Stochastic Permanent Income Model and Government Fiscal Policy
Honours Intermediate Macro
Stochastic Permanent Income
Basic Setup
Linear State Space + Normal Shock:
Let
\[ \begin{aligned} x_{t+1} &= Ax_{t} + Cw_{t+1} \\ y_t &= G\cdot x_t \end{aligned} \]
where \(A\) is \(n \times n\) matrix, \(x\) is \(n \times 1\) vector, \(C\) is \(n \times m\) matrix, \(w_{t+1} \sim N(0,I_{m \times m})\) are i.i.d. normal shocks; \(G\) is \(1 \times n\) vector, \(y_t\) is a scalar representing labor income.
Consumer’s Budget Constraint (assuming \(\beta R = 1\)):
\[ F_{t+1} = \underbrace{\frac{1}{\beta}}_{\substack{\text{gross}\\ \text{interest} \\ \text{rate}}}(\underbrace{F_t}_{\substack{\text{Financial}\\\text{wealth}}}+y_t-c_t) \tag{1}\]
Recall if \(\{y_t\}\) is deterministic, and \(R = 1/\beta\), then for any strictly concave \(u(c)\) they achieved perfect consumption smoothing:
\[ c_t = (1-\beta)\Bigl( \underbrace{F_t}_{\substack{\text{Financial}\\\text{wealth}}}+\underbrace{\sum_{j=0}^{\infty}\beta^j y_{t+j}}_{\substack{\text{PDV of}\\\text{human}\\\text{wealth}}}\Bigr) \]
If \(y_t\) is stochastic, can we just replace the above equation with expected value?
\[ c_t = (1-\beta)\Bigl(F_t + \underbrace{{\mathbb{E}_{{t}}\left[ {\sum_{j=0}^{\infty}\beta^j y_{t+j}} \right]}}_{\substack{\text{expected PDV} \\\text{of human wealth} \\\text{with information at} \\\text{time } t}}\Bigr) \tag{2}\]
Note: If \(u'(c)\) is not linear, then this is only an approximation.
Combine Equation 1 and Equation 2:
\[ F_{t+1} = \frac{1}{\beta}\left[F_t+y_t-(1-\beta)\left(F_t+{\mathbb{E}_{{t}}\left[ {\sum_{j=0}^{\infty}\beta^j y_{t+j}} \right]}\right)\right] \]
\[ = \frac{1}{\beta}\left[\beta F_t+y_t-(1-\beta){\mathbb{E}_{{t}}\left[ {\sum_{j=0}^{\infty}\beta^j y_{t+j}} \right]} \right] \]
\[ \Rightarrow F_{t+1}-F_t = \frac{1}{\beta}\left[y_t-(1-\beta){\mathbb{E}_{{t}}\left[ {\sum_{j=0}^{\infty}\beta^j y_{t+j}} \right]}\right] \tag{3}\]
That is, agents add the difference between \(y_t\) and permanent income. Now use Equation 2 at \(t\) and \(t+1\):
\[ \begin{aligned} c_{t+1} &= (1-\beta)\left[F_{t+1}+{\mathbb{E}_{{t+1}}\left[ {\sum_{j=0}^{\infty}\beta^j y_{t+j+1}} \right]}\right] \\ c_t &= (1-\beta)\left[F_t+{\mathbb{E}_{{t}}\left[ {\sum_{j=0}^{\infty}\beta^j y_{t+j}} \right]}\right] \end{aligned} \]
\[ \Rightarrow c_{t+1}-c_t = (1-\beta)(F_{t+1}-F_t)+(1-\beta)\left[{\mathbb{E}_{{t+1}}\left[ {\sum_{j=0}^{\infty}\beta^j y_{t+j+1}} \right]}-{\mathbb{E}_{{t}}\left[ {\sum_{j=0}^{\infty}\beta^j y_{t+j}} \right]}\right] \]
Use Equation 3 to find (after many steps):
\[ \boxed{c_{t+1}-c_t =(1-\beta)\sum_{j=0}^{\infty}\beta^j \Bigl(\underbrace{\mathbb{E}_{t+1}(y_{t+j+1})}_{\substack{\text{Forecast of } t+1, t+2, \ldots\\\text{with time } t+1 \\\text{information}}}-\underbrace{\mathbb{E}_t(y_{t+j+1})}_{\substack{\text{With time } t\\\text{information}}}\Bigr)} \]
- Consumption only changes due to “surprise” of new information changing expected value
- Only unanticipated changes in \(y_{t+j}, \ldots\) or other information which changes forecasts
- Could be unanticipated changes in government policy or shock realizations
Finally, for a shock between \(t \rightarrow t+1\) with our linear state space model:
\[ \begin{aligned} c_{t+1}-c_t &= (1-\beta)\left[\sum_{j=0}^{\infty}\beta^j\left(\mathbb{E}_{t+1}(y_{t+j+1})-\mathbb{E}_t (y_{t+j+1})\right) \right] \\ &=(1-\beta)\left[G(I-\beta A)^{-1}x_{t+1}-G(I-\beta A)^{-1}Ax_t \right] \\ &=(1-\beta)G(I-\beta A)^{-1}\left[ \underbrace{Ax_t+C w_{t+1}}_{x_{t+1}}-Ax_t \right] \end{aligned} \]
\[ \boxed{c_{t+1}-c_t =\underbrace{(1-\beta)}_{\substack{\text{Propensity to}\\ \text{consume}}} \underbrace{G(I-\beta A)^{-1}\cdot Cw_{t+1}}_{\substack{\text{PDV of impulse response to} \\\text{a shock to } x_{t+1}}}} \]
That is, the PDV of changes to forecasts from the realized shock.
Special Case of Quadratic Preferences
Recall Euler equation for Permanent Income Model:
\[ u'(c_t) = \beta(1+r)u'(c_{t+1}), \quad \forall t=0, \ldots, T-1 \]
If stochastic consumption and \(\beta = \frac{1}{1+r}\), just replace with expectation?
\[ \underbrace{u'(c_t)}_{\substack{\text{Marginal utility} \\\text{this period}}} = \underbrace{{\mathbb{E}_{{t}}\left[ {u'(c_{t+1})} \right]}}_{\substack{\text{Expectation of} \\\text{marginal utility} \\\text{next period}}} \]
Let \(u(c) = \frac{a_1}{2}c^2 + a_2 c+a_3 \Rightarrow u'(c) = a_1 c+a_2\).
In the Euler equation:
\[ a_1 c_t+a_2 = \mathbb{E}_t(a_1 c_{t+1}+a_2) \implies c_t = \mathbb{E}_t(c_{t+1}) \]
That is, the Euler equation implying perfect consumption smoothing with a deterministic process translates to consumption being a martingale if stochastic!
Notes:
- In general, \(\mathbb{E}_t(u(c)) \neq u\left(\mathbb{E}_t(c) \right)\)
- Then, we can use the linear-stochastic state space model for forecasting \(\mathbb{E}_t(c_{t+1})\)
- Due to linearity, it simply forecasts the mean
- This is a general result called Certainty Equivalence of optimizing a quadratic objective subject to a linear-Gaussian state space model
- The decision is identical in a model with or without the uncertainty
- However, the realized sequence contingent on the shocks, and utility, are not the same
Certainty Equivalence, Risk, and Prudence
A clean way to see what is special about quadratic utility is to start from the stochastic Euler equation (risk-free return, interior solution). Assume \(\beta R = 1\).
\[ \boxed{u'(c_t)=\mathbb{E}_t\left[u'(c_{t+1})\right]} \]
If \(u'(\cdot)\) is linear (equivalently \(u'''(\cdot)=0\), i.e. \(u\) is quadratic), then
\[ \mathbb{E}_t\left[u'(c_{t+1})\right]=u'\left(\mathbb{E}_t[c_{t+1}]\right) \quad\Rightarrow\quad c_t=\mathbb{E}_t[c_{t+1}], \]
so the conditional variance of \(c_{t+1}\) does not enter the consumption choice (though it still affects realized utility).
If \(u'''(c)>0\), then \(u'(\cdot)\) is convex. By Jensen,
\[ \mathbb{E}_t\left[u'(c_{t+1})\right] \ge u'\left(\mathbb{E}_t[c_{t+1}]\right), \]
with strict inequality when there is risk. Since \(u'\) is decreasing, holding the conditional mean fixed this pushes the Euler equation toward lower \(c_t\) (higher saving): the precautionary saving motive.
To quantify the effect, take a Taylor expansion of \(u'(c_{t+1})\) around \(c_t\) and define \(\Delta c_{t+1}:=c_{t+1}-c_t\):
\[ u'(c_{t+1}) \approx u'(c_t) + u''(c_t)\Delta c_{t+1} + \frac{1}{2}u'''(c_t)\Delta c_{t+1}^2. \]
Taking conditional expectations and substituting into the Euler equation gives
\[ 0 \approx u''(c_t)\,\mathbb{E}_t[\Delta c_{t+1}] + \frac{1}{2}u'''(c_t)\,\mathbb{E}_t[\Delta c_{t+1}^2]. \]
Note that \(\mathbb{E}_t[\Delta c_{t+1}^2]=\operatorname{Var}_t(\Delta c_{t+1})+\left(\mathbb{E}_t[\Delta c_{t+1}]\right)^2\). Ignoring the small \((\mathbb{E}_t[\Delta c_{t+1}])^2\) term (a higher-order term in this approximation), we obtain
Define the coefficient of relative prudence (appropriate for homothetic preferences):
\[ P_R(c)\equiv -\frac{c\,u'''(c)}{u''(c)}. \]
Divide both sides of the approximation by \(c_t\) and rewrite the variance in terms of relative consumption growth:
\[ \frac{\mathbb{E}_t[\Delta c_{t+1}]}{c_t} \approx \frac{1}{2}P_R(c_t)\,\operatorname{Var}_t\!\left(\frac{\Delta c_{t+1}}{c_t}\right). \]
Since \(u''<0\) for concave utility, if \(u'''(c)>0\) then \(P_R(c)>0\). Higher relative risk therefore raises expected consumption growth, implying lower \(c_t\) today and more saving (precautionary saving).
For quadratic utility \(u'''=0\) (so \(P_R=0\)), the approximation reduces to
\[ \mathbb{E}_t[\Delta c_{t+1}] = 0, \]
independent of risk: the classic certainty-equivalence intuition.
Example (log utility): If \(u(c)=\log c\), then
\[ P_R(c) = -\frac{c \cdot (2/c^3)}{-1/c^2} = 2. \]
The relative form becomes
\[ \mathbb{E}_t\!\left[\frac{\Delta c_{t+1}}{c_t}\right] \approx \operatorname{Var}_t\!\left(\frac{\Delta c_{t+1}}{c_t}\right). \]
Intuitively, the consumer wants to save more today (lower \(c_t\) so higher \(c_{t+1}/c_t\)) when there is more risk to future consumption growth.
Examples
Pre-announced Tax Cut
This will use a shock to knowledge about deterministic income processes, rather than a constant stream of shocks to income.
Setup:
- Government announces at \(t=0\) that at \(t=1\) it will borrow \(\alpha\) from international markets at interest rate \((1+r)\) per period and give it to each consumer
- They also announce that to eventually balance the budget, they will pay it back at \(t=2\) for simplicity by increasing taxation that period
- Assume consumers had deterministic \(y_{t+j}\). What happens to consumption?
Using our result:
\[ c_{t+1}-c_t = (1-\beta)\sum_{j=0}^{\infty}\beta^j\left[\mathbb{E}_{t+1}(y_{t+j+1})-\mathbb{E}_t(y_{t+j+1})\right] \]
Define: \(\{\hat{y}_{t+1}\}_{j=0}^{\infty} = \{y_t,\underbrace{y_{t+1}+\alpha,y_{t+2}-\alpha(1+r)}_{\text{Only difference}},y_{t+3},\ldots ,y_{t+j}\ldots\}\)
- Note that from \(t\) to \(t+1\), the agent has the news that \(\{y_{t+j}\} \rightarrow \{\hat{y}_{t+j}\}\)
- This is a change in expectations:
\[ \begin{aligned} c_1-c_0 &= (1-\beta)\sum_{j=0}^{\infty}\beta^j\left[\mathbb{E}_1(y_{j+1})-\mathbb{E}_0(y_{j+1})\right] = (1-\beta)\sum_{j=0}^{\infty}\beta^j(\hat{y}_{j+1}-y_{j+1}) \\ &= (1-\beta)\sum_{j=0}^{\infty}\beta^j (y_{j+1}-y_{j+1})+(1-\beta)\left[\alpha-\beta(1+r)\alpha \right] \end{aligned} \]
Result: If \(\beta=\frac{1}{1+r}\), then \(c_1-c_0 = 0\).
That is, the tax cut has no effect because of the anticipated rise in taxes. Later, we will investigate cases why \(\beta = \frac{1}{1+r}\) comes out of general equilibrium.
Timing of Tax Cuts
Setup:
- Between time 0 and 1, government announces that it will cut taxes to give \(\alpha\) to each individual at a deterministic time \(T \geq 1\)
- Assume they do not need to pay it back and taxes will not rise to compensate
- What happens to consumption at time \(\{0, \ldots, T, T+1, \ldots\}\)?
- Assume \(y_{t+j+1}\) are deterministic
Solve:
\[ \begin{aligned} c_1-c_0 &= (1-\beta)\sum_{j=0}^{\infty}\beta^j\left[\mathbb{E}_1(y_{j+1})-\mathbb{E}_0(y_{j+1}) \right] \\ &=(1-\beta)\sum_{j=0}^{\infty}\beta^j\left[y_{j+1}-y_{j+1} \right]+(1-\beta)\cdot\beta^{T-1}\cdot \alpha \\ &=\underbrace{(1-\beta)}_{\text{MPC out of wealth}} \underbrace{\beta^{T-1}\cdot \alpha}_{\substack{\text{Change in} \\\text{permanent income}}} \end{aligned} \]
For \(t \geq 1\):
\[ \mathbb{E}_{t+1}(y_{t+j+1}) = \mathbb{E}_t(y_{t+j+1}) \implies c_{t+1}-c_t = 0, \quad \forall t \geq 1 \]
That is:
- Changes only happen at announcement, not at tax cut time \(T\)
- A similar approach with stochastic income would yield the same result
Variation: The only reason that \(T\) enters the above is that the PDV of the \(\alpha\) delivery is discounted for \(T\) periods. If instead, the government announces they will set aside \(\alpha\), put it in the bank at \(R\) interest, and then deliver the \(\alpha\) with interest at time \(T\). In that case, interest compounds for \(T-1\) periods, which means that
\[ c_1 - c_0 = (1-\beta)\beta^{T-1} \left(R^{T-1} \alpha \right) = (1-\beta)\alpha \]
That is, the tax break (no matter when it is actually implemented) adds \(\alpha\) to the PDV of lifetime earnings.
Example from Friedman-Muth
Setup:
\[ \begin{aligned} y_t &= z_t+u_t \\ z_{t+1} &= z_t+\sigma_1 w_{1t+1} \\ u_{t+1} &= \sigma_2 w_{2t+1} \end{aligned} \]
where \(y_t\) is income, \(z_t\) is the persistent or “permanent income”, \(u_t\) is transitory changes in income.
- Which one is a martingale (i.e., random walk here)?
- Define the vector of shocks \(w_{t+1} = \begin{bmatrix}w_{1t+1} \\ w_{2t+1}\end{bmatrix} \sim N\left(0_2,I_{2\times 2}\right)\), i.e., iid normal distributed, mean 0, variance 1.
Setup in linear state space form:
Since \(x_t = \begin{bmatrix}z_t \\ u_t \end{bmatrix}\), we have:
\[ \underbrace{\begin{bmatrix}z_{t+1} \\ u_{t+1}\end{bmatrix}}_{x_{t+1}}=\underbrace{\begin{bmatrix}1&0 \\ 0&0 \end{bmatrix}}_{A}\cdot \underbrace{\begin{bmatrix}z_t \\u_t \end{bmatrix}}_{x_t}+\underbrace{\begin{bmatrix}\sigma_1&0\\0&\sigma_2 \end{bmatrix}}_{C} \underbrace{\begin{bmatrix}w_{1t+1}\\ w_{2t+1} \end{bmatrix}}_{w_{t+1}} \]
\[ y_t = \underbrace{\begin{bmatrix}1&1 \end{bmatrix}}_{G}\cdot\underbrace{\begin{bmatrix}z_t \\u_t \end{bmatrix}}_{x_t} \]
Computing the key matrices:
\[ I-\beta A = \begin{bmatrix}1&0\\0&1 \end{bmatrix}-\begin{bmatrix}\beta&0 \\0&0 \end{bmatrix}=\begin{bmatrix}1-\beta & 0 \\0&1 \end{bmatrix} \]
\[ (I-\beta A)^{-1} = \begin{bmatrix}\frac{1}{1-\beta}&0 \\ 0&1 \end{bmatrix} \]
(Since diagonal matrix, its inverse is just 1 over each element)
\[ G(I-\beta A)^{-1} = \begin{bmatrix}1&1 \end{bmatrix}\begin{bmatrix}\frac{1}{1-\beta}&0 \\ 0&1 \end{bmatrix}=\begin{bmatrix}\frac{1}{1-\beta}&1 \end{bmatrix} \]
Consumption:
Recall:
\[ \begin{aligned} c_t &= (1-\beta)\left[F_t+\mathbb{E}_t\left(\sum_{j=0}^{\infty}\beta^j y_{t+j}\right) \right] \\ &= (1-\beta)\left[F_t+G(I-\beta A)^{-1}x_t \right] \\ &= (1-\beta)\left[F_t+\begin{bmatrix}\frac{1}{1-\beta}&1\end{bmatrix} \cdot \begin{bmatrix}z_t\\u_t\end{bmatrix} \right] \\ &= (1-\beta)\left[F_t+\frac{1}{1-\beta}z_t+u_t \right] \end{aligned} \]
\[ \boxed{c_t = (1-\beta)F_t +z_t+(1-\beta)u_t} \]
Note: The coefficient on \(u_t\) is \((1-\beta)\), the marginal propensity to consume (MPC) out of transitory income; the coefficient on \(z_t\) is 1, which is the MPC out of permanent income. The marginal propensity to consume out of financial wealth \(F_t\) is the same as before.
Consumption changes:
Recall:
\[ \begin{aligned} c_{t+1}-c_t &= (1-\beta)G(I-\beta A)^{-1}\cdot C \cdot w_{t+1} \\ &=(1-\beta)\begin{bmatrix}\frac{1}{1-\beta}&1 \end{bmatrix}\begin{bmatrix}\sigma_1&0\\0&\sigma_2 \end{bmatrix}\cdot\begin{bmatrix}w_{1t+1}\\w_{2t+1} \end{bmatrix} \end{aligned} \]
\[ \boxed{c_{t+1}-c_t = \sigma_1 w_{1t+1}+(1-\beta)\sigma_2 w_{2t+1}} \]
That is, the consumer consumes all of the permanent shock, and the MPC out of the transitory shock.
Savings:
Recall:
\[ \begin{aligned} F_{t+1}-F_t &= \frac{1}{\beta}\left[y_t-(1-\beta)\mathbb{E}_t\sum_{j=0}^{\infty}\beta^j y_{t+j} \right] \\ &=\frac{1}{\beta}\left[G\cdot x_t-(1-\beta)G(I-\beta A)^{-1}x_t \right] \\ &=\frac{1}{\beta}G\left[I-(1-\beta)(I-\beta A)^{-1}\right]x_t \end{aligned} \]
Computing:
\[ G\cdot I = \begin{bmatrix}1&1 \end{bmatrix}\begin{bmatrix}1&0 \\ 0&1 \end{bmatrix}=\begin{bmatrix}1&1 \end{bmatrix} \]
\[ (1-\beta)G(I-\beta A)^{-1} = \begin{bmatrix}1& 1-\beta \end{bmatrix} \]
Therefore:
\[ \begin{aligned} F_{t+1}-F_t &= \frac{1}{\beta}\left[\begin{bmatrix}1&1 \end{bmatrix}-\begin{bmatrix}1& 1-\beta \end{bmatrix} \right]\begin{bmatrix}z_t\\u_t \end{bmatrix} \\ &=\frac{1}{\beta}\begin{bmatrix}0&\beta \end{bmatrix}\begin{bmatrix}z_t \\u_t \end{bmatrix} \\ &=\begin{bmatrix}0&1 \end{bmatrix}\begin{bmatrix}z_t\\u_t \end{bmatrix} \end{aligned} \]
\[ \boxed{F_{t+1}-F_t = u_t} \]
That is, the consumer spends all of \(z_t\) and saves nothing, but saves a fraction of transitory income (which returns on savings to \(F_{t+1}\)). The fraction of \(u_t\) consumed is the annuity value \(\frac{R-1}{R} u_t\) since \(R (1 - \frac{R-1}{R})u_t = u_t\) for the rest of the income.