134 CHAPTER 4. THE LUCAS MODEL
Appendix—Markov Chains
Let Xt be a random variable and xt be a particular realization of Xt. A Markov chain is a stochastic process {Xt}∞t=0 with the property that the information in the current realized value of Xt = xt summarizes the entire past history of the process. That is,
P[Xt+1 = xt+1|Xt = xt, Xt−1 = xt−1, . . . , X0 = x0] = P[Xt+1 = xt+1|Xt = xt].
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(4.64) |
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A key result that simpliÞes probability calculations of Markov chains is, |
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Property 1 If {Xt}t∞=0 is a Markov chain, then |
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P[Xt = xt ∩ Xt−1 = xt−1 ∩ · · · ∩ X0 = x0] = |
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P[Xt = xt|Xt−1 = xt−1] · · · P[X1 = x1|X0 = x0]P[X0 = x0]. |
(4.65) |
Proof: Let Aj be the event (Xj = xj). You can write the left side of (4.65) as,
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P(At ∩ At−1 ∩ · · · ∩ A0) = |
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P(At| Aj)P( |
Aj) |
(multiplication rule) |
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t−1 |
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j\ |
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= P(At|At−1)P( |
Aj) (Markov chain property) |
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t−2 |
t−2 |
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j\ |
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Aj) (mult. rule) |
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P(At|At−1)P(At−1| |
Aj)P( |
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j=0 |
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t−2 |
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j\ |
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P(At|At−1)P(At−1|At−2)P( ) |
(Markov chain) |
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= P(At|At−1)P(At−1|At−2) · · · P(A1|A0)P(A0)
Let λj, j = 1, . . . , N denote the possible states for Xt. A Markov chain has stationary probabilities if the transition probabilities from state λi to λj are time-invariant. That is,
P[Xt+1 = λj|Xt = λi] = pij
4.5. CALIBRATING THE LUCAS MODEL |
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Notice that in Markov chain analysis the Þrst subscript denotes the state that you condition on. For concreteness, consider a Markov chain with two possible states, λ1 and λ2, with transition matrix,
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P = |
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p12 |
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p22 |
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where the rows of P sum to 1.
Property 2 The transition matrix over k steps is
Pk = PP · · · P
| {z } k
Proof. For the two state process, deÞne
p(2)ij = P[Xt+2 = λj|Xt = λi]
=P[Xt+2 = λj ∩ Xt+1 = λ1|Xt = λi] + P[Xt+1 = λj ∩ Xt+1 = λ2|Xt = λi]
X2
=P[Xt+1 = λj ∩ Xt+1 = λk|Xt = λi]
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P[Xt+1 = λj ∩ Xt+1 = λk ∩ Xt = λi] |
(4.66) |
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P(Xt = λi) |
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Now by property 1, the numerator in last equality can be decomposed as, |
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P[Xt+2 = λj|Xt+1 = λk]P[Xt+1 = λk|Xt = λi]P[Xt = λi] |
(4.67) |
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Substituting (4.67) into (4.66) gives, |
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pij(2) = |
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P[Xt+1 = λj|Xt+1 = λk]P[Xt+1 = λk|Xt = λi] |
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k=1
X2
=pkjpik
k=1
which is seen to be the ij−th element of the matrix PP. |
The extension to |
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any arbitrary number of steps forward is straightforward. |
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136 |
CHAPTER 4. THE LUCAS MODEL |
Problems
1. Risk sharing in the Lucas model [Cole-Obstfeld (1991)]. Let the period utility function be u(cx, cy) = θ ln cx + (1 − θ) ln cy for the home agent and u(cx, cy) = θ ln cx + (1 −θ) ln cy for the foreign agent. Suppose That capital is internationally immobile. The home agent owns all of the x−endowment (φx = 1), the foreign agent owns all of the y−endowment (φy = 1). Show that in the equilibrium under portfolio autarchy, trade in goods alone is su cient to achieve e cient risk sharing.
2.Consider now the single-good model. Let xt be the home endowment and xt be the foreign endowment of the same good. The planner’s problem is to maximize
φ ln ct + (1 − φ) ln ct
subject to ct + ct = xt + xt .
Under zero capital mobility, the home agent’s problem is to maximize ln(ct) subject to ct = xt. The foreign agent maximizes ln(ct ) subject to ct = xt . Show that asset trade is necessary in this case to achieve e cient risk sharing.
3.Nontraded goods. Let x and y be traded as in the model of this chapter. In addition, let N be a nonstorable nontraded domestic good
generated by an exogenous endowment, and let N be a nonstorable (100) nontraded foreign good also generated by exogenous endowment. Let the domestic agent’s utility function be u(cxt, cyt, cN ) = (C1−γ)/(1−γ) where C = cθx1 cθy2 cθN3 with θ1 + θ2 + θ3 = 1. The foreign agent has the same utility function. Show that trade in goods under zero capital
mobility does not achieve e cient risk sharing.
4. Derive the exchange rate in the Lucas model under log utility, U(cxt, cyt) = (101) θ ln(cxt) + (1 − θ) ln(cyt) and compare it with the solution under con-
stant relative risk aversion utility.
5. Use the high and low growth states and the transition matrix given (102) in section 4.5 to solve for the price-dividend ratios for equities. What does the Lucas model have to say about the volatility of stock prices?
How does the behavior of equity prices in the monetary economy di er from the behavior of equity prices in the barter economy?