ECON 1110: Intermediate Macro
Lecture 7 + 8 Supplement
Government Budgeting
At any given time, let the government surplus be $S_t$. We will assume that the surplus is some fraction $a$ of GDP, so that
S_t = a Y_t
Note that $a$ can be either positive (a surplus) or negative (a deficit). The evolution of the debt level will thus be
D_t = (1+r) D_{t-1} - S_t
Now we want to find the evolution of the normalized debt to GDP ratio $d_t = D_t/Y_t$. To do this we divide by $Y_t$
\align \frac{D_t}{Y_t} = (1+r) \frac{D_{t-1}}{Y_t} - \frac{S_t}{Y_t} \\
\Rightarrow\ \align \frac{D_t}{Y_t} = (1+r) \frac{D_{t-1}}{Y_t} \frac{Y_{t-1}}{Y_{t-1}} - \frac{S_t}{Y_t} \\
\Rightarrow\ \align \frac{D_t}{Y_t} = (1+r) \frac{D_{t-1}}{Y_{t-1}} \frac{Y_{t-1}}{Y_t} - \frac{S_t}{Y_t} \\
\Rightarrow\ \align d_t = (1+r) d_{t-1} \frac{1}{1+g} - a \\
\Rightarrow\ \align d_t = \left(\frac{1+r}{1+g}\right) d_{t-1} - a
To find the steady state value, we impose $d_t = d_{t-1} = d^{\ast}$ this yields
\align d^{\ast} = \left(\frac{1+r}{1+g}\right) d^{\ast} - a \\
\Rightarrow\ \align \left[1-\left(\frac{1+r}{1+g}\right)\right] d^{\ast} = -a \\
\Rightarrow\ \align d^{\ast} = \frac{-a}{1-\left(\frac{1+r}{1+g}\right)} \\
\Rightarrow\ \align d^{\ast} = \frac{(-a)(1+g)}{g-r}
For the steady state to exist, we then need $r \lt g$.
Social Security
When thinking about social security, we have population growth at rate $n$ so that
N^{\prime} = (1+n) N
The young are taxed at rate $t$ and the old receive benefits $b$. As such, a balanced government budget requires
\align b N = t N^{\prime} = t (1+n) N \\
\Rightarrow\ \align b = (1+n) t
Upon introduction of this policy, the old are clearly better off. The change in the present value of wealth for the young is
\Delta we = -t + \frac{b}{1+r} = -t + \frac{(1+n)t}{1+r} = t \left[\frac{1+n}{1+r}-1\right] = \frac{t(n-r)}{1+r}
Thus they are better off only if $r \le n$ and they are worse off otherwise.
Savings and Investment
The profits of the firm are
\pi \align = z F(K,N) - w N - I \\
\pi^{\prime} \align = z^{\prime} F(K^{\prime},N^{\prime}) - w^{\prime} N^{\prime} + (1-d) K^{\prime}
and the present value of their income is
V = \pi + \frac{\pi^{\prime}}{1+r}
and capital evolves according to
K^{\prime} = (1-d) K + I
Find the optimal choice for investment then amounts to
\align \frac{\partial V}{\partial I} = -1 + \frac{z^{\prime} F_K(K^{\prime},N^{\prime}) + (1-d)}{1+r} = 0 \\
\Rightarrow\ \align z^{\prime} F_K(K^{\prime},N^{\prime}) + (1-d) = 1 + r \\
\Rightarrow\ \align MPK^{\prime} + (1-d) = 1 + r
The optimality condition for labor utilization is
z F_N(K,N) = w \quad \align \Rightarrow \quad MPL = w \\
z^{\prime} F_N(K^{\prime},N^{\prime}) = w^{\prime} \quad \align \Rightarrow \quad MPL^{\prime} = w^{\prime}
As we've derived previously, the optimality conditions for the consumer are
MRS_{c,c^{\prime}} \align = 1 + r \\
MRS_{\ell,c} \align = w \\
MRS_{\ell^{\prime},c^{\prime}} \align = w^{\prime}
Combining these with the firm conditions yields
MRS_{c,c^{\prime}} \align = MPK^{\prime} + (1-d) \\
MRS_{\ell,c} \align = MPL \\
MRS_{\ell^{\prime},c^{\prime}} \align = MPL^{\prime}
Standard Case
Let's use our usual utility function
U(c,c^{\prime}) = u(c,\ell) + \beta u(c^{\prime},\ell^{\prime}) \\
\text{where} \quad u(c,\ell) = \log(c) + \eta \log(\ell)
and production function
z F(K,N) \align = z K^{\alpha} N^{1-\alpha} \\
z^{\prime} F(K^{\prime},N^{\prime}) \align = z^{\prime} (K^{\prime})^{\alpha} (N^{\prime})^{1-\alpha}
Keep in mind that $\ell + N = 1$. Our equilibrium conditions are then
\frac{c^{\prime}}{\beta c} = \alpha z^{\prime} \left(\frac{N^{\prime}}{K^{\prime}}\right)^{1-\alpha} + (1-d) \\
\frac{\eta c}{1-N} = (1-\alpha) z \left(\frac{K}{N}\right)^{\alpha} \\
\frac{\eta c^{\prime}}{1-N^{\prime}} = (1-\alpha) z^{\prime} \left(\frac{K^{\prime}}{N^{\prime}}\right)^{\alpha}
Full Depreciation
Suppose that capital fully depreciates each period so that $d=1$ and $K^{\prime} = I$. Note that $Y = c + I$ and $Y^{\prime} = c^{\prime}$. Then we have
\align \frac{c^{\prime}}{\beta c} = \alpha z^{\prime} \left(\frac{N^{\prime}}{K^{\prime}}\right)^{1-\alpha} \\
\Rightarrow\ \align \frac{c^{\prime}}{\beta c} = \alpha \frac{Y^{\prime}}{K^{\prime}} \\
\Rightarrow\ \align \frac{1}{\beta (Y-I)} = \frac{\alpha}{I} \\
\Rightarrow\ \align \frac{I}{Y-I} = \alpha \beta \\
\Rightarrow\ \align \frac{I}{Y} = \frac{\alpha\beta}{1+\alpha\beta}
This is the savings rate we had assumed as exogenous in the Solow model. Additionally we can show
\frac{c}{Y} = \frac{1}{\alpha\beta+1}
Finally, from the labor market condition we can see
\align \frac{\eta c}{1-N} = (1-\alpha) z \left(\frac{K}{N}\right)^{\alpha} \\
\Rightarrow\ \align \frac{\eta c}{1-N} = (1-\alpha) \frac{Y}{N} \\
\Rightarrow\ \align \frac{\eta N}{1-N} = (1-\alpha) \frac{Y}{c} \\
\Rightarrow\ \align \frac{\eta N}{1-N} = (1-\alpha) (\alpha\beta+1) \\
\Rightarrow\ \align N = \frac{(1-\alpha) (\alpha\beta+1)}{\eta+(1-\alpha) (\alpha\beta+1)} \\
\Rightarrow\ \align N = \frac{1-\alpha}{\frac{\eta}{\alpha\beta+1}+(1-\alpha)}
Extra Problem
Suppose now that there is not labor, only capital, so that
z F(K) = z K^{\alpha} \quad \text{and} \quad z^{\prime} F(K^{\prime}) = z^{\prime} (K^{\prime})^{\alpha}
Further, we have a new utility function
u(c) = 1 - \exp(-c)
The implied marginal utility here is
u^{\prime}(c) = \exp(-c)
and the consumer optimality condition is
\align MRS_{c,c^{\prime}} = 1 + r \\
\Rightarrow\ \align \exp(c^{\prime}-c) = 1 + r
Combining this with the firm, we find
\align \exp(c^{\prime}-c) = \alpha z^{\prime} (K^{\prime})^{\alpha-1} \\
\Rightarrow\ \align \exp(z^{\prime} I^{\alpha} + I - z K^{\alpha}) = \frac{\alpha z^{\prime}}{I^{1-\alpha}}
Since the left-hand side is increasing and the right-hand side is decreasing, it is clear that this has a unique solution for $I$. To make things tractable, let's also assume that $\alpha = 1$ so that
\align \exp((1+z^{\prime}) I - z K) = z^{\prime} \\
\Rightarrow\ \align (1+z^{\prime}) I - z K = \log(z^{\prime}) \\
\Rightarrow\ \align (1+z^{\prime}) I = z K + \log(z^{\prime}) \\
\Rightarrow\ \align K^{\prime} = I = \frac{zK+\log(z^{\prime})}{1+z^{\prime}}