{i}_{t}}{E}_{t}\left\{In{c}_{a+1,t+1}\right\}\\ +\frac{1}{\theta }\left[{i}_{t-1}{B}_{a-1,t-1}+\frac{{i}_{t-1}}{\left(1+{\xi }_{t-1}^{b}\right)}\left({B}_{a-1,t-1}^{N}+{B}_{a-1,t-1}^{T}\right)+{i}_{t-1}\left(\stackrel{˜}{N}\right){\epsilon }_{t}{F}_{a-1,t-1}\left(1+{\xi }_{t-1}^{f}\right)\right],\end{array}$

and successively substitute forward in the same fashion. We impose the following no-Ponzi condition on the household’s optimization problem:

$\begin{array}{ccc}\underset{s\to \infty }{\mathrm{lim}}{E}_{t}{\stackrel{̃}{R}}_{t,s}\left[{B}_{a+s,t+s}+{B}_{a+s,t+s}^{N}+{B}_{a+s,t+s}^{T}+{ℰ}_{t+s}{F}_{a+s,t+s}\right]=0.& \phantom{\rule{7.0em}{0ex}}& \left(311\right)\end{array}$

Furthermore, we let

This expression denotes nominal financial wealth inherited from period t – 1. Next we define a variable HWa,t denoting lifetime human wealth, which equals the present discounted value of future incomes Inct. We have

$\begin{array}{ccc}H{W}_{a,t}={E}_{t}{\Sigma }_{s=0}^{\infty }{\stackrel{̃}{R}}_{t,s}In{c}_{a+s,t+s}.& \phantom{\rule{7.0em}{0ex}}& \left(313\right)\end{array}$

Further forward substitutions on (310), and application of the transversality condition (311), then yields the following:

$\begin{array}{ccc}{E}_{t}{\Sigma }_{s=0}^{\infty }{\stackrel{̃}{R}}_{t,s}\left[{P}_{t+s}{c}_{a+s,t+s}^{OLG}\frac{\left({p}_{t+s}^{R}+{p}_{t+s}^{C}{\tau }_{c,t+s}\right)}{{\eta }^{OLG}}\right]=H{W}_{a,t}+F{W}_{a-1,t-1}.& \phantom{\rule{7.0em}{0ex}}& \left(314\right)\end{array}$

The left-hand side of this expression can be further evaluated by using (301) for all future consumption terms. We let

Then we can write

$\begin{array}{ccc}{P}_{t}{c}_{a,t}^{OLG}{E}_{t}\left({\Sigma }_{s=0}^{\infty }{\stackrel{̃}{r}}_{t,s}{j}_{t,s}\frac{\left({p}_{t+s}^{R}+{p}_{t+s}^{C}{\tau }_{c,t+s}\right)}{{\eta }^{OLG}}\right)=H{W}_{a,t}+F{W}_{a-1,t-1}.& \phantom{\rule{7.0em}{0ex}}& \left(316\right)\end{array}$

The infinite summation on the left-hand side is recursive and can be written as

$\begin{array}{ccc}{\Theta }_{t}={E}_{t}{\Sigma }_{s=0}^{\infty }{\stackrel{̃}{r}}_{t,s}{j}_{t,s}\frac{\left({p}_{t+s}^{R}+{p}_{t+s}^{C}{\tau }_{c,t+s}\right)}{{\eta }^{OLG}}=\frac{\left({p}_{t}^{R}+{p}_{t}^{C}{\tau }_{c,t}\right)}{{\eta }^{OLG}}+{E}_{t}\frac{\theta {j}_{t}}{{\stackrel{ˇ}{r}}_{t}}{\Theta }_{t+1},& \phantom{\rule{7.0em}{0ex}}& \left(317\right)\end{array}$

so we finally obtain

$\begin{array}{ccc}{P}_{t}{c}_{a,t}^{OLG}{\Theta }_{t}=H{W}_{a,t}+F{W}_{a-1,t-1}.& \phantom{\rule{7.0em}{0ex}}& \left(318\right)\end{array}$

We want to express this equation in real aggregate terms. We begin with real aggregate human wealth, denoted by hwt:

$\begin{array}{ccc}h{w}_{t}=N{n}^{t}\left(1-\psi \right)\left(1-\frac{\theta }{n}\right){\Sigma }_{a=0}^{\infty }{\left(\frac{\theta }{n}\right)}^{a}\frac{H{W}_{a,t}}{{P}_{t}}.& \phantom{\rule{7.0em}{0ex}}& \left(319\right)\end{array}$

We break this down into its labor income and dividend income components $h{w}_{t}^{L}$ and $h{w}_{t}^{K}$. For $h{w}_{t}^{L}$ we have

$h{w}_{t}^{L}={E}_{t}{\Sigma }_{s=0}^{\infty }{\stackrel{̃}{r}}_{t,s}{\chi }^{s}\left(N{n}^{t}\left(1-\psi \right){w}_{t+s}\left(1-{\tau }_{L,t+s}\right){S}_{t+s}^{L}\right),$

where we have used (293) and (295). In recursive form, and scaling by technology, the last equation equals

$\begin{array}{ccc}\stackrel{ˇ}{h}{w}_{t}^{L}=\left(N\left(1-\psi \right)\stackrel{ˇ}{w}\left(1-{\tau }_{L,t}\right){S}_{t}^{L}\right)+{E}_{t}\frac{\theta \chi g}{\stackrel{ˇ}{r}}\stackrel{ˇ}{h}{w}_{t+1}^{L}.& \phantom{\rule{7.0em}{0ex}}& \left(320\right)\end{array}$

For $h{w}_{t}^{K}$ we have, using (307) and letting ${d}_{t}^{j}={D}_{t}^{j}/{P}_{t}$,

$h{w}_{t}^{K}={E}_{t}{\Sigma }_{s=0}^{\infty }{\stackrel{̃}{r}}_{t,s}\left(\left({\Sigma }_{j=N,T,D,C,I,M,X,F,K,EP}{d}_{t}^{j}+rb{r}_{t}\right)\left(1-\iota \right)+\frac{{\stackrel{ˇ}{c}}_{t}^{OLG}}{{\stackrel{ˇ}{C}}_{t}}\left({d}_{t}^{R}+{{\gamma }}_{t}-{\tau }_{t}^{ls}\right)+\frac{{\stackrel{ˇ}{\ell }}_{t}^{OLG}}{{\stackrel{ˇ}{L}}_{t}}{d}_{t}^{U}\right),$

which has the recursive representation, again after scaling by technology, of

$\begin{array}{ll}\stackrel{ˇ}{h}{w}_{t}^{K}=\left(\left({\Sigma }_{j=N,T,D,C,I,M,X,F,K,EP}{\stackrel{ˇ}{d}}_{t}^{j}+r\stackrel{ˇ}{b}{r}_{t}\right)\left(1-\iota \right)+\frac{{\stackrel{ˇ}{c}}_{t}^{OLG}}{{\stackrel{ˇ}{C}}_{t}}\left({\stackrel{ˇ}{d}}_{t}^{R}+{\stackrel{ˇ}{{\gamma }}}_{t}-{\stackrel{ˇ}{\tau }}_{t}^{ls}\right)+\frac{{\stackrel{ˇ}{\ell }}_{t}^{OLG}}{{\stackrel{ˇ}{L}}_{t}}{\stackrel{ˇ}{d}}_{t}^{U}\right)+{E}_{t}\frac{\theta g}{{\stackrel{ˇ}{r}}_{t}}\stackrel{ˇ}{h}{w}_{t+1}^{K}.& \text{ }\left(321\right)\end{array}$

Finally, we have

$\begin{array}{ccc}\stackrel{ˇ}{h}{w}_{t}=\stackrel{ˇ}{h}{w}_{t}^{L}+\stackrel{ˇ}{h}{w}_{t}^{K}.& \phantom{\rule{7.0em}{0ex}}& \left(322\right)\end{array}$

Next we aggregate over the financial wealth of different age groups. We note here that aggregation cancels the 1/θ term in front of the bracket in (312). This is because the period by period budget constraint (303) from which (312) was derived is the budget constraint of the agents that have in fact survived from period t – 1 to t. Aggregation has to take account of the fact that (1 – θ) agents did not survive and their wealth passed, through the insurance company, to surviving agents. Noting that B−1, t−1 = 0, we therefore have34

${B}_{t-1}=N{n}^{t}\left(1-\psi \right)\left(1-\frac{\theta }{n}\right){\Sigma }_{a=0}^{\infty }{\left(\frac{\theta }{n}\right)}^{a-1}{B}_{a-1,t-1}.$

For total nominal financial wealth, we therefore have

$F{W}_{t-1}=\left[{i}_{t-1}{B}_{t-1}+\frac{{i}_{t-1}}{\left(1+{\xi }_{t-1}^{b}\right)}\left({B}_{t-1}^{N}+{B}_{t-1}^{T}\right)+{i}_{t-1}\left(\stackrel{̃}{N}\right){ℰ}_{t}{F}_{t-1}\left(1+{\xi }_{t-1}^{f}\right)\right].$

To express this in real terms, we define the real domestic currency asset stock as bt = Bt/Pt. We adopt the convention that each nominal asset is deflated by the consumption based price index of the currency of its denomination, so that ft = Ft/Pt(Ñ). With the real exchange rate in terms of final output denoted by et = εtPt(Ñ)/Pt, and after scaling by technology and population, we can then write

$\begin{array}{lll}\stackrel{ˇ}{f}{w}_{t}=\frac{F{W}_{t-1}}{{P}_{t}{T}_{t}{n}^{t}}=\frac{1}{{\pi }_{t}gn}\left[{i}_{t-1}{\stackrel{ˇ}{b}}_{t-1}+\frac{{i}_{t-1}}{\left(1+{\xi }_{t-1}^{b}\right)}\left({\stackrel{ˇ}{b}}_{t-1}^{N}+{\stackrel{ˇ}{b}}_{t-1}^{T}\right)+{i}_{t-1}\left(\stackrel{̃}{N}\right){\epsilon }_{t}{\stackrel{ˇ}{f}}_{t-1}{e}_{t-1}\left(1+{\xi }_{t-1}^{f}\right)\right].& \phantom{\rule{7.0em}{0ex}}& \left(323\right)\end{array}$

Finally, using (318)-(323) we arrive at our final expression for current period consumption:

$\begin{array}{ccc}{\stackrel{ˇ}{c}}_{t}^{OLG}{\Theta }_{t}=\stackrel{ˇ}{h}{w}_{t}+\stackrel{ˇ}{f}{w}_{t}.& \phantom{\rule{7.0em}{0ex}}& \left(324\right)\end{array}$

The linearized form of the aggregate equation (324) can instead be derived by linearizing an individual age group’s budget constraint, using its linearized optimality conditions, and then aggregating over all generations. As mentioned above, it is therefore appropriate to use the expectations operator Et in nonlinear equations as long as it is understood that this is valid only up to first-order approximations of the system.

### APPENDIX 4

#### Manufacturers

The objective function facing each manufacturing firm in sectors J ∈ {N, T} is

$\underset{{P}_{s}^{J}\left(i\right),{U}_{s}^{J}\left(i\right),{I}_{s}^{J}\left(i\right),{K}_{s}^{J}\left(i\right)}{Max}{E}_{t}{\sum }_{s=t}^{\infty }{\stackrel{˜}{R}}_{t,s}{D}_{t+s}^{J}\left(i\right).$

The price (and inflation) terms in the two sectors will be indexed with $\stackrel{˜}{J}\in \left\{N,TH\right\}$. Then dividend terms are given by

${D}_{t}^{J}\left(i\right)={P}_{t}^{\stackrel{̃}{J}}\left(i\right){Z}_{t}^{J}\left(i\right)-{V}_{t}{U}_{t}^{J}\left(i\right)-{P}_{t}^{X}{X}_{t}^{J}\left(i\right)-{R}_{k,t}^{J}{K}_{t}^{J}\left(i\right)-{P}_{t}^{\stackrel{̃}{J}}{G}_{P,t}^{J}\left(i\right)-{P}_{t}^{\stackrel{̃}{J}}{T}_{t}{\omega }^{J}.$

Optimization is subject to the equality of output with demand

$\begin{array}{c}F\left({K}_{t}^{J}\left(i\right),{U}_{t}^{J}\left(i\right),{X}_{t}^{J}\left(i\right)\right)={Z}_{t}^{J}\left(i\right),\text{ }{where}\\ F\left({K}_{t}^{J}\left(i\right),{U}_{t}^{J}\left(i\right),{X}_{t}^{J}\left(i\right)\right)=\\ {𝔗{\left({\left(1-{\alpha }_{{J}_{t}}^{X}\right)}^{\frac{1}{{\xi }_{XJ}}}{\left({M}_{t}^{J}\left(i\right)\right)}^{\frac{{\xi }_{XJ}-1}{{\xi }_{XJ}}}+{\left({\alpha }_{{J}_{t}}^{X}\right)}^{\frac{1}{{\xi }_{XJ}}}\left({X}_{t}^{J}\left(i\right)1-{G}_{X,t}^{J}\left(i\right)\right)\right)}^{\frac{{\xi }_{XJ}-1}{{\xi }_{XJ}}}\right)}^{\frac{{\xi }_{XJ}}{{\xi }_{XJ}-1}},\\ {M}_{t}^{J}\left(i\right)={\left({\left(1-{\alpha }_{J}^{U}\right)}^{\frac{1}{{\xi }_{ZJ}}}{\left({K}_{t}^{J}\left(i\right)\right)}^{\frac{{\xi }_{ZJ}-1}{{\xi }_{ZJ}}}+{\left({\alpha }_{J}^{U}\right)}^{\frac{1}{{\xi }_{ZJ}}}{\left({T}_{t}{A}_{t}^{J}{U}_{t}^{J}\left(i\right)\right)}^{\frac{{\xi }_{ZJ}-1}{{\xi }_{ZJ}}}\right)}^{\frac{{\xi }_{ZJ}}{{\xi }_{ZJ}-1}}.\\ {Z}_{t}^{J}\left(i\right)={\left(\frac{{P}_{t}^{\stackrel{˜}{J}}\left(i\right)}{{P}_{t}^{\stackrel{˜}{J}}}\right)}^{-{\sigma }_{J}}{Z}_{t}^{J}.\end{array}$

We also have the following adjustment costs:

$\begin{array}{c}{G}_{P,t}^{J}\left(i\right)=\frac{{\phi }_{{P}^{J}}}{2}{Z}_{t}^{J}{\left(\frac{\frac{{P}_{t}^{\stackrel{˜}{J}}\left(i\right)}{{P}_{t-1}^{\stackrel{˜}{J}}\left(i\right)}}{\frac{{P}_{t-1}^{\stackrel{˜}{J}}}{{P}_{t-2}^{\stackrel{˜}{J}}}}-1\right)}^{2},\\ {G}_{X,t}^{J}\left(i\right)=\frac{{\phi }_{X}^{J}}{2}{\left(\frac{\left({X}_{t}^{J}\left(i\right)/\left(gn\right)\right)-{X}_{t-1}^{J}}{{X}_{t-1}^{J}}\right)}^{2}.\end{array}$

We write out the profit maximization problem of a representative manufacturing firm in Lagrangian form. Terms pertaining to period t and t +1 are sufficient. We introduce a multiplier ${\Lambda }_{t}^{J}$ for the market-clearing condition $F\left({K}_{t}^{J}\left(i\right),{U}_{t}^{J}\left(i\right),{X}_{t}^{J}\left(i\right)\right)={\left(\frac{{P}_{t}^{\stackrel{˜}{J}}\left(i\right)}{{P}_{t}^{\stackrel{˜}{J}}}\right)}^{-{\sigma }_{J}}{Z}_{t}^{J}$. The variable ${\Lambda }_{t}^{J}$ equals the nominal marginal cost of producing one more unit of good i in sector J. We have

We take the first-order condition with respect to ${P}_{t}^{\stackrel{̃}{J}}\left(i\right)$ and then impose symmetry by setting ${P}_{t}^{\stackrel{̃}{J}}\left(i\right)={P}_{t}^{\stackrel{̃}{J}}$ and ${Z}_{t}^{J}\left(i\right)={Z}_{t}^{J}$ because all firms face an identical problem. We let ${\lambda }_{t}^{J}={\Lambda }_{t}^{J}/{P}_{t}$ and rescale by technology. Then we obtain

$\begin{array}{l}\begin{array}{ccc}\left[\frac{{\sigma }_{J}}{{\sigma }_{J}-1}\frac{{\lambda }_{t}^{J}}{{p}_{t}^{\stackrel{̃}{J}}}-1\right]=\frac{{\phi }_{{P}^{J}}}{{\sigma }_{J}-1}\left(\frac{{\pi }_{t}^{\stackrel{̃}{J}}}{{\pi }_{t-1}^{\stackrel{̃}{J}}}\right)\left(\frac{{\pi }_{t}^{\stackrel{̃}{J}}}{{\pi }_{t-1}^{\stackrel{̃}{J}}}-1\right)& \phantom{\rule{7.0em}{0ex}}& \left(326\right)\end{array}\\ -{E}_{t}\frac{\theta gn}{{\stackrel{ˇ}{r}}_{t}}\frac{{\phi }_{{P}^{J}}}{{\sigma }_{J}-1}\left\{\frac{{p}_{t+1}^{\stackrel{̃}{J}}}{{p}_{t}^{\stackrel{̃}{J}}}\frac{{\stackrel{ˇ}{Z}}_{t+1}^{j}}{{\stackrel{ˇ}{Z}}_{t}^{j}}\left(\frac{{\pi }_{t+1}^{\stackrel{̃}{J}}}{{\pi }_{t}^{\stackrel{̃}{J}}}\right)\left(\frac{{\pi }_{t+1}^{\stackrel{̃}{J}}}{{\pi }_{t}^{\stackrel{̃}{J}}}-1\right)\right\}.\end{array}$

For ${U}_{t}^{J}\left(i\right)$, ${X}_{t}^{J}\left(i\right)$, ${I}_{t}^{J}\left(i\right)$, and ${K}_{t}^{J}\left(i\right)$ we have

$\begin{array}{ccc}{\stackrel{ˇ}{\upsilon }}_{t}={\lambda }_{t}^{J}{\stackrel{ˇ}{F}}_{U,t}^{J},& \phantom{\rule{7.0em}{0ex}}& \left(327\right)\end{array}$
$\begin{array}{ccc}{p}_{t}^{X}={\lambda }_{t}^{J}{\stackrel{ˇ}{F}}_{X,t}^{J},& \phantom{\rule{7.0em}{0ex}}& \left(328\right)\end{array}$
$\begin{array}{ccc}{r}_{k,t}^{J}={\stackrel{ˇ}{\lambda }}_{t}^{J}{\stackrel{ˇ}{F}}_{K,t}^{J},& \phantom{\rule{7.0em}{0ex}}& \left(329\right)\end{array}$

where we have used

$\begin{array}{ccc}{\stackrel{ˇ}{F}}_{U,t}^{J}=T{\left(\frac{\left(1-{\alpha }_{{J}_{t}}^{X}\right){\stackrel{ˇ}{Z}}_{t}^{J}}{T{\stackrel{ˇ}{M}}_{t}^{J}}\right)}^{\frac{1}{{\xi }_{XJ}}}{A}_{t}^{J}{\left(\frac{{\alpha }_{J}^{U}{\stackrel{ˇ}{M}}_{t}^{J}}{{A}_{t}^{J}{\stackrel{ˇ}{U}}_{t}^{J}}\right)}^{\frac{1}{{\xi }_{ZJ}}},& \phantom{\rule{7.0em}{0ex}}& \left(330\right)\end{array}$

### APPENDIX 5

#### Entrepreneur’s Problem - Lognormal Distribution

##### Basic Properties of Γ and G

We first repeat the expressions for Γ and G here for ease of reference:

Then we have

##### Basic Properties of the Lognormal Distribution

The assumption is that ${\omega }_{t}^{J}$ is lognormally distributed with $E\left({\omega }_{t}^{J}\right)=1$ and $Var\left({\omega }_{t}^{J}\right)={\left({\sigma }_{t}^{J}\right)}^{2}$. This implies the following:

##### Derivations

We will change integrands at various points in order to obtain solutions that can be expressed in terms of the cumulative distribution function Φ of the standard normal distribution. We begin by defining terms:

Manipulating the second expression in each case gives the following expressions:

Using (338)-(342) we can now evaluate the expressions determining Γ and G in terms of the c.d.f. Φ(.). We start with

Next we have

$\begin{array}{c}{\int }_{{\overline{\omega }}_{t+1}^{J}}^{\infty }{\omega }_{t+1}^{J}f\left({\omega }_{t+1}^{J}\right)d{\omega }_{t+1}^{J}={\int }_{{\overline{\omega }}_{t+1}^{J}}^{\infty }\frac{1}{\sqrt{2\pi }{\sigma }_{t+1}^{J}}{exp}\left\{-\frac{1}{2}{\left(\frac{{ln}\left({\omega }_{t+1}^{J}\right)+\frac{1}{2}{\left({\sigma }_{t+1}^{J}\right)}^{2}}{{\sigma }_{t+1}^{J}}\right)}^{2}\right\}d{\omega }_{t+1}^{J}\\ ={\int }_{{\stackrel{˜}{z}}_{t+1}^{J}}^{\infty }\frac{{\sigma }_{t+1}^{J}}{\sqrt{2\pi }{\sigma }_{t+1}^{J}}\mathrm{exp}\left\{-\frac{1}{2}{\left({\stackrel{˜}{y}}_{t+1}^{J}+{\sigma }_{t+1}^{J}\right)}^{2}\right\}\mathrm{exp}\left\{{\stackrel{˜}{y}}_{t+1}^{J}{\sigma }_{t+1}^{J}+\frac{1}{2}{\left({\sigma }_{t+1}^{J}\right)}^{2}\right\}d{\stackrel{˜}{y}}_{t+1}^{J}\\ ={\int }_{{\stackrel{˜}{z}}_{t+1}^{J}}^{\infty }\frac{1}{\sqrt{2\pi }}\mathrm{exp}\left\{-\frac{1}{2}{\left({\stackrel{˜}{y}}_{t+1}^{J}\right)}^{2}-\frac{1}{2}{\left({\sigma }_{t+1}^{J}\right)}^{2}-{\stackrel{˜}{y}}_{t+1}^{J}{\sigma }_{t+1}^{J}+{\stackrel{˜}{y}}_{t+1}^{J}{\sigma }_{t+1}^{J}+\frac{1}{2}{\left({\sigma }_{t+1}^{J}\right)}^{2}\right\}d{\stackrel{˜}{y}}_{t+1}^{J}\\ ={\int }_{{\stackrel{˜}{z}}_{t+1}^{J}}^{\infty }\frac{1}{\sqrt{2\pi }}\mathrm{exp}\left\{-\frac{1}{2}{\left({\stackrel{˜}{y}}_{t+1}^{J}\right)}^{2}\right\}d{\stackrel{˜}{y}}_{t+1}^{J}=1-\Phi \left({\stackrel{˜}{z}}_{t+1}^{J}\right)=1-\Phi \left({\overline{z}}_{t+1}^{J}-{\sigma }_{t+1}^{J}\right)\end{array}$

To summarize:

##### Final Equation System

The entrepreneur’s optimal loan contract condition (100) determines the equilibrium return to capital $r\stackrel{ˇ}{e}{t}_{k,t}^{J}$, the lender’s zero profit condition (101) determines the lender’s gross profit share ${\Gamma }_{t+1}^{J}$, and the net worth accumulation condition (109) determines the entrepreneur’s net worth ${\stackrel{ˇ}{n}}_{t}^{J}$. The conditions derived in this appendix close the system. To summarize, we have:

## References

• Bernanke, B.S., Gertler, M. and Gilchrist, S. (1999), “The Financial Accelerator in a Quantitative Business Cycle Framework”, in: John B. Taylor and Michael Woodford, eds., Handbook of Macroeconomics, Volume 1C. Amsterdam: Elsevier.

• Crossref
• Export Citation
• Blanchard, O.J. (1985), “Debt, Deficits, and Finite Horizons”, Journal of Political Economy, Vol. 93, pp. 223-247.

• Christiano, L., Motto, R. and Rostagno, M. (2007), “Financial Factors in Business Cycles”, Working Paper.

• Engen, E.M. and Hubbard, R.G. (2004), “Federal Government Debt and Interest Rates”, NBER Macroeconomics Annual, Vol. 19, pp. 83-138.

• Faruqee, H. and Laxton, D. (2000), “Life-Cycles, Dynasties, Saving: Implications for Closed and Small, Open Economies”, IMF Working Paper Series, WP/00/126.

• Crossref
• Export Citation
• Gale, W. and Orszag, P. (2004), “Budget Deficits, National Saving, and Interest Rates”, Brookings Papers on Economic Activity, Vol. 2, pp. 101-187.

• Crossref
• Export Citation
• Kamps, C. (2004), “New Estimates of Government Net Capital Stocks for 22 OECD Countries 1960-2001”, IMF Working Paper Series, WP/04/67.

• Laubach, T. (2003), “New Evidence on the Interest Rate Effects of Budget Deficits and Debt”, Finance and Economics Discussion Series 2003-12, Board of Governors of the Federal Reserve System.

• Crossref
• Export Citation
• Ligthart, J.E. and Suárez, R.M.M. (2005), “The Productivity of Public Capital: A Meta Analysis”, Working Paper, Tilburg University.

• Export Citation

In general we allow for the possibility that agents may be more myopic than what would be suggested by a planning horizon based on a biological probability of death.

We use the term liquidity-constrained agents, but this could also include agents that simply choose to consume all of their income. In the literature these agents are commonly referred to as rule-of-thumb consumers or hand-to-mouth consumers. This is important for interpreting the calibration of the model because we will be using higher estimates of the shares of these agents than is consistent with micro data on the share of agents in the economy that do not have any access to credit markets.

The alternative of using habit persistence to generate consumption inertia is not available in our setup. This is because we face two constraints in our choice of household preferences. The first is that preferences must be consistent with balanced growth. The second is the necessity of being able to aggregate across generations of households. We are left with preferences that, while commonly used, do not allow for a powerful role of habit persistence.

For flexible model calibration we allow for the possibility that OLG households attach a different weight ηOLG to consumption than liquidity constrained households. This allows us to model both groups as working during an equal share of their time endowment in steady state, while OLG households have much higher consumption due to their accumulated wealth.

Except for the special case of lump-sum taxation.

The most recent version of GIMF is symmetric in that it also allows for a nonzero foreign exchange risk premium payable by country Ñ.

The turnover in the population is assumed to be large enough that the income receipts of the insurance companies exactly equal their payouts.

Declining income profiles are necessary to eliminate excessive sensitivity of human wealth to changes in the real interest rate, see Faruqee and Laxton (2000). In models with exogenous labor supply and stationary population shares it can also be shown that declining productivity profiles can be calibrated to produce identical macro behavior as more plausible hump-shaped life-cycle productivity profiles.

It is sometimes convenient to keep these two items separate when trying to account for a country’s overall fiscal structure, and when allowing for targeted transfers to LIQ agents.

Without this assumption consumption tax revenue could become too volatile in the short run.

We adopt the convention throughout the paper that all nominal price level variables are written in upper case letters, and that all relative price variables are written in lower case letters.

The distinction of generations could be dropped as all agents must act identically.

There are also some small sales of aggregate manufacturing output back to manufacturing firms, related to manufacturers’ need for resources to pay for adjustment costs.

Note that, for the sake of clarity, we make a notational distinction between two types of elasticities of substitution. Elasticities between continua of goods varieties, which give rise to market and pricing power, are denoted by a σ subscripted by the respective sectorial indicator. Elasticities between factors of production, both in manufacturing and in final goods distribution, are denoted by a ξ subscripted by the respective sectorial indicator.

The factor T is a constant that can be set different from one to obtain different levels of GDP per capita across countries.

In all other instances of nominal rigidities that follow, GIMF offers this as one option. It will however not be mentioned again in this document.

Note that, unlike other adjustment costs, this expression treats lagged inputs as external. This has proved more useful than the alternatives in our applied work.

The tradables market clearing condition is reported for the example of country 1.

Any value of capital is profit maximizing.

This follows Christiano, Motto and Rostagno (2007), “Financial Factors in Business Cycles”. Papers where the model is linearized prior to solving it only require the elasticity σa of the function a(ut). Because for some applications GIMF is solved in nonlinear form we require a full functional form.

Note the absence of expectations operators because this equation has to hold in each state of nature. Likewise for subsequent equations.

In DYNARE this will have to be replaced by the complementary error function unless the Statistical Toolbox is available.

Dividend related shocks are easier to calibrate as they are already in terms of a share of gross returns on net worth.

Home bias in tradables use depends on the parameter αTH and on a similar parameter αDH at the level of final goods imports.

For the ratio ${ℛ}_{t}^{T}$ we assume as usual that the distributor takes the lagged denominator term as given in his optimization.

The presence of the growth terms ensures that adjustment costs are zero along the balanced growth path.

For applications of the model where unit root processes are not allowed for, potential GDP (and potential tax bases) can simply be evaluated at their non-stochastic steady state.

Under many calibrations of GIMF such rules exhibit superior properties to automatic stabilizers.

In quarterly versions of GIMF this is replaced by a one-year-ahead four-quarter geometric moving average of inflation.

Inflation persistence in the model is therefore exclusively due to inflation adjustment costs.

The trade spillover parameters spillT, spillI and spillC are calibrated at 0.175.

For simplicity we ignore money given the cashless limit assumption. We also treat some stochastic parameters like βt as constants.

With ${\xi }_{t}^{b}=0$ the condition for government bonds is identical. When ${\xi }_{t}^{b}\ne 0$ we assume that the private sector will absorb all government bonds irrespective of their return relative to private sector bonds. Recent versions of GIMF have not used shocks to ${\xi }_{t}^{b}$ because an external financing premium arises endogenously with a financial accelerator sector.

Take the example of bonds held by those of age 0 at time t−1. Only θ of those agents survive into period t, but those that do survive obtain 1/θ units of currency for every unit they held in t − 1. Their weight in period t bonds aggregation is therefore $\theta \frac{1}{\theta }=1$.

The Global Integrated Monetary and Fiscal Model (GIMF) – Theoretical Structure
Author: Mr. Douglas Laxton, Susanna Mursula, Mr. Michael Kumhof, and Mr. Dirk V Muir