15. Lucas Tree Model (1978, Econometrica)

The basic idea of asset pricing is that we want to establish a general equilibrium model that is as simple as possible to discuss the prices of assets. This model bridges the finance literature and macro literature in the way that it introduces general equilibrium and dynamic optimization framework to asset pricing topics.

15.1 Set-up

To make the model simplest, we want to have: - only one consumption good: fruit. - no production (i.e. endowment economy). - fruit is the numeraire. - we don't have money in this economy because money pays no dividends and thus has zero value, which means no one wants to hold it. - trees are the assets. - each tree gives some amount of fruit as dividends (the dividends are endowments, not production) in each period. Dividends are a function of state variable. - fruits cannot be stored to next period. - a continuum of identical households, so we only consider a representative agent. The representative agent's portfolio of each asset is characterized by vector \(\mathbf z=(z_1,z_2,\dots,z_N)\), and he gets one unit of each asset (each type of trees) at beginning as endowment, i.e. \(\mathbf z=(1,1,\dots,1)\). The agent consumes the dividends (fruits) of their assets (trees) in each period. - we give the agent some latitude to trade, so the allowable portfolios fall into the range \([1-\varepsilon,1+\varepsilon]^N\) with \(\varepsilon>0\). - agents have time invariant preferences, each agent has his preferences represented by \(u(c)\), where \(u\) is continuous, strictly increasing, strictly concave and differentiable, and the utility discounting parameter \(\beta\in(0,1)\). - shocks: \(J\) states in total, indexed by \(j=1,2,\dots,J\). Shocks are a first-order Markov process with transition matrix \(Q=[q_{jk}]_{J\times J}\) where \(q_{jk}\) is the probability of moving from state \(j\) to state \(k\) for \(\forall j,k\in\{1,\dots,J\}\). - assets: \(N\) types of trees, indexed by \(n=1,2,\dots,N\). The dividend matrix \(Y=[y_{jn}]_{J\times N}\) where \(y_{jn}\ge0\) is the dividends of asset \(n\) in state \(j\); the \(n\)th column is the dividends of asset \(n\) in all states and the \(j\)th row is the dividends of all assets in state \(j\). - asset prices: \(P=[p_{jn}]_{J\times N}\) where \(p_{jn}>0\) is the price of asset \(n\) in state \(j\).

• Assumptions on \(Y\):
• in every state \(j\), at least one asset has positive dividends: \(\sum_{n=1}^N y_{jn}>0\) for \(\forall j=1,\dots,J\).
• every asset has positive dividends at least in one state: \(\sum_{j=1}^J y_{jn}>0\) for \(\forall n=1,\dots,N\).
• Assumptions on \(Q\): for simplicity, the process has only one ergodic set and no transient states (cycles are allowed). Therefore, all states have strictly positive possibility to be reached in the future and prices for all assets are thus strictly positive.
• Assumptions on the economy: a perfectly competitive endowment economy, no other income, no taxes, no externality, etc. Therefore, we can solve the social planner's problem to achieve the competitive equilibrium.
• in each period, current consumption of goods are traded for the shares of assets.
• timing: assume assets are traded after the current dividends are paid.

15.3 The Only Possible Competitive Equilibrium: No Trading

Note that the only general equilibrium would be no trading at all. The logic is that since we have identical agents, if anyone wants to trade, then everyone else wants to do that same trade, thus there is no way for the market to clear.

So, the goal is to find the prices of assets (as a function of states) such that every agent (the representative agent) wants to hold the original portfolio. Therefore, we will proceed by taking the following steps: - Look at the representative household's maximization problem at any given price (the representative household is a price taker). - get the f.o.c. of his problem. - impose the market clearing condition (no trading condition).

15.4 Representative Agent's Maximization Problem

15.4.1 The state variable

The state variable of the agent's problem is \(S=(\mathbf z,j)\) where \(\mathbf z=(z_1,z_2,\dots,z_N)\) is the vector of assets and \(j\) is the state of shocks. The state space for \(j\) is \(\{1,\dots,J\}\), and the state space for \(\mathbf z\) is \(\mathbf Z=[1-\varepsilon,1+\varepsilon]^N\) with \(0<\varepsilon\le1\). Since we know that in equilibrium there is no trade, the state space for \(\mathbf z\) is not important as long as it has some latitude for trade (\(\varepsilon>0\)) and the space is compact (closed and bounded).

15.4.2 Bellman equation

Denote the current portfolio by vector \(\mathbf z=(z_1,\dots,z_N)\) and the next period portfolio by vector \(\mathbf x=(x_1,\dots,x_N)\). The Bellman equation for the consumer's value function at any given price vector \(\mathbf p\) is $$V^p(\mathbf z,j)=\max_{\mathbf x\in\Gamma(\mathbf z,j)}\left\{u\left(\sum_{n=1}^N z_n y_{jn}-\sum_{n=1}^N p_{jn}(x_n-z_n)\right)+\beta\sum_{k=1}^J q_{jk}V^p(\mathbf x,k)\right\},\ \forall j$$ where \(\Gamma(\mathbf z,j)=\left\{\mathbf x\in\mathbf Z:\sum_{n=1}^N p_{jn}(x_n-z_n)\le\sum_{n=1}^N z_n y_{jn}\right\}\). Note that \(\sum_n p_{jn}(x_n-z_n)\) is the net asset purchases and \(\sum_n z_n y_{jn}\) is dividends income.

15.4.3 First-order condition and Envelop condition

The f.o.c. of \(x_n\) is $$u'\left(\sum_{n=1}^N z_n y_{jn}-\sum_{n=1}^N p_{jn}(x_n-z_n)\right)p_{jn}=\beta\sum_{k=1}^J q_{jk}V^p_{x_n}(\mathbf x,k),\ \forall n,j\tag{15.1}$$ The Envelop condition is $$V^p_{z_n}(\mathbf z,j)=u'\left(\sum_{n=1}^N z_n y_{jn}-\sum_{n=1}^N p_{jn}(x^\star_n-z_n)\right)(y_{jn}+p_{jn}),\ \forall n,j\tag{15.2}$$

If there is a competitive equilibrium for this economy, then in order to have the market clearing condition, it must be that there is no trade in equilibrium, i.e. \(\mathbf x=\mathbf z\). We first assume such equilibrium exists to explicitly write down the equation that characterizes the competitive equilibrium, then in the next section we will show the existence and uniqueness of the equilibrium price vector \(\mathbf p^e\) such that the only possible no trading equilibrium is chosen by the agent as a utility maximization solution.

Denote: - the equilibrium asset price matrix by \(\mathbf p^e\). - the value function by \(V^e\), the optimal policy correspondence \(G^e(\mathbf z,j)\) s.t. \(V^e\). \(G^e(\mathbf z,j)\) solves the consumer's problem at given \(\mathbf p^e\). The only possible equilibrium is no trading, which means \(\mathbf 1=(1,1,\dots,1)\in G^e(\mathbf 1,j)\) for \(\forall j\). - consumption in state \(j\) in equilibrium by \(c^e_j\). No trading implies that \(c^e_j=\sum_{n=1}^N y_{jn}\), \(\forall j\).

Combine the f.o.c. in (15.1) and the Envelop condition in (15.2) both evaluated at \(\mathbf z=\mathbf x=\mathbf 1\) and \(\mathbf p=\mathbf p^e\) to get $$u'(c^e_j)p^e_{jn}=\beta\sum_{k=1}^J q_{jk}u'(c^e_k)(y_{kn}+p^e_{kn}),\ \forall n,j\tag{15.3}$$ Equation (15.3) characterizes the only equilibrium. Clearly, equilibrium price \(p^e_{jn}\) (\(\forall n,j\)) depends on asset \(n\)'s dividends and price in each state and on other assets through \(\{c^e_j\}\).

15.5.1 The existence, uniqueness and strict positivity of the equilibrium price vector \(\mathbf p^e\)

Write (15.4) for \(\forall n\) in matrix form by stacking the utility costs of asset \(n\) in all states into a vector: \(\boldsymbol\psi_n=\mathbf h_n+\beta Q\boldsymbol\psi_n\) (15.5), where \(\boldsymbol\psi_n=(\psi_{1n},\dots,\psi_{Jn})'\), \(\mathbf h_n=(h_{1n},\dots,h_{Jn})'\), \(Q=[q_{jk}]\).

The only unknown in (15.5) is \(\boldsymbol\psi_n\). Define the operator \(T(\boldsymbol\psi_n)=\mathbf h_n+\beta Q\boldsymbol\psi_n\). First, show \(\boldsymbol\psi_n\in\Psi\) where \(\Psi\) is a compact space. Given finite endowment and dividends of each asset, it is reasonable to assume \(\forall j\), \(p^e_{jn}\in[0,\bar p_j]\) for some \(\bar p_j>0\). Since \(a_j\) is a known number, we have \(\psi_{jn}=a_j p^e_{jn}\in[0,a_j\bar p_j]\). Therefore \(\Psi=\prod_{j=1}^J[0,a_j\bar p_j]\) is a compact space. For \(\boldsymbol\psi^a_n,\boldsymbol\psi^b_n\in\Psi\), consider \(d^e(\boldsymbol\psi^a_n,\boldsymbol\psi^b_n)=\max\{|\psi^a_{1n}-\psi^b_{1n}|,\dots,|\psi^a_{Jn}-\psi^b_{Jn}|\}\); we can easily verify \(d^e\) satisfies the four criteria in Definition 40.1 (non-negativity, identity of indiscernibles, symmetry, and triangular inequality). Therefore \(d^e\) is a metric on space \(\Psi\). By Theorem 40.1, \((\Psi,d^e)\) is a complete metric space.

Now show \(T\) is a contraction mapping.

Then \(d^e(T(\boldsymbol\psi^a_n),T(\boldsymbol\psi^b_n))=d^e(\beta Q\boldsymbol\psi^a_n,\beta Q\boldsymbol\psi^b_n)=\beta d^e(Q\boldsymbol\psi^a_n,Q\boldsymbol\psi^b_n)\le\beta d^e(\boldsymbol\psi^a_n,\boldsymbol\psi^b_n)\), i.e. \(T\) is a contraction mapping. By the Contraction Mapping Theorem (Theorem 41.1), there exists a unique fixed point \(\boldsymbol\psi^e_n\) such that \(T(\boldsymbol\psi^e_n)=\boldsymbol\psi^e_n\). Since each component \(\psi_{jn}\) is well-defined and unique and \(a_j=u'(c^e_j)>0\), we can uniquely pin down \(p^e_{jn}=\psi_{jn}/a_j\), \(\forall j\). Repeat for all assets \(n=1,\dots,N\) to uniquely pin down \(p^e_{jn}\) (\(\forall j,n\)). So we have shown the existence and uniqueness of \(\mathbf p^e\). \(\blacksquare\)

15.5.2 Factors determining the equilibrium price matrix

The equilibrium prices are characterized by \(a_j p^e_{jn}=\beta\sum_{k=1}^J q_{jk}a_k(y_{kn}+p^e_{kn})\) (\(\forall n,j\)). Write it in matrix form (\(A=\text{diag}(a_1,\dots,a_J)\)): $$A\mathbf p^e=\beta QA(Y+\mathbf p^e)\tag{15.6}$$ $$\Rightarrow A\mathbf p^e=\beta QAY+\beta QA\mathbf p^e\Rightarrow(I-\beta Q)A\mathbf p^e=\beta QAY\Rightarrow\mathbf p^e=\beta A^{-1}(I-\beta Q)^{-1}QAY\tag{15.7}$$ From (15.7), we can conclude that \(\mathbf p^e\) is affected by: - \(A^{-1}\) and \(A\): the marginal utility of each state. - \(Y\): the dividends of assets. - \(Q\): the variation on rows of \(Q\). - \(AY\): the interaction (covariance) between marginal utilities and dividends.

15.5.3 Linearly dependent assets

15.5.4 Spanning the state space

Consider a state contingent claim \(\mathbf e_j=(0,\dots,0,\underset{j\text{th}}{1},0,\dots,0)'\). The question is whether we can construct such a portfolio using our current assets \(1,\dots,N\).

If the current assets span the state space, then any new asset with certain payoff patterns can always be constructed by the set of state contingent claims \((\mathbf e_1,\mathbf e_2,\dots,\mathbf e_J)\), in which \(\mathbf e_j\)'s price in different states is the elements in the vector \(\mathbf p^e\Lambda_j\).

15.6.1 \(N=J=1\)

Suppose an economy has \(N=J=1\), then in (15.7) every item is scalar and \(I=Q=1\), \(Y=y\), so $$p^e=\frac{\beta}{1-\beta}y$$ where the price \(p^e\) is simply a sum of discounted deterministic dividends.

15.6.2 \(N=1\), \(J>1\) with i.i.d. dividends

Suppose an economy has \(N=1\), \(J>1\) and dividends are i.i.d. (i.i.d. dividends simply means \(q_{jk}=q_k\), i.e. \(Q\) has same rows). Then in (15.6) \(A\mathbf p^e=\beta QA(Y+\mathbf p^e)\), \(Y\) and \(\mathbf p^e\) are \(J\times1\) column vectors and \(Q\) has same rows, so each component \(a_j p^e_j=\beta\sum_k q_k a_k(y_k+p^e_k)\) is the same for all \(j\). So the RHS is a column vector with same rows, which means the LHS also has same rows. Since \(a_j=u'(c^e_j)=u'(y_j)\), we can conclude: High \(y_j\) ⟹ Low \(a_j\) ⟹ High \(p^e_j\).

15.6.3 Large \(N\), large \(J\) with i.i.d. assets and i.i.d. dividends

Suppose an economy has large \(N\), large \(J\), i.i.d. assets and i.i.d. dividends. Then in (15.6) \(A\mathbf p^e=\beta QA(Y+\mathbf p^e)\), \(QA\) has same rows, which implies the RHS also has same rows. So since the diagonal matrix \(A\) has same diagonal elements (large \(N\) ⟹ aggregate dividends constant across states), \(\mathbf p^e\) must also have same rows, i.e. price doesn't vary across states. Also note that all assets have exactly the same payoff patterns looking forward, so the price should not vary across assets. In conclusion, price neither varies across assets nor across states.

15.6.4 \(N=1\), \(J>1\), dividends are not i.i.d. (positive serial correlation)

Index the subscripts of states such that \(y_1j\)). So high current dividends implies higher probability of having high dividends in future periods. Then we can expand (15.7) to an infinite geometric series: $$\mathbf p^e=\beta A^{-1}(I-\beta Q)^{-1}QAY=A^{-1}\underbrace{(\beta Q+\beta^2 Q^2+\beta^3 Q^3+\cdots)}_{Q^\star\equiv[q^\star_{jk}]}AY\Rightarrow A\mathbf p^e=Q^\star AY$$ Since positive serial correlation (monotonicity of \(Q\)) implies \(Q^n\) (transition probability in \(n\) steps) is monotone, so \(Q^\star\) is also monotone. \(Y\) and \(\mathbf p^e\) are \(J\times1\) column vectors, expanding \(a_j p^e_j=\sum_l q^\star_{jl}a_l y_l\). Note that \(a_j=u'(y_j)\) is decreasing in \(j\) while \(y_j\) is increasing in \(j\). So, as long as \(a_j y_j\) is increasing in \(j\), monotone \(Q^\star\) will yield a vector increasing from top to bottom, which means \(a_j p^e_j\) is increasing in \(j\); since \(a_j\) decreases in \(j\), it must be that \(p^e_j\) increases in \(j\), i.e. higher current dividends mean higher price. However, if \(a_j y_j\) is decreasing in \(j\), then the effects of monotonicity of \(Q^\star\) and \(a_j\)'s decreasing in \(j\) are conflicting, which makes the change in \(p^e_j\) across \(j\) unclear.

For a specific utility function, consider the CRRA family \(u(c)=\frac{c^{1-\theta}}{1-\theta}\), then \(a_j y_j=y_j^{1-\theta}\): - \(\theta<1\): \(a_j y_j\) is increasing in \(j\), so \(p^e_j\) is increasing in \(j\). - \(\theta=1\): \(a_j y_j=1\) is constant across \(j\), so \(p^e_j\) is increasing in \(j\). - \(\theta>1\): \(a_j y_j\) is decreasing in \(j\), so the direction of change in \(p^e_j\) across \(j\) is unclear.

15.6.5 Large \(N\), large \(J\) with i.i.d. assets but not i.i.d. dividends (positive serial correlation)

Suppose an economy has large \(N\), large \(J\), i.i.d. assets but not i.i.d. dividends (positive serial correlation). Since the price of one asset is affected by other assets only through aggregate dividends, and large \(N\) makes the aggregate dividends constant across states, all assets with current dividends \(y_j\) should have the same price \(p^e_j\) due to their i.i.d. property — even though for any two assets the same dividends may happen in different states. So we need only talk about one asset's price and all other assets should have the same price in states with same dividends. Since \(a_j\) is constant across \(j\) (large \(N\) ⟹ aggregate constant ⟹ \(A\) has same diagonal elements), we know \(a_j y_j\) is increasing in \(j\); by the result in 15.6.4, \(p^e_j\) increases in \(j\). Therefore, assets with high current dividends have higher price; assets with same dividends (even in different states) have the same price.

15.6.6 \(N=1\), \(J=4\) with not i.i.d. dividends (positive serial correlation)

Suppose an economy has \(N=1\), \(J=4\) and dividends have positive serial correlation. Specifically, \(Y=(L,L,H,H)'\) (\(L

15.6.7 Two similar economies

Consider two economies \(M\) and \(S\) with the same set of states, i.e. \(J^M=J^S\), and same \(\beta\), \(u\) and \(Q\). The aggregate dividends in each state is the same between \(M\) and \(S\) (so matrix \(A\) is the same for the two economies), but dividends of individual assets are not the same, i.e. \(Y^M\ne Y^S\). - Scenario 1: Pick two particular assets: asset \(h\) in economy \(M\) and asset \(k\) in economy \(S\). Suppose they have the same dividends in every state, \(\mathbf y^M_h=\mathbf y^S_k\), then their price in every state is the same, i.e. \(\mathbf p^M_h=\mathbf p^S_k\). To see why, recall \(\mathbf p^e=\beta A^{-1}(I-\beta Q)^{-1}QAY\). The two economies have the same \(\beta A^{-1}(I-\beta Q)^{-1}QA\). Note \(\mathbf p^M_h\) is this matrix times the \(h\)th column of \(Y^M\) (i.e. \(\mathbf y^M_h\)) and \(\mathbf p^S_k\) is this matrix times the \(k\)th column of \(Y^S\) (i.e. \(\mathbf y^S_k\)). Since \(\mathbf y^M_h=\mathbf y^S_k\), we conclude \(\mathbf p^M_h=\mathbf p^S_k\). Again, the price of an asset is affected by dividends of other assets only through aggregate dividends displayed in \(A^{-1}\) and \(A\). - Scenario 2: Suppose \(\mathbf y^M_h=\sum_{n=1}^N\theta_n\mathbf y^S_n\), then \(\mathbf p^M_h=\sum_{n=1}^N\theta_n\mathbf p^S_n\), which is true by the result in Scenario 1. - Scenario 3: Suppose some asset \(k\) exists in \(S\) but not in \(M\). Introduce a new asset \(h\) to \(M\) with dividends \(\varepsilon\mathbf y^S_k\), then \(\mathbf p^M_h=\varepsilon\mathbf p^S_k\), which is true by the result in Scenario 1.