PROBLEMS

1. Using the notation from the text, answer the following questions. You may assume that net labor income from abroad is zero, there are no capital gains on external wealth, and there are no unilateral transfers.
   • Express the change in external wealth (ΔW₀) at the end of period 0 as a function of the economy’s trade balance (TB), the real interest rate (a constant r^*), and initial external wealth (W₋₁).
   • Using (a), write an expression for the stock of external wealth at the end of period 0 (W₀). This should be written as a function of the economy’s trade balance (TB₀), the real interest rate, and initial external wealth (W₋₁).
   • Using (a) and (b), write an expression for the stock of external wealth at the end of period 1 (W₁). This should be written as a function of the economy’s trade balance (TB) each period, the real interest rate, and initial external wealth (W₋₁).
   • Using your answers from (a), (b), and (c), write an expression for the stock of external wealth at the end of period 2 (W₂). This should be written as a function of the economy’s trade balance (TB) each period, the real interest rate, and initial external wealth (W₋₁).
   • Suppose we require that W₂ equal zero. Write down the condition that the three trade balances (in periods 0, 1, and 2) must satisfy. Arrange the terms in present value form.
2. Using the assumptions and answers from the previous question, complete the following:
   • Write an expression for the future value of the stock of external wealth in period N (W_N). This should be written as a function of the economy’s trade balance (TB) each period, the real interest rate r^*, and initial external wealth.
   • Using the answer from (a), write an expression for the present value of the stock of external wealth in period N (W_N).
   • The “no Ponzi game” conditions force the present value of W_N to tend to zero as N gets large. Explain why this implies that the economy’s initial external wealth is equal to the present value of future trade deficits.
   • How would the expressions in parts (a) and (b) change if the economy had net labor income (positive or negative) to/from abroad or net unilateral transfers? Explain briefly.
3. In this question, assume all dollar units are real dollars in billions, so, for example, $150 means $150 billion. It is year 0. Argentina thinks it can find $150 of domestic investment projects with an MPK of 10% (each $1 invested pays off $0.10 in every later year). Argentina invests $84 in year 0 by borrowing $84 from the rest of the world at a world real interest rate r^* of 5%. There is no further borrowing or investment after this.

   Use the standard assumptions: Assume initial external wealth W(W in year −1) is 0. Assume G = 0 always; and assume I = 0 except in year 0. Also, assume NUT = KA = 0 and that there is no net labor income so NFIA = r^*W.

   The projects start to pay off in year 1 and continue to pay off all years thereafter. Interest is paid in perpetuity, in year 1 and every year thereafter. In addition, assume that if the projects are not done, then GDP = Q = C = $200 in all years, so that PV(Q) = PV(C) = 200 + 200/0.05 = 4,200.

   • Should Argentina fund the $84 worth of projects? Explain your answer.
   • Why might Argentina be able to borrow only $84 and not $150?
   • From this point forward, assume the projects totaling $84 are funded and completed in year 0. If the MPK is 10%, what is the total payoff from the projects in future years?
   • Assume this is added to the $200 of GDP in all years starting in year 1. In dollars, what is Argentina’s Q = GDP in year 0, year 1, and later years?
   • At year 0, what is the new PV(Q) in dollars? Hint: To ease computation, calculate the value of the increment in PV(Q) due to the extra output in later years.
   • At year 0, what is the new PV(I) in dollars? Therefore, what does the LRBC say is the new PV(C) in dollars?
   • Assume that Argentina is consumption smoothing. What is the percent change in PV(C)? What is the new level of C in all years? Is Argentina better off?
   • For the year the projects go ahead, year 0, explain Argentina’s balance of payments as follows: state the levels of CA, TB, NFIA, and FA.
   • What happens in later years? State the levels of CA, TB, NFIA, and FA in year 1 and every later year.
4. Continuing from the previous question, we now consider Argentina’s external wealth position.
   • What is Argentina’s external wealth W in year 0 and later? Suppose Argentina has a one-year debt (i.e., not a perpetual loan) that must be rolled over every year. After a few years, in year N, the world interest rate rises to 15%. Can Argentina stick to its original plan? What are the interest payments due on the debt if r^* = 15%? If I = G = 0, what must Argentina do to meet those payments?
   • Suppose Argentina decides to unilaterally default on its debt. Why might Argentina do this? State the levels of CA, TB, NFIA, and FA in year N and all subsequent years. What happens to the Argentine level of C in this case?
   • When the default occurs, what is the change in Argentina’s external wealth W? What happens to the rest of the world’s external wealth?
   • External wealth data for Argentina and Rest of the World are recorded in the account known as the net international investment position. Is this change in wealth recorded as a financial flow, a price effect, or an exchange rate effect?
5. Using production function and MPK diagrams, answer the following questions. For simplicity, assume there are two countries: a poor country (with low living standards) and a rich country (with high living standards).
   • Assuming that poor and rich countries have the same production function, illustrate how the poor country will converge with the rich country. Describe how this mechanism works.
   • In the data, countries with low living standards have capital-to-worker ratios that are too high to be consistent with the model used in (a). Describe and illustrate how we can modify the model used in (a) to be consistent with the data.
   • Given your assumptions from (b), what does this suggest about the ability of poor countries to converge with rich countries? What do we expect to happen to the gap between rich and poor countries over time? Explain.
     Using the model from (b), explain and illustrate how convergence works in the following cases.
   • The poor country has a marginal product of capital that is higher than that of the rich country.
   • The marginal products in each country are equal. Then, the poor country experiences an increase in human capital through government funding of education.
   • The marginal products in each country are equal. Then, the poor country experiences political instability such that investors require a risk premium to invest in the poor country.
6. Assume that Brazil and the United States have different production functions q = f (k), where q is output per worker and k is capital per worker. Let q = Ak^1/3. You are told that relative to the U.S. = 1, Brazil has an output per worker of 0.32 and capital per worker of 0.24. Can A be the same in Brazil as in the United States? If not, compute the level of A for Brazil. What is Brazil’s MPK relative to the United States?
7. Use production function and MPK diagrams to examine Turkey and the EU. Assume that Turkey and the EU have different production functions q = f (k), where q is output per worker and k is capital per worker. Let q = Ak^1/3. Assume that the productivity level A in Turkey is lower than that in the EU.
   • Draw a production function diagram (with output per worker y as a function of capital per worker k) and MPK diagram (MPK versus k) for the EU. (Hint: Be sure to draw the two diagrams with the production function directly above the MPK diagram so that the level of capital per worker k is consistent on your two diagrams.)
   • For now, assume capital cannot flow freely in and out of Turkey. On the same diagrams, plot Turkish production function and MPK curves, assuming that the productivity level A in Turkey is half the EU level and that Turkish MPK exceeds EU MPK. Label the EU position in each chart EU and the Turkish position T1.
   • Assume capital can now flow freely between Turkey and the EU and the rest of the world, and that the EU is already at the point where MPK = r^*. Label r^* on the vertical axis of the MPK diagram. Assume no risk premium. What will Turkey’s capital per worker level k be? Label this outcome point T2 in each diagram. Will Turkey converge to the EU level of q? Explain.
8. This question continues from the previous problem, focusing on how risk premiums explain the gaps in living standards across countries.
   • Investors worry about the rule of law in Turkey and also about the potential for hyperinflation and other bad macroeconomic policies. Because of these worries, the initial gap between MPK in Turkey and r^* is a risk premium RP. Label RP on the vertical axis of the MPK diagram. Now where does Turkey end up in terms of k and q?
   • In light of (a), why might Turkey be keen to join the EU?
   • Some EU countries are keen to exclude Turkey from the EU. What might be the economic arguments for that position?
9. In this chapter, we saw that financial market integration is necessary for countries to smooth consumption through borrowing and lending. Consider two economies: those of the Czech Republic and France. For each of the following shocks, explain how and to what extent each country can trade capital to better smooth consumption.
   • The Czech Republic and France each experience an EU-wide recession.
   • A strike in France leads to a reduction in French income.
   • Floods destroy a portion of the Czech capital stock, lowering Czech income.
10. Assume that a country produces an output Q of 50 every year. The world interest rate is 10%. Consumption C is 50 every year, and I = G = 0. There is an unexpected drop in output in year 0, so output falls to 39 and is then expected to return to 50 in every future year. If the country desires to smooth consumption, how much should it borrow in period 0? What will the new level of consumption be from then on?
11. Assume that a country produces an output Q of 50 every year. The world interest rate is 10%. Consumption C is 50 every year, and I = G = 0. There is an unexpected war in year 0, which costs 11 units and is predicted to last one year. If the country desires to smooth consumption, how much should it borrow in period 0? What will the new level of consumption be from then on?

    The country wakes up in year 1 and discovers that the war is still going on and will eat up another 11 units of expenditure in year 1. If the country still desires to smooth consumption looking forward from year 1, how much should it borrow in period 1? What will the new level of consumption be from then on?

12. Consider a world of two countries: Highland (H) and Lowland (L). Each country has an average output of 9 and desires to smooth consumption. All income takes the form of capital income and is fully consumed each period.
    • Initially, there are two states of the world: Pestilence (P) and Flood (F). Each happens with 50% probability. Pestilence affects Highland and lowers the output there to 8, leaving Lowland unaffected with an output of 10. Flood affects Lowland and lowers the output there to 8, leaving Highland unaffected with an output of 10. Devise a table with two rows corresponding to each state (rows marked P and F). In three columns, show income to three portfolios: the portfolio of 100% H capital, the portfolio of 100% L capital, and the portfolio of 50% H + 50% L capital.
    • Two more states of world appear: Armageddon (A) and Utopia (U). Each happens with 50% probability but is uncorrelated with the P–F state. Armageddon affects both countries equally and lowers income in each country by a further four units, whatever the P–F state. Utopia leaves each country unaffected. Devise a table with four rows corresponding to each state (rows marked PA, PU, FA, and FU). In three columns, show income to three portfolios: the portfolio of 100% H capital, the portfolio of 100% L capital, and the portfolio of 50% H + 50% L capital.

    Compare your answers to parts (a) and (b), and consider the optimal portfolio choices. Does diversification eliminate consumption risk in each case? Explain.