As stated in an earlier lecture, an asset is a claim on a stream of prospective payments
What is the correct price to pay for such a claim?
The elegant asset pricing model of Lucas [Luc78] attempts to answer this question in an equilibrium setting with risk averse agents
While we mentioned some consequences of Lucas’ model earlier, it is now time to work through the model more carefully, and try to understand where the fundamental asset pricing equation comes from
A side benefit of studying Lucas’ model is that it provides a beautiful illustration of model building in general and equilibrium pricing in competitive models in particular
Another difference to our first asset pricing lecture is that the state space and shock will be continous rather than discrete
Lucas studied a pure exchange economy with a representative consumer (or household), where
Pure exchange means that all endowments are exogenous
Representative consumer means that either
Either way, the assumption of a representative agent means that prices adjust to eradicate desires to trade
This makes it very easy to compute competitive equilibrium prices
Let’s review the set up
There is a single “productive unit” that costlessly generates a sequence of consumption goods $ \{y_t\}_{t=0}^{\infty} $
Another way to view $ \{y_t\}_{t=0}^{\infty} $ is as a consumption endowment for this economy
We will assume that this endowment is Markovian, following the exogenous process
$$ y_{t+1} = G(y_t, \xi_{t+1}) $$Here $ \{ \xi_t \} $ is an iid shock sequence with known distribution $ \phi $ and $ y_t \geq 0 $
An asset is a claim on all or part of this endowment stream
The consumption goods $ \{y_t\}_{t=0}^{\infty} $ are nonstorable, so holding assets is the only way to transfer wealth into the future
For the purposes of intuition, it’s common to think of the productive unit as a “tree” that produces fruit
Based on this idea, a “Lucas tree” is a claim on the consumption endowment
A representative consumer ranks consumption streams $ \{c_t\} $ according to the time separable utility functional
$$ \mathbb{E} \sum_{t=0}^\infty \beta^t u(c_t) \tag{1} $$
Here
What is an appropriate price for a claim on the consumption endowment?
We’ll price an ex dividend claim, meaning that
the seller retains this period’s dividend
the buyer pays $ p_t $ today to purchase a claim on
Since this is a competitive model, the first step is to pin down consumer behavior, taking prices as given
Next we’ll impose equilibrium constraints and try to back out prices
In the consumer problem, the consumer’s control variable is the share $ \pi_t $ of the claim held in each period
Thus, the consumer problem is to maximize (1) subject to
$$ c_t + \pi_{t+1} p_t \leq \pi_t y_t + \pi_t p_t $$along with $ c_t \geq 0 $ and $ 0 \leq \pi_t \leq 1 $ at each $ t $
The decision to hold share $ \pi_t $ is actually made at time $ t-1 $
But this value is inherited as a state variable at time $ t $, which explains the choice of subscript
We can write the consumer problem as a dynamic programming problem
Our first observation is that prices depend on current information, and current information is really just the endowment process up until the current period
In fact the endowment process is Markovian, so that the only relevant information is the current state $ y \in \mathbb R_+ $ (dropping the time subscript)
This leads us to guess an equilibrium where price is a function $ p $ of $ y $
Remarks on the solution method
Using the assumption that price is a given function $ p $ of $ y $, we write the value function and constraint as
$$ v(\pi, y) = \max_{c, \pi'} \left\{ u(c) + \beta \int v(\pi', G(y, z)) \phi(dz) \right\} $$subject to
$$ c + \pi' p(y) \leq \pi y + \pi p(y) \tag{2} $$
We can invoke the fact that utility is increasing to claim equality in (2) and hence eliminate the constraint, obtaining
$$ v(\pi, y) = \max_{\pi'} \left\{ u[\pi (y + p(y)) - \pi' p(y) ] + \beta \int v(\pi', G(y, z)) \phi(dz) \right\} \tag{3} $$
The solution to this dynamic programming problem is an optimal policy expressing either $ \pi' $ or $ c $ as a function of the state $ (\pi, y) $
What we need to do now is determine equilibrium prices
It seems that to obtain these, we will have to
However, as Lucas showed, there is a related but more straightforward way to do this
Since the consumption good is not storable, in equilibrium we must have $ c_t = y_t $ for all $ t $
In addition, since there is one representative consumer (alternatively, since all consumers are identical), there should be no trade in equilibrium
In particular, the representative consumer owns the whole tree in every period, so $ \pi_t = 1 $ for all $ t $
Prices must adjust to satisfy these two constraints
Now observe that the first order condition for (3) can be written as
$$ u'(c) p(y) = \beta \int v_1'(\pi', G(y, z)) \phi(dz) $$where $ v'_1 $ is the derivative of $ v $ with respect to its first argument
To obtain $ v'_1 $ we can simply differentiate the right hand side of (3) with respect to $ \pi $, yielding
$$ v'_1(\pi, y) = u'(c) (y + p(y)) $$Next we impose the equilibrium constraints while combining the last two equations to get
$$ p(y) = \beta \int \frac{u'[G(y, z)]}{u'(y)} [G(y, z) + p(G(y, z))] \phi(dz) \tag{4} $$
In sequential rather than functional notation, we can also write this as
$$ p_t = \mathbb{E}_t \left[ \beta \frac{u'(c_{t+1})}{u'(c_t)} ( y_{t+1} + p_{t+1} ) \right] \tag{5} $$
This is the famous consumption-based asset pricing equation
Before discussing it further we want to solve out for prices
Instead of solving for it directly we’ll follow Lucas’ indirect approach, first setting
$$ f(y) := u'(y) p(y) \tag{6} $$
so that (4) becomes
$$ f(y) = h(y) + \beta \int f[G(y, z)] \phi(dz) \tag{7} $$
Here $ h(y) := \beta \int u'[G(y, z)] G(y, z) \phi(dz) $ is a function that depends only on the primitives
Equation (7) is a functional equation in $ f $
The plan is to solve out for $ f $ and convert back to $ p $ via (6)
To solve (7) we’ll use a standard method: convert it to a fixed point problem
First we introduce the operator $ T $ mapping $ f $ into $ Tf $ as defined by
$$ (Tf)(y) = h(y) + \beta \int f[G(y, z)] \phi(dz) \tag{8} $$
The reason we do this is that a solution to (7) now corresponds to a function $ f^* $ satisfying $ (Tf^*)(y) = f^*(y) $ for all $ y $
In other words, a solution is a fixed point of $ T $
This means that we can use fixed point theory to obtain and compute the solution
Let $ cb\mathbb{R}_+ $ be the set of continuous bounded functions $ f \colon \mathbb{R}_+ \to \mathbb{R}_+ $
We now show that
uniformly to $ f^* $
(Note: If you find the mathematics heavy going you can take 1–2 as given and skip to the next section)
Recall the Banach contraction mapping theorem
It tells us that the previous statements will be true if we can find an $ \alpha < 1 $ such that
$$ \| Tf - Tg \| \leq \alpha \| f - g \|, \qquad \forall \, f, g \in cb\mathbb{R}_+ \tag{9} $$
Here $ \|h\| := \sup_{x \in \mathbb{R}_+} |h(x)| $
To see that (9) is valid, pick any $ f,g \in cb\mathbb{R}_+ $ and any $ y \in \mathbb{R}_+ $
Observe that, since integrals get larger when absolute values are moved to the inside,
$$ \begin{aligned} |Tf(y) - Tg(y)| & = \left| \beta \int f[G(y, z)] \phi(dz) - \beta \int g[G(y, z)] \phi(dz) \right| \\ & \leq \beta \int \left| f[G(y, z)] - g[G(y, z)] \right| \phi(dz) \\ & \leq \beta \int \| f - g \| \phi(dz) \\ & = \beta \| f - g \| \end{aligned} $$Since the right hand side is an upper bound, taking the sup over all $ y $ on the left hand side gives (9) with $ \alpha := \beta $
The preceding discussion tells that we can compute $ f^* $ by picking any arbitrary $ f \in cb\mathbb{R}_+ $ and then iterating with $ T $
The equilibrium price function $ p^* $ can then be recovered by $ p^*(y) = f^*(y) / u'(y) $
Let’s try this when $ \ln y_{t+1} = \alpha \ln y_t + \sigma \epsilon_{t+1} $ where $ \{\epsilon_t\} $ is iid and standard normal
Utility will take the isoelastic form $ u(c) = c^{1-\gamma}/(1-\gamma) $, where $ \gamma > 0 $ is the coefficient of relative risk aversion
Some code to implement the iterative computational procedure can be found below:
using InstantiateFromURL
github_project("QuantEcon/quantecon-notebooks-julia", version = "0.2.0")
using LinearAlgebra, Statistics, Compat
using Distributions, Interpolations, Parameters, Plots, QuantEcon, Random
gr(fmt = :png);
# model
function LucasTree(;γ = 2.0,
β = 0.95,
α = 0.9,
σ = 0.1,
grid_size = 100)
ϕ = LogNormal(0.0, σ)
shocks = rand(ϕ, 500)
# build a grid with mass around stationary distribution
ssd = σ / sqrt(1 - α^2)
grid_min, grid_max = exp(-4ssd), exp(4ssd)
grid = range(grid_min, grid_max, length = grid_size)
# set h(y) = β * int u'(G(y,z)) G(y,z) ϕ(dz)
h = similar(grid)
for (i, y) in enumerate(grid)
h[i] = β * mean((y^α .* shocks).^(1 - γ))
end
return (γ = γ, β = β, α = α, σ = σ, ϕ = ϕ, grid = grid, shocks = shocks, h = h)
end
# approximate Lucas operator, which returns the updated function Tf on the grid
function lucas_operator(lt, f)
# unpack input
@unpack grid, α, β, h = lt
z = lt.shocks
Af = LinearInterpolation(grid, f, extrapolation_bc=Line())
Tf = [ h[i] + β * mean(Af.(grid[i]^α .* z)) for i in 1:length(grid) ]
return Tf
end
# get equilibrium price for Lucas tree
function solve_lucas_model(lt;
tol = 1e-6,
max_iter = 500)
@unpack grid, γ = lt
i = 0
f = zero(grid) # Initial guess of f
error = tol + 1
while (error > tol) && (i < max_iter)
f_new = lucas_operator(lt, f)
error = maximum(abs, f_new - f)
f = f_new
i += 1
end
# p(y) = f(y) * y ^ γ
price = f .* grid.^γ
return price
end
An example of usage is given in the docstring and repeated here
Random.seed!(42) # For reproducible results.
tree = LucasTree(γ = 2.0, β = 0.95, α = 0.90, σ = 0.1)
price_vals = solve_lucas_model(tree);
Here’s the resulting price function
plot(tree.grid, price_vals, lw = 2, label = "p*(y)")
plot!(xlabel = "y", ylabel = "price", legend = :topleft)
The price is increasing, even if we remove all serial correlation from the endowment process
The reason is that a larger current endowment reduces current marginal utility
The price must therefore rise to induce the household to consume the entire endowment (and hence satisfy the resource constraint)
What happens with a more patient consumer?
Here the orange line corresponds to the previous parameters and the green line is price when $ \beta = 0.98 $
We see that when consumers are more patient the asset becomes more valuable, and the price of the Lucas tree shifts up
Exercise 1 asks you to replicate this figure
Replicate the figure to show how discount rates affect prices
plot()
for β in (.95, 0.98)
tree = LucasTree(;β = β)
grid = tree.grid
price_vals = solve_lucas_model(tree)
plot!(grid, price_vals, lw = 2, label = "beta = beta_var")
end
plot!(xlabel = "y", ylabel = "price", legend = :topleft)