This lecture describes two types of consumption-smoothing and tax-smoothing models
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
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):
We’ll spend most of this lecture studying the finite-state Markov specification, but will briefly treat the linear state space specification before concluding
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
So these later lectures are partly about how the government should manipulate prices of government debt
Outcomes in consumption-smoothing (or tax-smoothing) models emerge from two sources:
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} $
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
$$ \mathbb E \left[ \sum_{t=0}^\infty \beta^t u(c_t) \right] \quad \text{where} \quad u(c_t) = - (c_t -\gamma)^2 \quad \text{and} \quad 0 < \beta < 1 \tag{1} $$
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
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,
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 $
The preceding calculations indicate that in the complete markets version of our model, we obtain the following striking results:
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 $
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
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
using InstantiateFromURL
github_project("QuantEcon/quantecon-notebooks-julia", version = "0.1.0")
using LinearAlgebra, Statistics, Compat
using Parameters, Plots, QuantEcon, Random
gr(fmt = :png);
ConsumptionProblem = @with_kw (β = 0.96,
y = [2.0, 1.5],
b0 = 3.0,
P = [0.8 0.2
0.4 0.6])
function consumption_complete(cp)
@unpack β, P, y, b0 = cp # Unpack
y1, y2 = y # extract income levels
b1 = b0 # b1 is known to be equal to b0
Q = β * P # assumed price system
# Using equation (7) calculate b2
b2 = (y2 - y1 - (Q[1, 1] - Q[2, 1] - 1) * b1) / (Q[1, 2] + 1 - Q[2, 2])
# Using equation (5) calculae c̄
c̄ = y1 - b0 + ([b1 b2] * Q[1, :])[1]
return c̄, b1, b2
end
function consumption_incomplete(cp; N_simul = 150)
@unpack β, P, y, b0 = cp # unpack
# for the simulation use the MarkovChain type
mc = MarkovChain(P)
# useful variables
y = y
v = inv(I - β * P) * y
# simulate state path
s_path = simulate(mc, N_simul, init=1)
# store consumption and debt path
b_path, c_path = ones(N_simul + 1), ones(N_simul)
b_path[1] = b0
# optimal decisions from (12) and (13)
db = ((1 - β) * v - y) / β
for (i, s) in enumerate(s_path)
c_path[i] = (1 - β) * (v[s, 1] - b_path[i])
b_path[i + 1] = b_path[i] + db[s, 1]
end
return c_path, b_path[1:end - 1], y[s_path], s_path
end
Let’s test by checking that $ \bar c $ and $ b_2 $ satisfy the budget constraint
cp = ConsumptionProblem()
c̄, b1, b2 = consumption_complete(cp)
debt_complete = [b1, b2]
isapprox((c̄ + b2 - cp.y[2] - debt_complete' * (cp.β * cp.P)[2, :])[1], 0)
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
$$ b_{t+1} - b_t = \beta^{-1} [ (1- \beta) v(i) - y(i) ] \tag{14} $$
In contrast to outcomes in the complete markets model, in the incomplete markets model
The code above also contains a function called consumption_incomplete() that uses (13) and (14) to
Let’s try this, using the same parameters in both complete and incomplete markets economies
Random.seed!(42)
N_simul = 150
cp = ConsumptionProblem()
c̄, b1, b2 = consumption_complete(cp)
debt_complete = [b1, b2]
c_path, debt_path, y_path, s_path = consumption_incomplete(cp, N_simul=N_simul)
plt_cons = plot(title = "Consumption paths", xlabel = "Periods", ylim = [1.4,2.1])
plot!(plt_cons, 1:N_simul, c_path, label = "incomplete market", lw = 2)
plot!(plt_cons, 1:N_simul, fill(c̄, N_simul), label = "complete market", lw = 2)
plot!(plt_cons, 1:N_simul, y_path, label = "income", lw = 2, alpha = 0.6, linestyle = :dash)
plot!(plt_cons, legend = :bottom)
plt_debt = plot(title = "Debt paths", xlabel = "Periods")
plot!(plt_debt, 1:N_simul, debt_path, label = "incomplete market")
plot!(plt_debt, 1:N_simul, debt_complete[s_path], label = "complete market", lw = 2)
plot!(plt_debt, 1:N_simul, y_path, label = "income", lw = 2, alpha = 0.6, linestyle = :dash)
plot!(plt_debt, legend = :bottomleft)
plot(plt_cons, plt_debt, layout = (1,2), size = (800, 400))
In the graph on the left, for the same sample path of nonfinancial income $ y_t $, notice that
We can simply relabel variables to acquire tax-smoothing interpretations of our two models
plt_tax = plot(title = "Tax collection paths", x_label = "Periods", ylim = [1.4,2.1])
plot!(plt_tax, 1:N_simul, c_path, label = "incomplete market", lw = 2)
plot!(plt_tax, 1:N_simul, fill(c̄, N_simul), label = "complete market", lw = 2)
plot!(plt_tax, 1:N_simul, y_path, label = "govt expenditures", alpha = .6, linestyle = :dash,
lw = 2)
plt_gov = plot(title = "Government assets paths", x_label = "Periods")
plot!(plt_gov, 1:N_simul, debt_path, label = "incomplete market", lw = 2)
plot!(plt_gov, 1:N_simul, debt_complete[s_path], label = "complete market", lw = 2)
plot!(plt_gov, 1:N_simul, y_path, label = "govt expenditures", alpha = .6, linestyle = :dash,
lw = 2)
hline!(plt_gov, [0], linestyle = :dash, color = :black, lw = 2, label = "")
plot(plt_tax, plt_gov, layout = (1,2), size = (800, 400))
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:
# Parameters
β = .96
y = [1.0, 2.0]
b0 = 0.0
P = [0.8 0.2;
0.4 0.6]
cp = ConsumptionProblem(β, y, b0, P)
Q = β * P
N_simul = 150
c̄, b1, b2 = consumption_complete(cp)
debt_complete = [b1, b2]
println("P = $P")
println("Q = $Q")
println("Govt expenditures in peace and war = $y")
println("Constant tax collections = $c̄")
println("Govt assets in two states = $debt_complete")
msg = """
Now let's check the government's budget constraint in peace and war.
Our assumptions imply that the government always purchases 0 units of the
Arrow peace security.
"""
println(msg)
AS1 = Q[1, 2] * b2
println("Spending on Arrow war security in peace = $AS1")
AS2 = Q[2, 2] * b2
println("Spending on Arrow war security in war = $AS2")
println("\n")
println("Government tax collections plus asset levels in peace and war")
TB1 = c̄ + b1
println("T+b in peace = $TB1")
TB2 = c̄ + b2
println("T+b in war = $TB2")
println("\n")
println("Total government spending in peace and war")
G1= y[1] + AS1
G2 = y[2] + AS2
println("total govt spending in peace = $G1")
println("total govt spending in war = $G2")
println("\n")
println("Let's see ex post and ex ante returns on Arrow securities")
Π = 1 ./ Q # reciprocal(Q)
exret = Π
println("Ex post returns to purchase of Arrow securities = $exret")
exant = Π .* P
println("Ex ante returns to purchase of Arrow securities = $exant")
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
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 $
Now we’ll use a setting like that in first lecture on the permanent income model
In that model, there were
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.
function complete_ss(β, b0, x0, A, C, S_y, T = 12)
# Create a linear state space for simulation purposes
# This adds "b" as a state to the linear state space system
# so that setting the seed places shocks in same place for
# both the complete and incomplete markets economy
# Atilde = vcat(hcat(A, zeros(size(A,1), 1)),
# zeros(1, size(A,2) + 1))
# Ctilde = vcat(C, zeros(1, 1))
# S_ytilde = hcat(S_y, zeros(1, 1))
lss = LSS(A, C, S_y, mu_0=x0)
# Add extra state to initial condition
# x0 = hcat(x0, 0)
# Compute the (I - β*A)^{-1}
rm = inv(I - β * A)
# Constant level of consumption
cbar = (1 - β) * (S_y * rm * x0 .- b0)
c_hist = ones(T) * cbar[1]
# Debt
x_hist, y_hist = simulate(lss, T)
b_hist = (S_y * rm * x_hist .- cbar[1] / (1.0 - β))
return c_hist, vec(b_hist), vec(y_hist), x_hist
end
N_simul = 150
# Define parameters
α, ρ1, ρ2 = 10.0, 0.9, 0.0
σ = 1.0
# N_simul = 1
# T = N_simul
A = [1.0 0.0 0.0;
α ρ1 ρ2;
0.0 1.0 0.0]
C = [0.0, σ, 0.0]
S_y = [1.0 1.0 0.0]
β, b0 = 0.95, -10.0
x0 = [1.0, α / (1 - ρ1), α / (1 - ρ1)]
# Do simulation for complete markets
out = complete_ss(β, b0, x0, A, C, S_y, 150)
c_hist_com, b_hist_com, y_hist_com, x_hist_com = out
# Consumption plots
plt_cons = plot(title = "Cons and income", xlabel = "Periods", ylim = [-5.0, 110])
plot!(plt_cons, 1:N_simul, c_hist_com, label = "consumption", lw = 2)
plot!(plt_cons, 1:N_simul, y_hist_com, label = "income",
lw = 2, alpha = 0.6, linestyle = :dash)
# Debt plots
plt_debt = plot(title = "Debt and income", xlabel = "Periods")
plot!(plt_debt, 1:N_simul, b_hist_com, label = "debt", lw = 2)
plot!(plt_debt, 1:N_simul, y_hist_com, label = "Income",
lw = 2, alpha = 0.6, linestyle = :dash)
hline!(plt_debt, [0], color = :black, linestyle = :dash, lw = 2, label = "")
plot(plt_cons, plt_debt, layout = (1,2), size = (800, 400))
plot!(legend = :bottomleft)
In the above graph, please note that:
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]
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