Assignment 3

Queen's University Economics Department
Economics 320A - Macroeconomic Theory II
Fall 1998
Assignment 3



Instructor: Marc-André Letendre

Deadline: November 12, 1998 (in class)



Question 1 - War Expenditures Financing 65 points

In class we learned how to solve for a perfect foresight competitive equilibrium when the government issues k-period bonds in period t and does not issue any more bonds before those bonds come to maturity. In this problem, we look at a more general case where the government does issue bonds before the k-period bonds issued in period t come to maturity.

Consider the case of a country engaged in a war in period 3. In period 1, the government does nothing (no taxes, no borrowing). In period 2, the government is preparing for the war. Therefore, government consumption is positive in period 2 and it is financed by issuing 10 3-period bonds. During the war, in period 3, the government still has positive consumption and finances it by issuing 7 2-period bonds. During the reconstruction, period 4, the government still has positive consumption and finances it by issuing 3 1-period bonds. Finally, in period 5, government consumption is zero and the government imposes taxes on the young to totally pay off its debt.

Let Bk(t) be the number of k-period government bonds that exist in period t. Let [^(B)]k(t) be the number of k-period government bonds issued in period t. Therefore we have [^(B)]3(2) = 10, [^(B)]2(3) = 7 and [^(B)]1(4) = 3.

Every generation has 100 members. All members of generation t has the utility function uht = ln(cht(t))+0.7 ln(cht(t+1)). There are two types of agents in every generation, the even and the odd. They differ according to their endowments

[wth(t), wth(t+1)] = ì
í
î
[4, 3]
for h even
[3, 3]
for h odd

We have the usual consumption and savings functions

cht(t) = wht(t)-tht(t)
1.7
 + wht(t+1)-tht(t+1)
1.7r(t)
sht(r(t)) = 0.7(wht(t)-tht(t))
1.7
 - wht(t+1)-tht(t+1)
1.7 r(t)
 .


(a) 10 points

Explain why B3(2) = 10, B2(3) = 17 and B1(4) = 20?


(b), Period 5 7 points

The government pays off it debt so th5(5) = B1(4)/100 = 20/100 = 0.20 "h. There is no government borrowing in period 5. Solve for the equilibrium interest rate in period 5. Since the bonds come to maturity in period 5, their price (before payoff) is p0(5) = 1.


(c), Period 4 10 points

Using the perfect foresight forecast pe0(5) = p0(5) = 1, solve for the temporary equilibrium interest rate in period 4 using the method presented in class. Also, compute p1(4) and G(4).


(d), Period 3 10 points

Using the perfect foresight forecast pe1(4) = p1(4) = 0.7335, solve for the temporary equilibrium interest rate in period 3 using the method presented in class. Also, compute p2(3) and G(3).


(e), Period 2 10 points

Using the perfect foresight forecast pe2(3) = p2(3) = 0.5595, solve for the temporary equilibrium interest rate in period 2 using the method presented in class. Also, compute p3(2) and G(2).


(f), Period 1 5 points

There are no taxes nor government borrowing in period 1. Solve for the equilibrium interest rate in period 1.

(g) 8 points

Give a complete description of the perfect foresight equilibrium for this economy.


(h) 5 points

Explain, in no more than 10 lines, why r(1) < r(2) < r(3) < r(4)?


(i) Bonus question 15 points

In period 3, the government issues [^(B)]2(3) = 7 2-period bonds. Why is it that one of the equilibrium condition we must use is S3(r(3)) = p2(3) B2(3) and not S3(r(3)) = p2(3) [^(B)]2(3)? After all, the government is simply borrowing p2(3) [^(B)]2(3) units of the time 3 good.



Question 2 - Testing the PFEH of the TSIR 18 points

The perfect foresight expectations hypothesis (PFEH) of the term structure of the interest rate (TSIR) predicted by our OLG model is

rk(t) = [r(t) r(t+1) r(t+2) ... r(t+k-1)]1/k,
where rk(t) is the internal rate of return (IRR), in period t, on a k-period government bond. Assume there exist 1-period, 2-period and 3-period government bonds in period 1 and that period 1 is January 30, period 2 is February 28, period 3 is March 31, period 4 is April 30. Using the quotations below:

(a) describe the TSIR in period 1.6 points

(b) test the PFEH of the TSIR.12 points

On January 30, you read the following quotations on internal rate of returns on government bonds in a newspaper.
Maturity date IRR on January 30
February 28 7.42%
March 31 7.73%
April 30 8.00%

On March 31, you read in the newspaper that the IRR on a government bond with maturity on April 30 is 8.00%.


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