Econ 2221 - Practice Problems 4
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Econ 2221 - Practice Problems 4
1 Default Risk and Interest Rate Premium
Consider a two-period small open economy with a representative consumer. The economy is endowed with Y units of output in the present, but it faces uncertainty over future output. Future output can by high (YH）with probability p, or low (YL） with probability 1 − p.
The consumer chooses a plan for total domestic demand in the present (C ＋G） and future (CH ＋GH, CL ＋GL）to maximize expected lifetime utility. The consumer chooses what to if future output is high or low. For simplicity I will relabel the variables as: {D, DH, DL}.
The consumer can also borrow and save in the international market. The con-
sumer saves at a rate r兴 = 0 and borrows at a rate rb .
The consumer can also default on the debt, but doing so would induce a reces- sion in the country that reduces output by v. Without the punishment (recession) the consumer would find it optimally to default always. The consumer has limited commitment.
Consider first the solution if there is no default:
max ln D ＋p ln DH ＋ (1 − p）ln DL
{D,DH ,DL,B≠ }
s.t. D = Y ＋B
DH ＋ ╱ 1 ＋rb、B≠ = YH
DL ＋ ╱ 1 ＋rb、B≠ = YL
1. Pose a Lagrangian for the problem. The Lagrangian has three constraints and
three multipliers (入, 入H, 入L）.
2. Get the FOC.
3. Combine the FOC to get an Euler equation that relates present demand (D）to future demand (DH, DL）.
4. Interpret the Euler equation.
Now consider the default case.
5. Because of limited commitment there will be a borrowing constraint for the agent. What is this constrained given by? Does the constraint depend on p, YH or YL? if so, how?
6. Because of the risk of having low income (output) in the future the consumer might still choose to default even if the debt satisfies the constraint.
Assume that the agent defaults if Y ≠ = YL there is default. In this case DL = YL − v and the agent only has to choose ,D, DH, B≠ 、. Amend the problem of the consumer and solve for the variables (now you only have two constraints because you know DL).
7. The international lending market is competitive and international lenders are risk neutral.
(a) What are the expected profits of a lender?
(b) Due to competition expected profits are zero. What is the borrowing in- terest rate ╱rb、that arises in equilibrium?
8. What is the solution of the consumer problem under the interest rate you just found?
2 Offshoring and Productivity
Consider an open economy with one representative firm (apple) and one represen- tative household.
The household consumes a final good (say i-phones) produced by the firm, works at the firm, owns physical capital that is rented to the firm, and receives the firm’s profits. The amount of capital is fixed at k and the amount of labor l . Labor is supplied inelastically.
The firm (apple) has a main office (m）, located inside the economy, and an affiliate (a）, located overseas. Production of i-phones takes place both in the main office and in the affiliate. Production uses capital (k）, labor (l）and intangible capital (h）. Capital and labor are immobile and rivalrous, so the main office can only use local capital and labor to produce. Intangible capital on the other hand is non-rivalrous, so its use by the main office does not prevent the affiliate to use it too. Output in the main office and the affiliate is:
ym = f (km, lm, h）= km(α)lm(奁)h1 −α −奁
ya = f (ka, la, h）= ka(α)la(奁)h1 −α −奁
Note that the production function f has constant returns to scale and that h has not subscript, since it can be use simultaneously by the main office and the affiliate.
The capital and labor markets are competitive and all factors are paid their marginal products. The price of labor is w, the price of capital is rk and the price of intangible capital is rh .
Intangible capital is owned by the firm, but the firm can declare it as part of the main office or the affiliate’s assets. Payments to intangible capital depend on which part of the firm owns it. We assume that a fraction 入 of the intangible capital is owned by the main office and 1 − 入 by the affiliate.
1. Get an expression for w, rk and rh for the main office.
2. Main office:
(a) What are the profits of the main office (πm）?
(b) How do they vary with 入? Explain.
(c) How do they vary with α? Explain and relate to current trends in the U.S. economy.
3. What are the profits of the affiliate (πa）? (the affiliate hires labor and rents capital overseas at wages w≠ and rental rate r ≠ ).
4. Compute GDP in the economy using the expenditure approach. All the domes- tic production of i-phones is consumed by the household. There is no invest- ment. Remember that when the affiliate uses intangible capital it has to import part of it from the main office. Similarly, when the main office uses intangible capital it has to import part of it from the affiliate.
5. Compute GDP in the economy using the income approach.
6. How does GDP change when 入 decreases? Explain.
7. For a given value of 入, how does GDP change when α decreases? Interpret what a decrease of α means for the economy and explain the effect on measured
GDP.
3 Reduction of CA deficits
Consider the model with production and international trade in a small open econ- omy (SOE).
The government in an SOE is concerned that the current account deficit is too high. One group of economic advisers to the government argues that high govern- ment deficits cause the current account deficit to be high and that the way to reduce the current account deficit is to increase taxes. A second group of economic advisers argues that the high current account deficit is caused by high domestic investment and proposes taxing domestic investment and returning these tax revenues to con- sumers as lump-sum transfers.
1. Which advice should the government take if its goal is to reduce the current account deficit? Explain.
2. Is the government’s goal of reducing the current account deficit sensible? Why or why not?
3. What will happen if the government takes the advice that achieves its goal, as in part (a)?
4 Business Cycle Facts
We have previously studied how persistent total factor productivity shocks in a closed economy can provide an explanation for business cycles. Now consider the model with production and international trade in a small open economy (SOE).
Determine the effects of a persistent increase in total factor productivity on do- mestic output, consumption, investment, and the current account surplus. Are the predictions of the model consistent with the correlation between output and net ex- ports for Canada (see figure below)? Explain why or why not.
5 Trade war and production
One type of trade barrier is an import quota, where a country sets a limit on the quantity of a particular good that can be imported. Suppose that there is a trade war and all trade stops, which is an extreme version of import quotas. In particular, analyze a small open economy (with production and investment) which bans all imports from the rest of the world, and the rest of the world bans all imports from the small open economy. There is still trade in assets so the interest rate faced by the small open economy does not change.
Determine the macroeconomic effects of the trade war using diagrams. Can the small open economy meet the internal demand? Is the small open economy produc- ing as much as it can? What happens to aggregate demand, aggregate supply, and output? Considering two cases:
1. The small open economy is initially a net importer of goods;
2. The small open economy is initially a net exporter of goods. Discuss your results.
2022-03-09