Consumption and Tax Smoothing with Complete and Incomplete Markets¶

Contents¶

• Consumption and Tax Smoothing with Complete and Incomplete Markets
  • Overview
  • Background
  • Model 1 (Complete Markets)
  • Model 2 (One-Period Risk Free Debt Only)
  • Example: Tax Smoothing with Complete Markets
  • Linear State Space Version of Complete Markets Model

Overview¶

This lecture describes two types of consumption-smoothing and tax-smoothing models

• One is in the complete markets tradition of Lucas and Stokey [LS83].
• The other is in the incomplete markets tradition of Hall [Hal78] and Barro [Bar79].
• Complete markets* allow a consumer or government to buy or sell claims contingent on all possible states of the world.
• Incomplete markets* allow a consumer or government to buy or sell only a limited set of securities, often only a single risk-free security.

Hall [Hal78] and Barro [Bar79] both assumed that the only asset that can be traded is a risk-free one period bond.

Hall assumed an exogenous stochastic process of nonfinancial income and an exogenous gross interest rate on one period risk-free debt that equals $ \beta^{-1} $, where $ \beta \in (0,1) $ is also a consumer’s intertemporal discount factor.

Barro [Bar79] made an analogous assumption about the risk-free interest rate in a tax-smoothing model that we regard as isomorphic to Hall’s consumption-smoothing model.

We maintain Hall and Barro’s assumption about the interest rate when we describe an incomplete markets version of our model.

In addition, we extend their assumption about the interest rate to an appropriate counterpart that we use in a “complete markets” model in the style of Lucas and Stokey [LS83].

While we are equally interested in consumption-smoothing and tax-smoothing models, for the most part we focus explicitly on consumption-smoothing versions of these models.

But for each version of the consumption-smoothing model there is a natural tax-smoothing counterpart obtained simply by

• relabeling consumption as tax collections and nonfinancial income as government expenditures
• relabeling the consumer’s debt as the government’s assets

For elaborations on this theme, please see Optimal Savings II: LQ Techniques and later parts of this lecture.

We’ll consider two closely related alternative assumptions about the consumer’s exogenous nonfinancial income process (or in the tax-smoothing interpretation, the government’s exogenous expenditure process):

• that it obeys a finite $ N $ state Markov chain (setting $ N=2 $ most of the time)
• that it is described by a linear state space model with a continuous state vector in $ {\mathbb R}^n $ driven by a Gaussian vector iid shock process

We’ll spend most of this lecture studying the finite-state Markov specification, but will briefly treat the linear state space specification before concluding.

Relationship to Other Lectures¶

This lecture can be viewed as a followup to Optimal Savings II: LQ Techniques and a warm up for a model of tax smoothing described in opt_tax_recur.

Linear-quadratic versions of the Lucas-Stokey tax-smoothing model are described in lqramsey.

The key differences between those lectures and this one are

• Here the decision maker takes all prices as exogenous, meaning that his decisions do not affect them.
• In lqramsey and opt_tax_recur, the decision maker – the government in the case of these lectures – recognizes that his decisions affect prices.

So these later lectures are partly about how the government should manipulate prices of government debt.

Background¶

Outcomes in consumption-smoothing (or tax-smoothing) models emerge from two sources:

• a decision maker – a consumer in the consumption-smoothing model or a government in the tax-smoothing model – who wants to maximize an intertemporal objective function that expresses its preference for paths of consumption (or tax collections) that are smooth in the sense of not varying across time and Markov states
• a set of trading opportunities that allow the optimizer to transform a possibly erratic nonfinancial income (or government expenditure) process into a smoother consumption (or tax collections) process by purchasing or selling financial securities

In the complete markets version of the model, each period the consumer can buy or sell one-period ahead state-contingent securities whose payoffs depend on next period’s realization of the Markov state.

In the two-state Markov chain case, there are two such securities each period.

In an $ N $ state Markov state version of the model, $ N $ such securities are traded each period.

These state-contingent securities are commonly called Arrow securities, after Kenneth Arrow who first theorized about them.

In the incomplete markets version of the model, the consumer can buy and sell only one security each period, a risk-free bond with gross return $ \beta^{-1} $.

Finite State Markov Income Process¶

In each version of the consumption-smoothing model, nonfinancial income is governed by a two-state Markov chain (it’s easy to generalize this to an $ N $ state Markov chain).

In particular, the state of the world is given by $ s_t $ that follows a Markov chain with transition probability matrix

$$ P_{ij} = \mathbb P \{s_{t+1} = \bar s_j \,|\, s_t = \bar s_i \} $$

Nonfinancial income $ \{y_t\} $ obeys

$$ y_t = \begin{cases} \bar y_1 & \quad \text{if } s_t = \bar s_1 \\ \bar y_2 & \quad \text{if } s_t = \bar s_2 \end{cases} $$

A consumer wishes to maximize

$$ \mathbb E \left[ \sum_{t=0}^\infty \beta^t u(c_t) \right] \quad \text{where} \quad u(c_t) = - (c_t -\gamma)^2 \quad \text{and} \quad 0 < \beta < 1 \tag{1} $$

Remark About Isomorphism¶

We can regard these as Barro [Bar79] tax-smoothing models if we set $ c_t = T_t $ and $ G_t = y_t $, where $ T_t $ is total tax collections and $ \{G_t\} $ is an exogenous government expenditures process.

Market Structure¶

The two models differ in how effectively the market structure allows the consumer to transfer resources across time and Markov states, there being more transfer opportunities in the complete markets setting than in the incomplete markets setting.

Watch how these differences in opportunities affect

• how smooth consumption is across time and Markov states
• how the consumer chooses to make his levels of indebtedness behave over time and across Markov states

Model 1 (Complete Markets)¶

At each date $ t \geq 0 $, the consumer trades one-period ahead Arrow securities.

We assume that prices of these securities are exogenous to the consumer (or in the tax-smoothing version of the model, to the government).

Exogenous means that they are unaffected by the decision maker.

In Markov state $ s_t $ at time $ t $, one unit of consumption in state $ s_{t+1} $ at time $ t+1 $ costs $ q(s_{t+1} \,|\, s_t) $ units of the time $ t $ consumption good.

At time $ t=0 $, the consumer starts with an inherited level of debt due at time $ 0 $ of $ b_0 $ units of time $ 0 $ consumption goods.

The consumer’s budget constraint at $ t \geq 0 $ in Markov state $ s_t $ is

$$ c_t + b_t \leq y(s_t) + \sum_j q(\bar s_j \,|\, s_t ) \, b_{t+1}(\bar s_j \,|\, s_t) $$

where $ b_t $ is the consumer’s one-period debt that falls due at time $ t $ and $ b_{t+1}(\bar s_j\,|\, s_t) $ are the consumer’s time $ t $ sales of the time $ t+1 $ consumption good in Markov state $ \bar s_j $, a source of time $ t $ revenues.

An analogue of Hall’s assumption that the one-period risk-free gross interest rate is $ \beta^{-1} $ is

$$ q(\bar s_j \,|\, \bar s_i) = \beta P_{ij} \tag{2} $$

To understand this, observe that in state $ \bar s_i $ it costs $ \sum_j q(\bar s_j \,|\, \bar s_i) $ to purchase one unit of consumption next period for sure, i.e., meaning no matter what state of the world occurs at $ t+1 $.

Hence the implied price of a risk-free claim on one unit of consumption next period is

$$ \sum_j q(\bar s_j \,|\, \bar s_i) = \sum_j \beta P_{ij} = \beta $$

This confirms that (2) is a natural analogue of Hall’s assumption about the risk-free one-period interest rate.

First-order necessary conditions for maximizing the consumer’s expected utility are

$$ \beta \frac{u'(c_{t+1})}{u'(c_t) } \mathbb P\{s_{t+1}\,|\, s_t \} = q(s_{t+1} \,|\, s_t) $$

or, under our assumption (2) on Arrow security prices,

$$ c_{t+1} = c_t \tag{3} $$

Thus, our consumer sets $ c_t = \bar c $ for all $ t \geq 0 $ for some value $ \bar c $ that it is our job now to determine.

Guess: We’ll make the plausible guess that

$$ b_{t+1}(\bar s_j \,|\, s_t = \bar s_i) = b(\bar s_j) , \quad i=1,2; \;\; j= 1,2 \tag{4} $$

so that the amount borrowed today turns out to depend only on tomorrow’s Markov state. (Why is this is a plausible guess?).

To determine $ \bar c $, we shall pursue the implications of the consumer’s budget constraints in each Markov state today and our guess (4) about the consumer’s debt level choices.

For $ t \geq 1 $, these imply

$$ \begin{aligned} \bar c + b(\bar s_1) & = y(\bar s_1) + q(\bar s_1\,|\, \bar s_1) b(\bar s_1) + q(\bar s_2 \,|\, \bar s_1) b(\bar s_2) \cr \bar c + b(\bar s_2) & = y(\bar s_2) + q(\bar s_1\,|\, \bar s_2) b(\bar s_1) + q(\bar s_2 \,|\, \bar s_2) b(\bar s_2), \end{aligned} \tag{5} $$

or

$$ \begin{bmatrix} b(\bar s_1) \cr b(\bar s_2) \end{bmatrix} + \begin{bmatrix} \bar c \cr \bar c \end{bmatrix} = \begin{bmatrix} y(\bar s_1) \cr y(\bar s_2) \end{bmatrix} + \beta \begin{bmatrix} P_{11} & P_{12} \cr P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} b(\bar s_1) \cr b(\bar s_2) \end{bmatrix} $$

These are $ 2 $ equations in the $ 3 $ unknowns $ \bar c, b(\bar s_1), b(\bar s_2) $.

To get a third equation, we assume that at time $ t=0 $, $ b_0 $ is the debt due; and we assume that at time $ t=0 $, the Markov state is $ \bar s_1 $.

Then the budget constraint at time $ t=0 $ is

$$ \bar c + b_0 = y(\bar s_1) + q(\bar s_1 \,|\, \bar s_1) b(\bar s_1) + q(\bar s_2\,|\,\bar s_1) b(\bar s_2) \tag{6} $$

If we substitute (6) into the first equation of (5) and rearrange, we discover that

$$ b(\bar s_1) = b_0 \tag{7} $$

We can then use the second equation of (5) to deduce the restriction

$$ y(\bar s_1) - y(\bar s_2) + [q(\bar s_1\,|\, \bar s_1) - q(\bar s_1\,|\, \bar s_2) - 1 ] b_0 + [q(\bar s_2\,|\,\bar s_1) + 1 - q(\bar s_2 \,|\, \bar s_2) ] b(\bar s_2) = 0 , \tag{8} $$

an equation in the unknown $ b(\bar s_2) $.

Knowing $ b(\bar s_1) $ and $ b(\bar s_2) $, we can solve equation (6) for the constant level of consumption $ \bar c $.

Key outcomes¶

The preceding calculations indicate that in the complete markets version of our model, we obtain the following striking results:

• The consumer chooses to make consumption perfectly constant across time and Markov states.

We computed the constant level of consumption $ \bar c $ and indicated how that level depends on the underlying specifications of preferences, Arrow securities prices, the stochastic process of exogenous nonfinancial income, and the initial debt level $ b_0 $

• The consumer’s debt neither accumulates, nor decumulates, nor drifts. Instead the debt level each period is an exact function of the Markov state, so in the two-state Markov case, it switches between two values.
• We have verified guess (4).

We computed how one of those debt levels depends entirely on initial debt – it equals it – and how the other value depends on virtually all remaining parameters of the model.

Code¶

Here’s some code that, among other things, contains a function called consumption_complete().

This function computes $ b(\bar s_1), b(\bar s_2), \bar c $ as outcomes given a set of parameters, under the assumption of complete markets.

Setup¶

Let’s test by checking that $ \bar c $ and $ b_2 $ satisfy the budget constraint

Below, we’ll take the outcomes produced by this code – in particular the implied consumption and debt paths – and compare them with outcomes from an incomplete markets model in the spirit of Hall [Hal78] and Barro [Bar79] (and also, for those who love history, Gallatin (1807) [Gal37]).

Model 2 (One-Period Risk Free Debt Only)¶

This is a version of the original models of Hall (1978) and Barro (1979) in which the decision maker’s ability to substitute intertemporally is constrained by his ability to buy or sell only one security, a risk-free one-period bond bearing a constant gross interest rate that equals $ \beta^{-1} $.

Given an initial debt $ b_0 $ at time $ 0 $, the consumer faces a sequence of budget constraints

$$ c_t + b_t = y_t + \beta b_{t+1}, \quad t \geq 0 $$

where $ \beta $ is the price at time $ t $ of a risk-free claim on one unit of time consumption at time $ t+1 $.

First-order conditions for the consumer’s problem are

$$ \sum_{j} u'(c_{t+1,j}) P_{ij} = u'(c_{t, i}) $$

For our assumed quadratic utility function this implies

$$ \sum_j c_{t+1,j} P_{ij} = c_{t,i} , \tag{9} $$

which is Hall’s (1978) conclusion that consumption follows a random walk.

As we saw in our first lecture on the permanent income model, this leads to

$$ b_t = \mathbb E_t \sum_{j=0}^\infty \beta^j y_{t+j} - (1 -\beta)^{-1} c_t \tag{10} $$

and

$$ c_t = (1-\beta) \left[ \mathbb E_t \sum_{j=0}^\infty \beta^j y_{t+j} - b_t \right] . \tag{11} $$

Equation (11) expresses $ c_t $ as a net interest rate factor $ 1 - \beta $ times the sum of the expected present value of nonfinancial income $ \mathbb E_t \sum_{j=0}^\infty \beta^j y_{t+j} $ and financial wealth $ -b_t $.

Substituting (11) into the one-period budget constraint and rearranging leads to

$$ b_{t+1} - b_t = \beta^{-1} \left[ (1-\beta) \mathbb E_t \sum_{j=0}^\infty\beta^j y_{t+j} - y_t \right] \tag{12} $$

Now let’s do a useful calculation that will yield a convenient expression for the key term $ \mathbb E_t \sum_{j=0}^\infty\beta^j y_{t+j} $ in our finite Markov chain setting.

Define

$$ v_t := \mathbb E_t \sum_{j=0}^\infty \beta^j y_{t+j} $$

In our finite Markov chain setting, $ v_t = v(1) $ when $ s_t= \bar s_1 $ and $ v_t = v(2) $ when $ s_t=\bar s_2 $.

Therefore, we can write

$$ \begin{aligned} v(1) & = y(1) + \beta P_{11} v(1) + \beta P_{12} v(2) \\ v(2) & = y(2) + \beta P_{21} v(1) + \beta P_{22} v(2) \end{aligned} $$

or

$$ \vec v = \vec y + \beta P \vec v $$

where $ \vec v = \begin{bmatrix} v(1) \cr v(2) \end{bmatrix} $ and $ \vec y = \begin{bmatrix} y(1) \cr y(2) \end{bmatrix} $.

We can also write the last expression as

$$ \vec v = (I - \beta P)^{-1} \vec y $$

In our finite Markov chain setting, from expression (11), consumption at date $ t $ when debt is $ b_t $ and the Markov state today is $ s_t = i $ is evidently

$$ c(b_t, i) = (1 - \beta) \left( [(I - \beta P)^{-1} \vec y]_i - b_t \right) \tag{13} $$

and the increment in debt is

$$ b_{t+1} - b_t = \beta^{-1} [ (1- \beta) v(i) - y(i) ] \tag{14} $$

Summary of Outcomes¶

In contrast to outcomes in the complete markets model, in the incomplete markets model

• consumption drifts over time as a random walk; the level of consumption at time $ t $ depends on the level of debt that the consumer brings into the period as well as the expected discounted present value of nonfinancial income at $ t $
• the consumer’s debt drifts upward over time in response to low realizations of nonfinancial income and drifts downward over time in response to high realizations of nonfinancial income
• the drift over time in the consumer’s debt and the dependence of current consumption on today’s debt level account for the drift over time in consumption

The Incomplete Markets Model¶

The code above also contains a function called consumption_incomplete() that uses (13) and (14) to

• simulate paths of $ y_t, c_t, b_{t+1} $
• plot these against values of of $ \bar c, b(s_1), b(s_2) $ found in a corresponding complete markets economy

Let’s try this, using the same parameters in both complete and incomplete markets economies

In the graph on the left, for the same sample path of nonfinancial income $ y_t $, notice that

• consumption is constant when there are complete markets, but it takes a random walk in the incomplete markets version of the model
• the consumer’s debt oscillates between two values that are functions of the Markov state in the complete markets model, while the consumer’s debt drifts in a “unit root” fashion in the incomplete markets economy

Using the Isomorphism¶

We can simply relabel variables to acquire tax-smoothing interpretations of our two models

Example: Tax Smoothing with Complete Markets¶

It is useful to focus on a simple tax-smoothing example with complete markets.

This example will illustrate how, in a complete markets model like that of Lucas and Stokey [LS83], the government purchases insurance from the private sector.

• Purchasing insurance protects the government against the need to raise taxes too high or issue too much debt in the high government expenditure event.

We assume that government expenditures move between two values $ G_1 < G_2 $, where Markov state $ 1 $ means “peace” and Markov state $ 2 $ means “war”.

The government budget constraint in Markov state $ i $ is

$$ T_i + b_i = G_i + \sum_j Q_{ij} b_j $$

where

$$ Q_{ij} = \beta P_{ij} $$

is the price of one unit of output next period in state $ j $ when today’s Markov state is $ i $ and $ b_i $ is the government’s level of assets in Markov state $ i $.

That is, $ b_i $ is the amount of the one-period loans owned by the government that fall due at time $ t $.

As above, we’ll assume that the initial Markov state is state $ 1 $.

In addition, to simplify our example, we’ll set the government’s initial asset level to $ 0 $, so that $ b_1 =0 $.

Here’s our code to compute a quantitative example with zero debt in peace time:

Explanation¶

In this example, the government always purchase $ 0 $ units of the Arrow security that pays off in peace time (Markov state $ 1 $).

But it purchases a positive amount of the security that pays off in war time (Markov state $ 2 $).

We recommend plugging the quantities computed above into the government budget constraints in the two Markov states and staring.

This is an example in which the government purchases insurance against the possibility that war breaks out or continues

• the insurance does not pay off so long as peace continues
• the insurance pays off when there is war

Exercise: try changing the Markov transition matrix so that

$$ P = \begin{bmatrix} 1 & 0 \\ .2 & .8 \end{bmatrix} $$

Also, start the system in Markov state $ 2 $ (war) with initial government assets $ - 10 $, so that the government starts the war in debt and $ b_2 = -10 $.

Linear State Space Version of Complete Markets Model¶

Now we’ll use a setting like that in first lecture on the permanent income model.

In that model, there were

• incomplete markets: the consumer could trade only a single risk-free one-period bond bearing gross one-period risk-free interest rate equal to $ \beta^{-1} $
• the consumer’s exogenous nonfinancial income was governed by a linear state space model driven by Gaussian shocks, the kind of model studied in an earlier lecture about linear state space models

We’ll write down a complete markets counterpart of that model.

So now we’ll suppose that nonfinancial income is governed by the state space system

$$ \begin{aligned} x_{t+1} & = A x_t + C w_{t+1} \cr y_t & = S_y x_t \end{aligned} $$

where $ x_t $ is an $ n \times 1 $ vector and $ w_{t+1} \sim {\cal N}(0,I) $ is IID over time.

Again, as a counterpart of the Hall-Barro assumption that the risk-free gross interest rate is $ \beta^{-1} $, we assume the scaled prices of one-period ahead Arrow securities are

$$ p_{t+1}(x_{t+1} \,|\, x_t) = \beta \phi(x_{t+1} \,|\, A x_t, CC') \tag{15} $$

where $ \phi(\cdot \,|\, \mu, \Sigma) $ is a multivariate Gaussian distribution with mean vector $ \mu $ and covariance matrix $ \Sigma $.

Let $ b(x_{t+1}) $ be a vector of state-contingent debt due at $ t+1 $ as a function of the $ t+1 $ state $ x_{t+1} $.

Using the pricing function assumed in (15), the value at $ t $ of $ b(x_{t+1}) $ is

$$ \beta \int b(x_{t+1}) \phi(x_{t+1} \,|\, A x_t, CC') d x_{t+1} = \beta \mathbb E_t b_{t+1} $$

In the complete markets setting, the consumer faces a sequence of budget constraints

$$ c_t + b_t = y_t + \beta \mathbb E_t b_{t+1}, t \geq 0 $$

We can solve the time $ t $ budget constraint forward to obtain

$$ b_t = \mathbb E_t \sum_{j=0}^\infty \beta^j (y_{t+j} - c_{t+j} ) $$

We assume as before that the consumer cares about the expected value of

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

In the incomplete markets version of the model, we assumed that $ u(c_t) = - (c_t -\gamma)^2 $, so that the above utility functional became

$$ -\sum_{t=0}^\infty \beta^t ( c_t - \gamma)^2, \quad 0 < \beta < 1 $$

But in the complete markets version, we can assume a more general form of utility function that satisfies $ u' > 0 $ and $ u'' < 0 $.

The first-order condition for the consumer’s problem with complete markets and our assumption about Arrow securities prices is

$$ u'(c_{t+1}) = u'(c_t) \quad \text{for all } t\geq 0 $$

which again implies $ c_t = \bar c $ for some $ \bar c $.

So it follows that

$$ b_t = \mathbb E_t \sum_{j=0}^\infty \beta^j (y_{t+j} - \bar c) $$

or

$$ b_t = S_y (I - \beta A)^{-1} x_t - \frac{1}{1-\beta} \bar c \tag{16} $$

where the value of $ \bar c $ satisfies

$$ \bar b_0 = S_y (I - \beta A)^{-1} x_0 - \frac{1}{1 - \beta } \bar c \tag{17} $$

where $ \bar b_0 $ is an initial level of the consumer’s debt, specified as a parameter of the problem.

Thus, in the complete markets version of the consumption-smoothing model, $ c_t = \bar c, \forall t \geq 0 $ is determined by (17) and the consumer’s debt is a fixed function of the state $ x_t $ described by (16).

Here’s an example that shows how in this setting the availability of insurance against fluctuating nonfinancial income allows the consumer completely to smooth consumption across time and across states of the world.

Interpretation of Graph¶

In the above graph, please note that:

• nonfinancial income fluctuates in a stationary manner
• consumption is completely constant
• the consumer’s debt fluctuates in a stationary manner; in fact, in this case because nonfinancial income is a first-order autoregressive process, the consumer’s debt is an exact affine function (meaning linear plus a constant) of the consumer’s nonfinancial income

Incomplete Markets Version¶

The incomplete markets version of the model with nonfinancial income being governed by a linear state space system is described in the first lecture on the permanent income model and the followup lecture on the permanent income model.

In that version, consumption follows a random walk and the consumer’s debt follows a process with a unit root.

We leave it to the reader to apply the usual isomorphism to deduce the corresponding implications for a tax-smoothing model like Barro’s [Bar79].

Government Manipulation of Arrow Securities Prices¶

In optimal taxation in an LQ economy and recursive optimal taxation, we study complete-markets models in which the government recognizes that it can manipulate Arrow securities prices.

In optimal taxation with incomplete markets, we study an incomplete-markets model in which the government manipulates asset prices.