Permanent Income Model

Chase Coleman and Thomas Sargent

This notebook maps instances of the linear-quadratic-Gaussian permanent income model with $\beta R = 1$ into a linear state space system, applies two different approaches to solving the model and compares outcomes from those two approaches. After confirming that answers produced by the two methods agree, it applies the quantecon LinearStateSpace class to illustrate various features of the model.

Besides being a workhorse model for analyzing consumption data, the model is good for illustrating the concepts of

* stationarity

* ergodicity

* ensemble moments and cross section observations

* cointegration

* linear-quadratic dynamic programming problems 

Background readings on the linear-quadratic-Gaussian permanent income model are Robert Hall's 1978 JPE paper ``Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence'' and chapter 2 of Recursive Macroeconomic Theory

Let's get started

Plan of the notebook

We study a version of the linear-quadratic-Gaussian model described in section 2.12 of chapter 2 of Ljungqvist and Sargent's Recursive Macroeconomic Theory

We solve the model in two ways:

• as an LQ dynamic programming problem, and

• as a system of expectational difference equations with boundary conditions that advise us to solve stable roots backwards and unstable roots forwards (see appendix A of chapter 2 of Ljungqvist and Sargent).

We confirm numerically that these two methods give rise to approximately the same solution. The adverb approximately is appropriate because we use a technical trick to map the problem into a well behaved LQ dynamic programming problem.

The model

The LQ permanent income model is an example of a ``savings problem.'' A consumer has preferences over consumption streams that are ordered by the utility functional

$$ E_0 \sum_{t=0}^\infty \beta^t u(c_t), \quad(1) $$

where $E_t$ is the mathematical expectation conditioned on the consumer's time $t$ information, $c_t$ is time $t$ consumption, $u(c)$ is a strictly concave one-period utility function, and $\beta \in (0,1)$ is a discount factor. The LQ model gets its name partly from assuming that the utility function $u$ is quadratic:

$$ u(c) = -.5(c - \gamma)^2 $$

where $\gamma>0$ is a bliss level of consumption.

The consumer maximizes the utility functional (1) by choosing a consumption, borrowing plan $\{c_t, b_{t+1}\}_{t=0}^\infty$ subject to the sequence of budget constraints

$$ c_t + b_t = R^{-1} b_{t+1} + y_t, t \geq 0, \quad(2) $$

where $y_t$ is an exogenous stationary endowment process, $R$ is a constant gross risk-free interest rate, $b_t$ is one-period risk-free debt maturing at $t$, and $b_0$ is a given initial condition. We shall assume that $R^{-1} = \beta$. Equation (2) is linear. We use another set of linear equations to model the endowment process. In particular, we assume that the endowment process has the state-space representation

$$ \eqalign{ z_{t+1} & = A_{22} z_t + C_2 w_{t+1} \cr y_t & = U_y z_t \cr} \quad (3) $$

where $w_{t+1}$ is an i.i.d. process with mean zero and identity contemporaneous covariance matrix, $A_{22}$ is a stable matrix, its eigenvalues being strictly below unity in modulus, and $U_y$ is a selection vector that identifies $y$ with a particular linear combination of the $z_t$.

We impose the following condition on the consumption, borrowing plan:

$$ E_0 \sum_{t=0}^\infty \beta^t b_t^2 < +\infty. \quad (4) $$

This condition suffices to rule out Ponzi schemes. (We impose this condition to rule out a borrow-more-and-more plan that would allow the household to enjoy bliss consumption forever.)

The state vector confronting the household at $t$ is $$ x_t = \left[\matrix{z_t \cr b_t\cr}\right]',$$ where $b_t$ is its one-period debt falling due at the beginning of period $t$ and $z_t$ contains all variables useful for forecasting its future endowment.

We shall solve the problem two ways.

• First, as a linear-quadratic control dynamic programming problem that we can solve using the LQ class.

• Second, as a set of expectational difference equations that we can solve with homemade programs.

Solution as an LQ problem

We can map the problem into a linear-quadratic dynamic programming problem, also known as an optimal linear regulator problem.

The stochastic discounted linear optimal regulator problem is to choose a decision rule for $u_t$ to maximize

$$ - E_0\sum_{t=0}^\infty \beta^t \{x'_t Rx_t+u'_tQu_t\},\quad 0<\beta<1,$$

subject to $x_0$ given, and the law of motion

$$x_{t+1} = A x_t+ Bu_t+ C w_{t+1},\qquad t\geq 0, $$

where $w_{t+1}$ is an $(n\times 1)$ vector of random variables that is independently and identically distributed according to the normal distribution with mean vector zero and covariance matrix $Ew_t w'_t= I .$

The value function for this problem is $v(x)= - x'Px-d,$ where $P$ is the unique positive semidefinite solution of the discounted algebraic matrix Riccati equation corresponding to the limit of iterations on matrix Riccati difference equation

$$P_{j+1} =R+\beta A'P_j A-\beta^2 A'P_jB(Q+\beta B'P_jB)^{-1} B'P_jA.$$

from $P_0=0$. The optimal policy is $u_t=-Fx_t$, where $F=\beta (Q+\beta B'PB)^{-1} B'PA$. The scalar $d$ is given by $ d=\beta(1-\beta)^{-1} {\rm trace} ( P C C') . $

Under an optimal decision rule $F$, the state vector $x_t$ evolves according to

$$ x_{t+1} = (A-BF) x_t + C w_{t+1} $$$$ \left[\matrix{z_{t+1} \cr b_{t+1} \cr}\right] = \left[\matrix{ A_{22} & 0 \cr R(U_\gamma - U_y) & R } \right]\left[\matrix{z_{t} \cr b_{t} \cr}\right] + \left[\matrix{0 \cr R}\right] (c_t - \gamma) + \left[\matrix{ C_t \cr 0 } \right] w_{t+1} $$

or

$$ x_{t+1} = A x_t + B u_t + C w_{t+1} $$

We form the quadratic form $x_t' \bar R x_t + u_t'Q u_t $ with $Q =1$ and $\bar R$ a $ 4 \times 4$ matrix with all elements zero except for a very small entry $\alpha >0$ in the $(4,4)$ position. (We put the $\bar \cdot$ over the $R$ to avoid ``recycling'' the $R$ notation!)

We begin by creating an instance of the state-space system (2) that governs the income $\{y_t\}$ process. We assume it is a second order univariate autoregressive process: $$ y_{t+1} = \alpha + \rho_1 y_t + \rho_2 y_{t-1} + \sigma w_{t+1} $$

It turns out that the bliss level of consumption $\gamma$ in the utility function $-.5 (c_t -\gamma)^2$ has no effect on the optimal decision rule. (We shall see why below when we inspect the Euler equation for consumption.)

Now create the objects for the optimal linear regulator.

Here we will use a trick to induce the Bellman equation to respect restriction (4) on the debt sequence $\{b_t\}$. To accomplish that, we'll put a very small penalty on $b_t^2$ in the criterion function.

That will induce a (hopefully) small approximation error in the decision rule. We'll check whether it really is small numerically soon.

Now create the appropriate instance of an LQ model

Now create the optimal policies using the analytic formulas.

We'll save the answers and will compare them with answers we get by employing an alternative solution method.

Solution via a system of expectational difference equations

Now we will solve the household's optimum problem by first deducing the Euler equations that are the first-order conditions with respect to consumption and savings, then using the budget constraints and the boundary condition (4) to complete a system of expectational linear difference equations that we'll solve for the optimal consumption, debt plan.

First-order conditions for the problem are

$$ E_t u'(c_{t+1}) = u'(c_t) , \ \ \forall t \geq 0. \quad (5) $$

In our linear-quadratic model, we assume the quadratic utility function $u(c_t) = -.5 (c_t - \gamma)^2$, where $\gamma$ is a bliss level of consumption. Then the consumption Euler equation becomes $$ E_t c_{t+1} = c_t . \quad (6) $$ Along with the quadratic utility specification, we allow consumption $c_t$ to be negative.

To deduce the optimal decision rule, we want to solve the system of difference equations formed by (2) and (6) subject to the boundary condition (4). To accomplish this, solve (2) forward and impose $\lim_{T\rightarrow +\infty} \beta^T b_{T+1} =0$ to get $$ b_t = \sum_{j=0}^\infty \beta^j (y_{t+j} - c_{t+j}) . \quad (7) $$ Imposing $\lim_{T\rightarrow +\infty} \beta^T b_{T+1} =0$ suffices to impose (4) on the debt path. Take conditional expectations on both sides of (7) and use (6) and the law of iterated expectations to deduce $$ b_t = \sum_{j=0}^\infty \beta^j E_t y_{t+j} - {1 \over 1-\beta} c_t \quad (8) $$ or $$ c_t = (1-\beta) \left[ \sum_{j=0}^\infty \beta^j E_t y_{t+j} - b_t\right]. \quad (9) $$ If we define the net rate of interest $r$ by $\beta ={1 \over 1+r}$, we can also express this equation as $$ c_t = {r \over 1+r} \left[ \sum_{j=0}^\infty \beta^j E_t y_{t+j} - b_t\right]. \quad (10) $$ Equation (9) or (10) asserts that consumption equals what Irving Fisher defined as economic income, namely, a constant marginal propensity to consume or interest factor ${r \over 1+r}$ times the sum of nonfinancial wealth $ \sum_{j=0}^\infty \beta^j E_t y_{t+j}$ and financial wealth $-b_t$. Notice that (9) or (10) represents $c_t$ as a function of the state $[b_t, z_t]$ confronting the household, where from $z_t$ contains all information useful for forecasting the endowment process.

Pulling together our preceding results, we can regard $z_t, b_t$ as the time $t$ state, where $z_t$ is an exogenous component of the state and $b_t$ is an endogenous component of the state vector. The system can be represented as

$$ \eqalign{ z_{t+1} & = A_{22} z_t + C_2 w_{t+1} \cr b_{t+1} & = b_t + U_y [ (I -\beta A_{22})^{-1} (A_{22} - I) ] z_t \cr y_t & = U_y z_t \cr c_t & = (1-\beta) [ U_y(I-\beta A_{22})^{-1} z_t - b_t ]. \cr } \quad (11) $$

Now we'll apply the formulas in equation system (11).

Later we shall use them to get objects needed to form the system (11) as an instance of a LinearStateSpace class that we'll use to exhibit features of the LQ permanent income model.

A_LSS calculated as we have here should equal ABF calculated above using the LQ model.

Here comes the check. The difference between ABF and A_LSS should be zero

Now compare pertinent elements of c_pol and -F

We have verified that the two methods give the same solution.

Now let's create an instance of a LinearStateSpace model.

To do this, we'll use the outcomes from out second method.

Two examples

Now we'll generate panels of consumers. We'll study two examples that are differentiated only by the initial states with which we endow consumers. All other parameter values are kept the same in the two examples.

• In the first example, all consumers begin with zero nonfinancial income and zero debt. The consumers are thus ex ante identical.
• In the second example, consumers are ex ante heterogeneous. While all of them begin with zero debt, we draw their initial income levels from the invariant distribution of financial income.

In the first example, consumers' nonfinancial income paths will display prounounced transients early in the sample that will affect outcomes in striking ways. Those transient effects will not be present in the second example.

Now we'll use methods that the LinearStateSpace class contains to simulate the model with our first set of intitial conditions.

• 25 paths of the exogenous non-financial income process and the associated consumption and debt paths. In the first set of graphs, the darker lines depict one particular sample path, while the lighter lines indicate the other 24 paths.

• A second graph that plots a collection of simulations against the population distribution that we extract from the LinearStateSpace instance LSS

Population and sample panels

In the code below, we use the LinearStateSpace class to

• compute and plot population quantiles of the distributions of consumption and debt for a population of consumers

• simulate a group of 25 consumers and plot sample paths on the same graph as the population distribution

First example

Here is what is going on in the above graphs.

Because we have set $y_{-1} = y_{-2} = 0$, nonfinancial income $y_t$ starts far below its stationary mean $\mu_{y, \infty}$ and rises early in each simulation.

To help interpret the behavior above graph, recall that we can represent the optimal decision rule for consumption in terms of the co-integrating relationship $$ (1-\beta) b_t + c_t = (1-\beta) E_t \sum_{j=0}^\infty \beta^j y_{t+j}, $$ For our simulation, we have set initial conditions $b_0 = y_{-1} = y_{-2} = 0$ (please see the code above). So at time $0$ we have $$ c_0 = (1-\beta) E_0 \sum_{t=0}^\infty \beta^j y_{t} . $$ This tells us that consumption starts at the value of an annuity from the expected discounted value of nonfinancial income. To support that level of consumption, the consumer borrows a lot early on, building up substantial debt. In fact, he or she incurs so much debt that eventually, in the stochastic steady state, he consumes less each period than his income. He uses the gap between consumption and income mostly to service the interest payments due on his debt.

Thus, when we look at the panel of debt in the accompanying graph, we see that this is a group of ex ante indentical people each of whom starts with zero debt. All of them accumulate debt in anticipation of rising nonfinancial income. The expect their nonfinancial income to rise toward the invariant distribution of income, a consequence of our having started them at $y_{-1} = y_{-2} = 0$.

Illustration of cointegration

The LQ permanent income model is a good one for illustrating the concept of cointegration.

The following figure plots realizations of the left side of $$ (1-\beta) b_t + c_t = (1-\beta) E_t \sum_{j=0}^\infty \beta^j y_{t+j}, \quad (12) $$ which is called the cointegrating residual.

Notice that it equals the right side, namely, $(1-\beta) E_t \sum_{j=0}^\infty \beta^j y_{t+j}$, which equals an annuity payment on the expected present value of future income $E_t \sum_{j=0}^\infty \beta^j y_{t+j}$.

Early along a realization, $c_t$ is approximately constant while $(1-\beta) b_t$ and $(1-\beta) E_t \sum_{j=0}^\infty \beta^j y_{t+j}$ both rise markedly as the household's present value of income and borrowing rise pretty much together.

Note: This example illustrates the following point: the definition of cointegration implies that the cointegrating residual is asymptotically covariance stationary, not covariance stationary. The cointegrating residual for the specification with zero income and zero debt initially has a notable transient component that dominates its behavior early in the sample. By specifying different initial conditions, we shall remove this transient in our second example to be presented below.

A "borrowers and lenders" closed economy

When we set $y_{-1} = y_{-2} = 0$ and $b_0 =0$ in the preceding exercise, we make debt "head north" early in the sample. Average debt rises and approaches asymptote.

We can regard these as outcomes of a ``small open economy'' that borrows from abroad at the fixed gross interest rate $R$ in anticipation of rising incomes.

So with the economic primitives set as above, the economy converges to a steady state in which there is an excess aggregate supply of risk-free loans at a gross interest rate of $R$. This excess supply is filled by ``foreigner lenders'' willing to make those loans.

We can use virtually the same code to rig a "poor man's Bewley model" in the following way.

• as before, we start everyone at $b_0 = 0$.

• But instead of starting everyone at $y_{-1} = y_{-2} = 0$, we draw $\begin{bmatrix} y_{-1} \cr y_{-2} \end{bmatrix}$ from the invariant distribution of the $\{y_t\}$ process.

This rigs a closed economy in which people are borrowing and lending with each other at a gross risk-free interest rate of $R = \beta^{-1}$. Here within the group of people being analyzed, risk-free loans are in zero excess supply. We have arranged primitives so that $R = \beta^{-1}$ clears the market for risk-free loans at zero aggregate excess supply. There is no need for foreigners to lend to our group.

The following graphs confirm the following outcomes:

• as before, the consumption distribution spreads out over time. But now there is some initial dispersion because there is ex ante heterogeneity in the initial draws of $\begin{bmatrix} y_{-1} \cr y_{-2} \end{bmatrix}$.

• as before, the cross-section distribution of debt spreads out over time.

• Unlike before, the average level of debt stays at zero, reflecting that this is a closed borrower-and-lender economy.

• Now the cointegrating residual seems stationary, and not just asymptotically stationary.