Optimal Consumption, Savings, and the Permanent Income Model
Undergraduate Computational Macro
Overview
Motivation
- Previously, the savings rate has been an exogenously given function of income
- In this section we will analyze optimal consumption and savings decisions in a simple model: the classic permanent income model of Milton Friedman and refined by Hall (1978)
- Given these decision processes, we could embed them into our previous models of income dynamics
Fort Knox
- Fort Knox holds about 13K tons of gold
- Secured in granite, with a steel door 21 inches thick weighing 20 tons
- The data:
- Fort Knox has been around since 1937 and has never been robbed
- Counterfactual: How much should we spend on protection?
- What if we only look at the historical data
- If it has never been robbed, why does it need to be secured at all?
- “Structural” models let us consider a broader class of counterfactuals
- Need to have a “deep” model of criminal incentives, detection ability, etc. to see the appropriate level of security
- This is used as a classic example of the Lucas Critique
Exogenous vs. Endogenous Savings
- Why do we need to bother with a model of savings and consumption? Couldn’t we just estimate it from the data use it empirically?
- The challenge is that this only leads to a limited number of counterfactuals
- For example, we can simulate a panel of agents living in a fixed economy spanned by the data
- But what about numerical experiments where the environment changes??
- A tax cut when future taxes balance budget
- Wouldn’t the savings rate change in response to these plans?
- This led early macro-economists to consider that the Marginal Propensity to Consume (MPC) might adjust based on information sets alone
Materials
- Adapted from QuantEcon lectures coauthored with John Stachurski and Thomas J. Sargent
- Optimal Savings I: The Permanent Income Model
- Note using \(F_t = -b_t\). i.e., financial assets rather than debt
- Optimal Savings I: The Permanent Income Model
Preferences
Welfare and Preferences
To introduce an endogenous choice, consider how an agent would compare alternative bundles of consumption goods. Assume agent:
- Lives for \(t=0,\ldots \infty\) (see our first lectures importance)
- Gains period utility \(u(c_t)\) from consumption \(c_t\). Previously we assumed this was linear
- Discounts future \(u(c_t)\) with discount factor \(\beta \in (0,1)\)
This leads to preferences that are additively separable and they compare \(\{c_t\}_{t=0}^\infty\) streams of consumption at time \(t\)
\[ \sum_{j=0}^\infty \beta^j u(c_{t+j}) \]
Period Utility
- Previously we had assumed a linear utility function (which we called “risk neutral”)
- Consider utility which is strictly concave where:
- \(u'(c) > 0\): More is better
- \(u''(c) < 0\): Diminishing Marginal Utility
- \(\lim_{c \to 0} u'(c) = \infty\): Infinite Marginal Utility at zero
- Examples include
- \(u(c) = \log(c)\) and \(u(c) = \frac{c^{1-\gamma}}{1-\gamma}\) for \(\gamma > 0\)
- \(u(c) = \frac{a_2}{2}c^2 + a_1 c + a_0\) for \(a_2 < 0\) as long as \(c\) is less than the “satiation point” where \(u'(c) = 0\)
Strictly Concave Utility
- Positive Marginal Utility of Consumption
- Diminishing Returns
- No (visible, at least) point of satiation
Uncertainty
What if the agent does not know \(\{c_t\}_{t=0}^\infty\) because it is random or uncertain?
In that case, we can instead have the agent compare expected utility streams
\[ \mathbb{E}_t\left[\sum_{j=0}^\infty \beta^j u(c_{t+j})\right] \]
- Where \(\mathbb{E}_t[\cdot] \equiv \mathbb{E}[\cdot|I_t]\) with \(I_t\) the information set we make available at time \(t\) for forecasting in our model
- This uses our model of expectation formation from the previous lecture
Risk Aversion vs. Inter-temporal Substitution
- If \(u(c)\) is strictly concave the agent:
- Risk Averse: Prefers more deterministic consumption to those with a higher variance
- Preferences for Consumption Smoothing: Will substitute between time periods rather than smoother consumption over time rather than large fluctuations
- One challenge in macroeconomics with these preferences is that the \(u(c)\) serves both purposes, which have different economic interpretations.
- To disentangle, can use recursive preferences such as Epstein-Zin which decouple these two concepts
Smoothing Incentives
- Consider a simpler case where they live for two periods and don’t discount the future: \(V(c_1, c_2) \equiv u(c_1) + u(c_2)\)
- Consider two possible bundles: \(\{c_t, c_{t+1}\}\) and \(\{\bar{c}, \bar{c}\}\) where \(c_t + c_{t+1} = 2 \bar{c}\)
- If the agent is risk-neutral, we see that \(V(c_t, c_{t+1}) = V(\bar{c}, \bar{c})\)
- However, if the agent if risk-averse, then \[
V(c_t, c_{t+1}) < V(\bar{c}, \bar{c})\quad \text{unless }c_t = c_{t+1} = \bar{c}
\]
- They strictly prefer smoother consumption over time
- i.e., would forgo consumption on average to gain smoother consumption
Smoothing and Concavity
- Recall \(\bar{c} \equiv (c_t + c_{t+1})/2\)
- 2 periods, \(\beta = 1\)
- Same “price” for \(c_t\) and \(c_{t+1}\)
- Two possible bundles:
- \(\{c_t, c_{t+1}\}\)
- \(\{\bar{c}, \bar{c}\}\)
- Later, \(\beta\) and prices will simply distort this exact tradeoff
Risk-Aversion Intuition
Consider a utility \(u(c)\) and a lottery which is a random variable
- \(C = \begin{cases} c_L & \text{with probability } \frac{1}{2} \\ c_H & \text{with probability } \frac{1}{2} \end{cases}\)
- Let \((c_L + c_H)/2 = \bar{c}\)
- We can form expected utility as \(\mathbb{E}[u(C)] = \frac{1}{2} u(c_L) + \frac{1}{2} u(c_H)\)
Note if risk-neutral then \(\mathbb{E}[C] = \frac{1}{2} c_L + \frac{1}{2}c_H = \bar{c} = u(\bar{c})\)
Then if an agent is risk-averse, \[ u(\mathbb{E}(C)) > \mathbb{E}[u(C)] \]
- i.e., would forgo consumption on average to avoid the risk
Risk Aversion and Concavity
- Interpretation as fair, risk-neutral prices for lotteries
- Then compare choice between lotteries:
- \(\mathbb{E}[u(C)] \equiv \frac{1}{2} u(c_L) + \frac{1}{2} u(c_2)\)
- \(u(\mathbb{E}(C)) = u(\frac{1}{2} c_L + \frac{1}{2} c_H)\)
- The strict concavity of \(u(c)\) shows you are better off with the deterministic consumption
The Decision Problem
Permanent Income Model
- The classic permanent income model explored the impact of these economic forces on consumption and savings decisions. The agent
- has an exogenous, potentially stochastic, income stream \(\{y_t\}_{t=0}^\infty\)
- chooses a consumption policy \(\{c_t\}_{t=0}^\infty\) to maximize expected welfare
- forecasts the random variables \(\{c_t\}_{t=0}^\infty\) and \(\{y_t\}_{t=0}^\infty\) streams using mathematical expectations
- has access to a risk-free bond market with interest rate \(R\) to either save or borrow, enabling them to smooth consumption over time or deal with uncertainty
- has financial assets at time \(t\) be \(F_t\) which also must be forecast
Period-By-Period Budgets
- Given income \(y_t\), consumption \(c_t\), and financial assets \(F_t\), the agent’s budget constraint is \[
F_{t+1} = R(F_t + \underbrace{y_t - c_t}_{\text{savings}})
\]
- where \(R > 1\) is the gross interest rate on saving or borrowing, and \(1 + r \equiv R\) would be the net interest rate
- the interpretation is simple: take their bank account value (positive or negative), add or subtract savings that period, and then they gain interest on the new balance (or the the debt grows if negative)
- If this was the only constraint, you might have infinite borrowing each period
Lifetime Budget Constraint (LBC)
Alternatively, if all of those accounting relationships must hold, substitute to form a single budget
Given forecasts of \(c_{t+j}\) and \(y_{t+j}\), the budget must fulfill
\[ \underbrace{\mathbb{E}_t\left[\sum_{j=0}^{\infty}\frac{c_{t+j}}{R^j}\right]}_{\text{EPDV of consumption}} = \underbrace{\mathbb{E}_t\left[\sum_{j=0}^{\infty} \frac{y_{t+j}}{R^j}\right]}_{\text{EPDV of income}} + F_t \]
The \(F_t\) is the current financial assets. Consider leaving in bank to pay for \(c_{t+j}\)
- Then \(R^{-j}\) enters because \(\$1\) today grows to \(\$1 \times R^j\) in \(j\) periods
- That allows you to buy \(1 \times R^{-j}\) units of \(c_{t+j}\)
A Special Case of \(R = 1/\beta\)
- Where does \(R\) come from? The decisions of other agents in the economy
- Lenders are asked to give up 1 unit of consumption today for \(R\) units of consumption tomorrow
- Hence the \(R\) should reflect the degree of impatience of the lender
- An important case is when \(\beta R = 1\)
- As we will discuss later, this will arise in equilibrium as the natural rate of interest when agent’s can smooth consumption fully
- The intuition is that the gross interest rate exactly offsets the impatience, as captured by the discount factor. Risk, etc. will enter later
Lifetime Budget Constraint when \(\beta R = 1\)
In that case, we see that the budget constraint simplifies to
\[ \underbrace{\mathbb{E}_t\left[\sum_{j=0}^{\infty}\beta^j c_{t+j}\right]}_{\text{EPDV of consumption}} = \underbrace{\mathbb{E}_t\left[\sum_{j=0}^{\infty} \beta^j y_{t+j}\right]}_{\text{EPDV of income}} + F_t \]
- This should give you hope on tractability: if \(c_{t+j}\) and \(y_{t+j}\) follow simple stochastic processes (e.g., the LSS) then we can can calculate these EPDV
Decision Problem
- Economists (usually) formalize decisions as optimization problems
- Taking an exogenous gross interest rate \(R > 1\) and \(\{F_t, y_t, y_{t-1}\ldots\}\)
\[ \begin{aligned} \max_{\{c_{t+j}, F_{t+j+1}\}_{j=0}^\infty} & \mathbb{E}_t\left[\sum_{j=0}^\infty \beta^j u(c_{t+j})\right] \\ \text{s.t.} \,& F_{t+j+1} = R(F_{t+j} + y_{t+j} - c_{t+j})\,\quad \text{ for all } j \geq 0\\ & \text{no-ponzi scheme/transversality condition} \end{aligned} \]
- No-ponzi condition treated informally: prevents the agent from borrowing too quickly, equivalent to not dying in debt if they had a finite life
Decision Problem (Alternative)
While we cannot in extensions with borrowing constraints, etc., here we can use the LBC
\[ \begin{aligned} \max_{\{c_{t+j}\}_{j=0}^\infty} & \mathbb{E}_t\left[\sum_{j=0}^\infty \beta^j u(c_{t+j})\right] \\ \text{s.t.} \,& \mathbb{E}_t\left[\sum_{j=0}^{\infty}R^{-j} (c_{t+j}-y_{t+j})\right] = F_t \end{aligned} \]
- rearranged to show that the EPDV of savings = initial financial assets. Implicitly uses the no-ponzi scheme condition
Consumption Plans, Information Sets, and Forecasts
- \(c_{t+j}\) and \(F_{t+j+1}\) are random variables for all \(j > 0\)
- The agent is making a plan for each realization of the random shocks
- To forecast the future, they are conditioning on their own decisions given the randomness inherent to the \(y_{t+j}\) process
- Formally modeling information to use \(\mathbb{E}_t[\cdot]=\mathbb{E}[\cdot\,|\,\mathcal{I}_t]\)
State-Contingent Plans
- At time \(t\), the agent chooses a \(c_t\,|\,F_t, y_t, F_{t-1}, y_{t-1},\ldots\)
- In doing so, they need to consider their future \(c_{t+j}\) for \(j > 0\)
- But they do not know \(c_{t+j}\) until they see \(y_{t+\tau}\) for \(\tau \leq j\)
- More generally, the agent is choosing a plan \(c_{t+j}\,|\,F_{t+j}, y_{t+j}, F_{t+j-1}, y_{t+j-1}, \ldots\)
- i.e. a contingent plan of how much to consume given all possible information states.
- They are choosing a random variable!
- This seems like an intractable problem? Is there even a sense of self-consistency?
Time-Consistency
Would they change their mind and choose a different \(c_{t+1}\) than their plan?
- \(\mathcal{I}_t = \{F_t, y_t, \ldots\}\) to new information \(\mathcal{I}_{t+1} = \{F_{t+1}, y_{t+1}, F_t, y_t, \ldots\}\)
Solve for \(c_t\) and plan \(c_{t+1}\) for all \(y_{t+1}\) realizations \[ \max_{\{c_{t+j}\}_{j=0}^\infty} \mathbb{E}\left[\sum_{j=0}^\infty \beta^j u(c_{t+j})\,|\,\mathcal{I}_t\right] \text{ s.t.} \, \mathbb{E}\left[\sum_{j=0}^{\infty}R^{-j} (c_{t+j}-y_{t+j})\,|\,\mathcal{I}_t\right] = F_t \]
See \(y_{t+1}\) and \(F_{t+1}\) and “reoptimize”, solving for \(c_{t+1}\) \[ \max_{\{c_{t+j+1}\}_{j=0}^\infty} \mathbb{E}\left[\sum_{j=0}^\infty \beta^j u(c_{t+j+1})\,|\,\mathcal{I}_{t+1}\right] \text{ s.t.} \, \mathbb{E}\left[\sum_{j=0}^{\infty}R^{-j} (c_{t+j+1}-y_{t+j+1})\,|\,\mathcal{I}_{t+1}\right] = F_{t+1} \]
Without proof, with these preferences the plan is time-consistent: they will not want to change their plan, even after seeing \(y_{t+j}\) for \(j > 0\)
First-Order Conditions
Without a full derivation, can show that the solution to this problem exists, for strictly concave \(u(c)\) where, given a \(F_0\) initial condition
\[ \begin{aligned} u'(c_t) &= \beta R\, \mathbb{E}_t[u'(c_{t+1})],\quad \text{ Euler equation} \\ F_{t+1} &= R(F_t + y_t - c_t),\quad \text{ Budget Constraint}\\ 0 &= \mathbb{E}_0\left[\lim_{j \to \infty} \beta^j F_{t+j}\right],\quad \text{ No-Ponzi Scheme} \end{aligned} \]
- Or, equivalently in this case, the LBC must also hold
- Note that we have switched from the \(t+j\) to just \(t\) for current period
Still challenging since we need to forecast optimal \(c_{t+1}\)
Interpreting the Euler Equation
Euler Equations are ubiquitous intertemporal optimality conditions
\[ u'(c_t) = \beta R\, \mathbb{E}_t[u'(c_{t+1})] \]
The cost consuming less today is the marginal utility today, \(u'(c_t)\)
The right-hand term is the benefit
- You gain the marginal utility (MU) of consuming a little more tomorrow
- Need to forecast MU tomorrow, considering risk aversion/smoothing
- A unit of utility tomorrow is only worth \(\beta\) times that of today
- However, you are compensated by the savings growing at interest rate \(R\) which increases the amount of units of consumption you can afford
(Optional) 2-Period Motivation
Motivating Derivation of the Euler Equation
- See here for a more complete derivation in the deterministic case
- Will derive Euler Equation for the simple case of 2-periods \(t\) and \(t+1\), which ends up nesting the general case
- Budget \(F_{t+1} = R(F_t + y_t - c_t)\) but assume \(F_{t+2} = 0\) since they “die”
- Then \(c_{t+1} = y_{t+1} + F_{t+1}\)
Decision Problem with 2 Periods
\[ \begin{aligned} \max_{c_t} & \left[u(c_t) + \beta \mathbb{E}_t[u(c_{t+1})]\right]\\ & \text{s.t.} \,c_{t+1} = y_{t+1} + R(F_t + y_t - c_t) \end{aligned} \]
- And substitute the budget constraint into the objective function
\[ \max_{c_t} \left[u(c_t) + \beta \mathbb{E}_t[u(y_{t+1} + R(F_t + y_t - c_t))]\right] \]
- Take the FONC, which can be rearranged to the Euler equation
\[ \begin{aligned} 0 &= u'(c_t) - \beta R \mathbb{E}_t[u'(y_{t+1} + R(F_t + y_t - c_t))]\\ u'(c_t) &= \beta R\, \mathbb{E}_t[u'(c_{t+1})] \end{aligned} \]
Special Case of Deterministic Income
Special Case of Deterministic Income and \(\beta R = 1\)
If \(y_t\) is deterministic, then this problem no longer requires forcasts
Furthermore, assume \(\beta R = 1\) (i.e. interest exactly offsets impatience)
\[ u'(c_t) = u'(c_{t+1}) \]
Moreover, given assumptions that \(u'(c) > 0\) and \(u''(c) < 0\), this implies that \(c_t = c_{t+1}\)
This is the classic Permanent Income Result
Reminder: Strictly Concave Utility
- Positive Marginal Utility of Consumption
- Diminishing Returns
- No (visible, at least) point of satiation
Marginal Utility
- \(u'(c) > 0\) but decreasing \(u''(c) < 0\)
- \(u'(c_1) = u'(c_2)\implies c_1 = c_2\)
- If \(u'(c_t) < u'(c_{t+1})\) then \(c_t > c_{t+1}\)
- The less they consume, the more valuable additional consumption in that period would be
Equating Marginal Utilities
- Euler here: \(u'(c_t) = u'(c_{t+1})\) for all \(t\)
- Exact for simple deterministic, \(\beta R = 1\) case
- By equating marginal utilities at all points, they gain a lower average marginal utility
Smoothing and Welfare
- The higher average marginal utility for the volatile consumption path corresponds to a lower average utility
- i.e. welfare here
- If “risk-neutral”, then the agent is indifferent between the two paths
- We see that since the utility function would be linear itself
Permanent Income Result for Strictly Concave \(u(c)\)
With \(\beta R = 1\), use \(c_t = \bar{c}\) with the LBC \[ \begin{aligned} \sum_{j=0}^{\infty}\beta^j c_{t+j} &= \sum_{j=0}^{\infty}\beta^j y_{t+j} + F_t\\ \bar{c} \sum_{j=0}^{\infty}\beta^j &= \sum_{j=0}^{\infty}\beta^j y_{t+j} + F_t\\ c_t = \bar{c} &= \underbrace{(1-\beta)}_{\text{MPC}}\bigg[\underbrace{\sum_{j=0}^{\infty}\beta^j y_{t+j}}_{\text{Human Wealth}} + F_t\bigg] \end{aligned} \]
- The consumer has a constant MPC out of total wealth
Annuity Values
- If \(R \equiv 1 + r = 1/\beta\) then notice that \(1-\beta = \frac{r}{1+r} = \frac{R - 1}{R}\)
- Consider \(F_t\) in financial assets and want to consume some and save the rest
- Assume no income, and consider if you consume \(\frac{R-1}{R}\) proportion of \(F_t\)
\[te F_{t+1} = R(F_t - c_t) = R\left(F_t - \frac{R-1}{R}F_t\right) = F_t \]
- So the annuity value is the amount I can take out and leave the bank account identical after interest
- Alternatively, the inverse of the price of an asset that payed out \(\$1\) each period forever
Stochastic Income and Consumption
What about Stochastic Income?
Leaving \(\beta R = 1\) for the remainder of the slides
Optimality: agents would LOVE to equate all marginal utilities
\[ u'(c_t) = \mathbb{E}_t[u'(c_{t+1})] \]
- Will do the best they can given information sets
With enough financial instruments to hedge all risks, they might!
- However, for an arbitrary \(u(c)\) function this is hard to achieve in our environment, where they only have a single, risk-free asset
Quadratic Utility
- A special case of these preferences is \(u(c) = \frac{a_2}{2} c^2 + a_1 c + a_0\) for \(a_2 < 0\)
- This is a quadratic utility function
- If \(a_2 < 0\) this is strictly concave
- However, \(u'(c) = a_2 c + a_1\) is always negative for large enough \(c\)
- i.e, satiation point for the \(c_{\max}\) where \(u'(c_{\max}) = 0\)
- Assume conditions such that \(c_t \ll c_{\max}\), and this is strictly concave in the relevant range
Euler Equation for Quadratic Utility
Since \(u'(c) = a_2 c + a_1\) we can write the euler equation as
\[ \begin{aligned} u'(c_t) &= \mathbb{E}_t[u'(c_{t+1})]\\ a_2 c_t + a_1 &= \mathbb{E}_t[a_2 c_{t+1} + a_1]\\ c_t &= \mathbb{E}_t[c_{t+1}] \end{aligned} \]
- That is, the agent will choose consumption so that the expected value of consumption next period is equal to the current period
With more general strictly concave preferences, often: \(c_t \approx \mathbb{E}_t[c_{t+1}]\)
Recall: Martingales
- Reminder: \(X_t\) is a martingale if \(\mathbb{E}_t[X_{t+1}] = X_t\)
- In other words, consumption is a martingale for any \(y_t\) stochastic process!
- Key feature of martingales: the history has no predictive power for the future
- This will also come up in macro-finance and asset pricing later
- i.e., if the past had systematic and consistent predictive power, then there would be systematic and consistent profits to be made
- If there were systematic profits to be made, wouldn’t prices adjust as people tried to make those profits?
Consumption is a Martingale
- Similar logic: Consumers use all available information to smooth consumption
- With the financial assets we give them, a martingale is the closest they can get to fully smoothed consumption
- The agent will look at their permanent income (i.e. EPDV of human wealth + financial assets) and plan to keep it constant on average
- This highlights that the changes in consumption come from “surprise”
- If any of that surprise was in their information sets, then they would have already adjusted
Some Implications
- As discussed, suggests that agents will adjust based on their forecasts of future income, so they are harder to trick. Tax cut now with tax increase later may have little to know effect?
- Limits the effectiveness of fiscal policy
- Makes it harder to interpret the data
- A rapidly increasing income process might have little or no effect on consumption if it is forecast
- Policies which smooth consumption will increase consumer welfare
- e.g., social security, unemployment insurance, etc.
- Financial assets allowing more intertemporal substitution (e.g. bonds) or across states (e.g. insurance, or risky assets like bonds)
Changes in Consumption
We know that \(c_t = \mathbb{E}_t[c_{t+1}]\) and there is no systematic bias in forecasting
Changes are driven by shocks. Without proof, can show
\[ c_{t+1} - c_t = (1-\beta)\sum_{j=0}^\infty \beta^j \left[\mathbb{E}_{t+1}[y_{t+j+1}] - \mathbb{E}_t[y_{t+j+1}]\right] \]
- Changes in consumption come only from information (\(\mathbb{E}_{t+1}[\cdot]\) vs. \(\mathbb{E}_t[\cdot]\))
- “Surprises” were anything they couldn’t forecast on average
- By law of iterated expectations we see this is mean zero, consistent with the martingale property
Linear State Space Models
Linear State Space Models for Income
While this theory applies to any stochastic process for income, consider a special case which is a Linear Gaussian State Space \[ \begin{aligned} x_{t+1} & = A x_t + C w_{t+1},&\text{ evolution equation} \\ y_t &= G x_t,&\text{ observation equation} \end{aligned} \]
- \(x_t \in \mathbb{R}^n, y\in \mathbb{R}\), and \(w_{t+1}\sim \mathcal{N}(0,I) \in \mathbb{R}^m\)
- \(A\in\mathbb{R}^{n\times n}, C \in \mathbb{R}^{n\times m}, G \in \mathbb{R}^{1\times n}\)
Key result if \(\beta\in(0,1)\) and \(\max\{|\text{ eigenvalue of } A|\} < 1/\beta\):
\[ \mathbb{E}_t\left[\sum_{j=0}^\infty \beta^j y_{t+j}\right] = G (I - \beta A)^{-1} x_t \]
Optimal Consumption with the LSS
Take the optimal consumption we derived earlier
\[ \begin{aligned} c_t &= (1-\beta)\left[\mathbb{E}_t\left[\sum_{j=0}^\infty \beta^j y_{t+j}\right] + F_t\right]\\ &= (1-\beta)\left[G (I - \beta A)^{-1} x_t + F_t\right] \end{aligned} \]
We can use this in the evolution of the financial assets, with \(R = 1/\beta\)
\[ \begin{aligned} F_{t+1} &= \beta^{-1}(F_t + y_t - c_t)\\ &= \beta^{-1}(F_t + G x_t - (1-\beta)\left[G (I - \beta A)^{-1} x_t + F_t\right])\\ &= F_t + G(I - \beta A)^{-1}(I - A)x_t,\quad \text{ after some algebra} \end{aligned} \]
Impulse Response Function (IRFs)
- A common tool in macro: look at the response of the system to a “shock”
- This is called the impulse response function
- Think of this as feeding in a one-time change to \(w_{t+1}\) and then seeing how that propagates for \(x_{t+j}\) and \(y_{t+j}\)
- These are especially easy in LSS models because you can solve the system feeding in zeros for all other shocks as the comparison
- Given this, we can also look at the present discounted value of the impulse response function, which will help us interpret the model
Impulse Response for a LSS
The impulse is a \(w_1\) shock, typically just zeros and ones depending on the experiment and then \(w_{t+1} = 0\) for all \(t > 1\)
Then for some \(x_0\) initial condition, compare the evolution with this shock relative to one with \(w_{t+1} = 0\) for all \(t\geq 1\)
Denote the version with zero shocks throughout as \(\bar{x}_t\), then
\[ \begin{aligned} x_1 - \bar{x}_1 &= A x_0 + C w_1 - (A x_0 + C\times 0) = C w_1\\ x_2 - \bar{x}_2 &= A x_1 + C \times 0 - (A \bar{x}_1 + C\times 0)= A C w_1 \end{aligned} \]
More generally, for any \(t > 0\)
- \(x_t - \bar{x}_t = A^{t-1} C w_1\)
- \(y_t - \bar{y}_t = G A^{t-1} C w_1\)
EPDV of an Impulse Response
Consider instead the expected present-discounted value of this “shock”
\[ \sum_{j=0}^\infty \beta^j y_{t+j} - \sum_{j=0}^\infty \beta^j \bar{y}_{t+j} = \sum_{j=0}^\infty \beta^j G A^{t+j-1} C w_1 = G(I - \beta A)^{-1} C w_1 \]
Going back to the change in consumption, we can show that
\[ \begin{aligned} c_{t+1} - c_t &= (1-\beta)\sum_{j=0}^\infty \beta^j \left[\mathbb{E}_{t+1}[y_{t+j+1}] - \mathbb{E}_t[y_{t+j+1}]\right]\\ &= (1-\beta)G(I - \beta A)^{-1} C w_{t+1} \end{aligned} \]
- Interpretation: change in \(c\) = MPC \(\times\) EPDV of IRF to “shock”
Consolidating into a Single LSS
- The “state” of the agent is then summarized by \(x_t, F_t\)
- The key observations are the \(y_t\) and \(c_t\), where the later is now the optimal decision
- Given that everything is still linear and Gaussian, we can combine these into a new LSS (note: could instead have used the \(c_{t+1}\) LOM)
- State: \(\tilde{x}_t \equiv \begin{bmatrix} x_t & F_t \end{bmatrix}^{\top}\)
- Observables: \(\tilde{y}_t \equiv \begin{bmatrix} y_t & c_t \end{bmatrix}^{\top}\)
- Then the evolution and observation equations just need to be stacked
Stacked LSS
- The stacked evolution equation (for \(\mathbf{0}\) a vector of zeros) \[
\begin{aligned}
\underbrace{\begin{bmatrix} x_{t+1} \\ F_{t+1} \end{bmatrix}}_{\tilde{x}_{t+1}} &= \underbrace{\begin{bmatrix} A & \mathbf{0} \\ G(I - \beta A)^{-1}(I - A) & 1 \end{bmatrix}}_{\tilde{A}} \underbrace{\begin{bmatrix} x_t \\ F_t \end{bmatrix}}_{\tilde{x}_t} + \underbrace{\begin{bmatrix} C \\ 0 \end{bmatrix}}_{\tilde{C}} w_{t+1}\\
\underbrace{\begin{bmatrix} y_t \\ c_t \end{bmatrix}}_{\tilde{y}_t} &= \underbrace{\begin{bmatrix} G & 0\\ (1-\beta)G(I - \beta A)^{-1} & 1-\beta \end{bmatrix}}_{\tilde{G}} \underbrace{\begin{bmatrix} x_t \\ F_t \end{bmatrix}}_{\tilde{x}_t}
\end{aligned}
\]
- Then use our LSS tools with \(\tilde{x}_{t+1} = \tilde{A} \tilde{x}_t + \tilde{C} w_{t+1}\) and \(\tilde{y}_t = \tilde{G} \tilde{x}_t\)
Examples
IID Income LSS
Consider the case where IID income: \(y_t \sim N(\mu, \sigma^2)\)
- Hence: \(y_t = \mu + \sigma w_t\) and \(y_{t+1} = \mu + \sigma w_{t+1}\) where \(w_{t+1}\sim N(0,1)\)
One way to write as a state-space model is: \(x_t \equiv \begin{bmatrix} w_t & 1 \end{bmatrix}^{\top}\)
\[ \begin{aligned} \underbrace{\begin{bmatrix} w_{t+1} \\ 1 \end{bmatrix}}_{x_{t+1}} &= \underbrace{\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}}_{A} \underbrace{\begin{bmatrix} w_t \\ 1 \end{bmatrix}}_{x_t} + \underbrace{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}_{C} w_{t+1}\\ y_t &= \underbrace{\begin{bmatrix} \sigma & \mu \end{bmatrix}}_{G} \underbrace{\begin{bmatrix} w_t \\ 1 \end{bmatrix}}_{x_t} \end{aligned} \]
- Note: \(G(I - \beta A)^{-1} = \begin{bmatrix} \sigma & \frac{\mu}{1-\beta} \end{bmatrix}, G(I-\beta A)^{-1}(I - A) = \begin{bmatrix} \sigma & 0 \end{bmatrix}\)
Stacked LSS for IID Income
\[ \begin{aligned} \underbrace{\begin{bmatrix} w_{t+1}\\ 1 \\ F_{t+1} \end{bmatrix}}_{\tilde{x}_{t+1}} &= \underbrace{\begin{bmatrix} 0 & 0 & 0\\ 0 & 1 & 0\\ \sigma & 0 & 1 \end{bmatrix}}_{\tilde{A}} \underbrace{\begin{bmatrix} w_t \\ 1 \\ F_t \end{bmatrix}}_{\tilde{x}_t} + \underbrace{\begin{bmatrix} 1 \\0 \\ 0 \end{bmatrix}}_{\tilde{C}} w_{t+1}\\ \underbrace{\begin{bmatrix} y_t \\ c_t \end{bmatrix}}_{\tilde{y}_t} &= \underbrace{\begin{bmatrix} \sigma & \mu & 0\\ (1-\beta)\sigma & \mu & 1-\beta \end{bmatrix}}_{\tilde{G}} \underbrace{\begin{bmatrix} w_t \\ 1 \\ F_t \end{bmatrix}}_{\tilde{x}_t} \end{aligned} \]
Analysis of Savings
From this, we see that
\[ F_{t+1} = F_t + \sigma w_t \]
- i.e., financial assets are a random walk, a martingale
Note that if \(F_0 = 0\) and \(w_0 = 0\) (i.e., \(y_t = \mu\))
\[ F_t = \sigma w_0 + \sigma w_1 + \ldots + \sigma w_{t-1} = \sigma \sum_{j=0}^{t-1} w_j \]
- i.e., financial assets = accumulated unanticipated income shocks
- Lack of persistence in the income process means agent does not need to smooth for predictable drifts
Analysis of Consumption
The row of \(\tilde{G}\) in the LSS row is \(c_t = \mu + (1-\beta)\left[\sigma w_t + F_t\right]\)
And with \(F_0 = 0\) and \(w_0 = 0\) (i.e., \(y_t = \mu\))
\[ c_t = \mu + (1-\beta)\left[\sigma w_t + \sigma \sum_{j=0}^{t-1} w_j\right] = \mu + (1-\beta)\sigma \sum_{j=0}^t w_j \]
- i.e., consumption is also a random walk and a martingale
Change in consumption is driven by IID shocks. MPC \(\times\) annuitized value of the shock \[ c_{t+1} - c_t = (1-\beta)\sigma w_{t+1} \]
Direct Simulation
function simulate_iid_income(p, T; w = randn(T))
w_sum = cumsum(w) #(w_1, w_1 + w_2, w_1 + w_2 + w_3, ... sum_{j=1}^T w_j))
c = p.mu .+ (1 - p.beta) * p.sigma * w_sum # (c_1, c_2, ... c_T)
y = p.mu .+ p.sigma * w # (y_1, y_2, ... y_T)
F = [0.0; p.sigma * w_sum[1:end-1]] #(F_1, F_2, ... F_T)
return (;w, F, c, y)
end
p = (;beta= 1.0 / (1.0 + 0.05), mu = 1.0, sigma = 0.15)
T = 60
res = simulate_iid_income(p, T)
plot(1:T, res.y, color = :green, label = L"non-financial income, $y_t$",
xlabel="Time", legend = :bottomright)
plot!(res.c, color = :black, label = L"consumption, $c_t$")
plot!(res.F, color = :blue, label = L"Assets, $F_t$")
hline!([p.mu], color = :black, linestyle = :dash, label = L"mean income, $\mu$")Direct Simulation
LSS Formulation
A = [0 0; 0 1]
G = [p.sigma p.mu]
C = [1; 0]
H = G*inv(I-p.beta*A)
A_tilde = [A zeros(2,1); H*(I-A) 1]
C_tilde = [C; 0]
G_tilde = [G 0; (1-p.beta)*H 1-p.beta]
x_tilde_0 = [0.0, 1, 0.0] #[w_0, 1, F_0]
lss_pi = LSS(A_tilde, C_tilde, G_tilde;
mu_0 = x_tilde_0)
x, y = simulate(lss_pi, T)
plot(1:T, y[1,:];label=L"y_t", size=(600,400))
plot!(1:T, y[2,:], label=L"c_t")
hline!([p.mu], color=:black, linestyle=:dash,
label = L"mean income, $\mu$")Permanent and Transitory Income
Classic model analyzes income with persistent and transitory shocks
Let \(x_{1t}\) be the permanent component and \(x_{2t}\) be the transitory component where \[ \begin{bmatrix} x_{1t+1}\\ x_{2t+1} \end{bmatrix} = \underbrace{\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}}_{A} \begin{bmatrix} x_{1t}\\ x_{2t} \end{bmatrix} + \underbrace{\begin{bmatrix} \sigma_1 & 0\\ 0 &\sigma_2 \end{bmatrix}}_{C} \underbrace{\begin{bmatrix}w_{1t+1}\\w_{2t+1} \end{bmatrix}}_{w_{t+1}} \]
And income itself is the sum of these two components
\[ y_t = \underbrace{\begin{bmatrix} 1 & 1 \end{bmatrix}}_{G}\begin{bmatrix} x_{1t}\\ x_{2t} \end{bmatrix} \]
Changes in Consumption
- Recall from previous slides: \(c_{t+1} - c_t = (1-\beta) G (I - \beta A)^{-1} C w_{t+1}\)
- For this LSS note that \(G(I-\beta A)^{-1} = \begin{bmatrix} \frac{1}{1-\beta} & 1\end{bmatrix}\). Substitute: \[
c_{t+1} - c_t = \sigma_1 w_{1t+1} + (1-\beta) \sigma_2 w_{2t+1}
\]
- Savings can show to follow: \[ F_{t+1} - F_t = \sigma w_{2t+1} \]
Interpretation
- After they see shock, the consumer:
- Consumes the entire permanent shock (i.e. MPC \(=1\))
- Consumers an annuitized fraction of transitory shock (i.e., MPC \(=1-\beta\))
- Given this, next periods savings:
- Do not change from the permanent shocks
- Add the transitory shock (after consuming the \(1-\beta\) fraction, and paid \(1/\beta\) gross interest)
Iteration and IRFs
function iterate_LSS(A, C, G, x_0, T; w = randn(size(C,2), T))
x = zeros(length(x_0), T + 1)
x[:, 1] = x_0
for t in 2:(T + 1)
x[:, t] = A * x[:, t - 1] + C * w[:, t - 1]
end
return x, G * x #x, y
end
function IRF(A, C, G, w_1, T)
h_x = zeros(size(A,1), T+1)
h_x[:,2] = C * w_1
for t in 3:T+1
h_x[:,t] = A * h_x[:,t-1]
end
return h_x, G * h_x #h_x, h_y
endLSS Formulation
sigma_1, sigma_2 = 0.15, 0.15
beta = 1.0/(1.0 + 0.05)
A = [1 0; 0 0]
G = [1 1]
C = [sigma_1 0; 0 sigma_2]
H = G*inv(I-beta*A)
A_tilde = [A zeros(2,1); H*(I-A) 1]
C_tilde = [C; zeros(1,2)]
G_tilde = [G 0; (1-beta)*H 1-beta]
x_tilde_0 = [1.0, 0.0, 0.0]#[x_10, x_20, F_0]
lss_pi = LSS(A_tilde, C_tilde, G_tilde;
mu_0 = x_tilde_0)
T = 20
x, y = simulate(lss_pi, T)
plot(1:T, y[1,:];label=L"y_t",size=(600,400))
plot!(1:T, y[2,:], label=L"c_t")