Question

5. Assume that a country produces an output Q of 50 every year and that r* = 10%. Consumption C is 50 every year, and I = G = 0. Suppose there is a temporary drop in output in year 0, so Q falls to 28. Q returns to 50 in every future year. If the country desires to smooth C, how much should it borrow in period 0? What will the new level of C from then on? (Hint: calculate the new present value of output and then estimate the new constant level of consumption.)

A. The nation borrows 20 in period 0 and the new level of C is 48.

B. The nation borrows 22 in period 0 and the new level of C is 28.

C. The nation borrows 28 in period 0 and the new level of C is 36.

D. The nation borrows 48 in period 0 and the new level of C is 22.

6. If a nation experiences an output shock and wishes to borrow to smooth consumption, how much of consumer spending must it forgo each year to achieve consumption smoothing and maintain the long-run budget constraint?

A. an amount equal to r*/(1 + r*) of the output shock

B. an amount equal to r* as a percent of the former level of GDP

C. an amount equal to 20% of its output shock

D. 95% of the output shock

7. Which expression below represents the change in wealth in period 0? Use the following notation: Δ denotes change, W_t is external wealth at time t, TB_t is the trade balance during time period t, and r* is the constant real interest rate. Assume that net labor income from abroad is zero, there are no capital gains on external wealth, and there are no unilateral transfers.

A. ΔTB₀ = W₀ + r*W_-1.

B. ΔW₀ = TB₀ + r*W_-1.

C. ΔW_-1 = ΔW₀ + TB₀.

D. ΔTB₀ = ΔW_-1 + r*W_-1.

Answer 1

Solution:

5. For the period 0, output Y0 = 28

For period 1, output Y1 = 50. Then, present value of this output level (that is its value for period 0) = Y1/(1 + r) = 50/(1 + 0.1)

Similarly, present value for output level at period 2 = Y2/(1 + r)² = 50/(1 + 0.1)²

The lifetime net present value of output then becomes = Y0 + Y1/(1 + r) + Y2/(1 + r)² + Y3/(1 + r)³ + ... for infinite time

NPV = 28 + 50/(1 + 0.1) + 50/(1 + 0.1)² + 50/(1 + 0.1)³ + ...

NPV = 28 + 50/1.1 + 50/1.1² + 50/1.1³ + ...

Note that this is a geometric progression series (GP) as every consecutive term increases by a common factor. Sum for infinite series of GP = first term/(1 - common ratio). Then, according to our series, first term = 50/1.1 ; and common ratio = 1/1.1

So, NPV of output = 28 + (50/1.1)/(1 - (1/1.1))

NPV of output = 28 + (50/1.1)/((1.1 - 1)/1.1)

NPV of output = 28 + 50/0.1 = 28 + 500 = 528

With consumption smoothing, consumption for every year is let's say C. Then, present value of lifetime consumption = C + C/(1 + r) + C/(1 + r)² + C/(1 + r)³ + ...

= C + C/(1 + 0.1) + C/(1 + 0.1)² + C/(1 + 0.1)³ + ...

= C + C/1.1+ C/1.1² + C/1.1³ + ...

This is also a GP series, with first term = C, and common ratio = 1/1.1

So, NPV of consumption = C/(1 - 1/1.1) = C*1.1/(1.1 - 1) = 1.1*C/0.1 or 11*C

As for lifetime, NPV of consumption = NPV of output

11*C = 528

C = 528/11 = 48

Finally, as output for period 0 = 28, however consumption = 48, we must borrow C - Y = 48 - 28 = 20 in period 0

Thus, the correct option is (A).

6. We have already seen a situation with output shock. Had there been no output shock and same level of output for lifetime, NPV of output = Y + Y/(1 + r) + Y/(1 + r)² + ...

With first term of Y, and common ratio of 1/(1 + r), NPV of output in this case = Y/(1 - (1/(1 + r)) = Y(1 + r)/r

So, NPV of consumption = C*(1 + r)/r (as already seen from above). This C equated to above NPV will give consumption level under no output shock, or former consumption level:

C*(1 + r)/r = Y*(1 + r)/r

So, C = Y. Let's call this consumption level (without output shock) as C* which equals the former GDP or output level.

The consumption level calculated in case of output shock, C'= (Y' + Y/r)*(r/(1 + r)) where Y' is the output after shock and Y is former output

So, C'= Y'r/(1+r) + Y/(1 + r)

So, the amount of consumption spending forgone = C* - C'

= Y - [Y'r/(1+r) + Y/(1 + r)]

= (Y(1 + r) - Y)/(1 + r) - Y'r/(1 + r)

= Yr/(1 + r) - Y'r/(1 + r)

= (r/(1 + r))*(Y - Y') ; where Y - Y' is the output shock. So, amount of consumption forgone is r/(1 + r) of output shock.

Thus, the correct option is (A).

7. Change in wealth in period 0 = Wealth in period 0 - wealth in previous period

Change in W0 = W0 - W-1

Wealth in period 0 = Trade balance in period 0 + (1 + r)*wealth in previous period

So, W0 = TB0 + (1 + r)W-1

So, Change in W0 = TB0 + (1 + r)W-1 - W-1

Change in W0 = TB0 + W-1 + r+W-1 - W-1

Change in W0 = TB0 + r*W-1

Thus, the correct option is (B).