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316-406/671 ADVANCED MACRO TECHNIQUES
This exam lasts 180 minutes and has two questions. The first question is worth 120 marks, while the second question is worth only 60 marks. Allocate your timeaccordingly. Within each question there are a number of parts and the weight given to each part is also indicated. Even if you cannot complete one part of a question, you should be able to move on ananswer other parts, so do not spend too much time. If you feel like you are getting stuck, move on to the next part. You also have 15 minutes perusal before you can start writing answers. Question 1.Term Structure of Interest Rates (120 marks): Consider a representative agent asset pricing model where preferences are E0
(∞ X
c1−γ βt t 1−γ t=0
)
,
0 < β < 1 and γ > 0
There are twokinds of assets. First, there is a "Lucas tree" with dividends {xt } that follow an autoregression of the form ¯ xt+1 = x1−φ xφ εt+1 , t 0 < φ < 1 and x > 0 ¯ (1)
where log(εt+1 ) are IID normal withmean 1 and variance σ 2 . Second, there are also bonds of various maturities. A j-period bond (j ≥ 1) is a riskless claim to one unit of consumption to be delivered in j-period’s time.
(a) (15marks): Suppose we let bonds up-to maturity J = 2 be traded. That is, one-period (j = 1) and two-period (j = 2) bonds are traded. Let qj (x) denote the price of a j-bond if the current aggregate state isx and let p(x) denote the price of a claim to the Lucas tree. Let V (w, x) denote the consumer’s value function if their individual wealth is w and the aggregate state is x. Write down a Bellmanequation for the consumer’s problem. Be careful to explain the Bellman equation and any constraints that you provide. (b) (15 marks): Define a recursive competitive equilibrium for this economy. (c) (15marks): Suppose we let bonds with a maturity up to an arbitrary J be traded. Explain
1
why in equilibrium the price of a j-bond satisfies the relationship qj (xt ) = Et β
(
j
µ
xt+j xt...
This exam lasts 180 minutes and has two questions. The first question is worth 120 marks, while the second question is worth only 60 marks. Allocate your timeaccordingly. Within each question there are a number of parts and the weight given to each part is also indicated. Even if you cannot complete one part of a question, you should be able to move on ananswer other parts, so do not spend too much time. If you feel like you are getting stuck, move on to the next part. You also have 15 minutes perusal before you can start writing answers. Question 1.Term Structure of Interest Rates (120 marks): Consider a representative agent asset pricing model where preferences are E0
(∞ X
c1−γ βt t 1−γ t=0
)
,
0 < β < 1 and γ > 0
There are twokinds of assets. First, there is a "Lucas tree" with dividends {xt } that follow an autoregression of the form ¯ xt+1 = x1−φ xφ εt+1 , t 0 < φ < 1 and x > 0 ¯ (1)
where log(εt+1 ) are IID normal withmean 1 and variance σ 2 . Second, there are also bonds of various maturities. A j-period bond (j ≥ 1) is a riskless claim to one unit of consumption to be delivered in j-period’s time.
(a) (15marks): Suppose we let bonds up-to maturity J = 2 be traded. That is, one-period (j = 1) and two-period (j = 2) bonds are traded. Let qj (x) denote the price of a j-bond if the current aggregate state isx and let p(x) denote the price of a claim to the Lucas tree. Let V (w, x) denote the consumer’s value function if their individual wealth is w and the aggregate state is x. Write down a Bellmanequation for the consumer’s problem. Be careful to explain the Bellman equation and any constraints that you provide. (b) (15 marks): Define a recursive competitive equilibrium for this economy. (c) (15marks): Suppose we let bonds with a maturity up to an arbitrary J be traded. Explain
1
why in equilibrium the price of a j-bond satisfies the relationship qj (xt ) = Et β
(
j
µ
xt+j xt...
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