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Homework 3 Intermediate Macroeconomics Economics 215 Assigned: Thursday, May 6th, 2004 Due: Friday, May 14th, 2004 1. Foreign Real Estate. You consider buying an apartment in London. The apartment will earn a rental income of ₤10,000 next year. The current HK$ interest rate is 5%. The spot exchange rate is HK$13.23 per British pound and the 1-year forward rate is 12. The price of real estate in London is expected to remain constant, so capital gains are zero. The depreciation rate of the rental property is 5%. In each period, the sum of tax and other costs is 2% of the value of the apartment. Calculate the price level of the apartment in HK$ using covered interest parity. Forward interest parity allows us to calculate interest rates in pounds. 1 itHK Ft 12 1 itUK 1.05 1 itUK 1 itUK St 13.23 (1.1) 13.23 1.05 1.157625 12 The rental yield should be equal to the cost of capital which is the sum of interest costs (i) plus depreciation costs (δ) plus taxes and other costs (tw+c) minus capital gains. Capital gains are zero so Rent 10000 RE .157625 .05 .02 .227625 P RE P 10000 P RE 43931.91 .227625 (1.2) 2. Predicting Exchange Rates The current exchange rate with Singapore is S = 4.44925 (i.e. HK$ per Sing$). Table 3 includes the yield curve for Singapore government securities and HK Exchange Fund bills. Assume that uncovered interest parity and the expectations theory of the term structure holds true for HK and Singapore. Calculate the markets expectation of the exchange rate in 1 year (T = 1) and 2,5,7, and 10 year (T = 2,5,7,10). Table 3 – Yield Curves for Hong Kong & Singapore T Nominal Intereest Rate HK Singapore StET 1 0.70% 1.31% 4.422461 2 0.87% 1.71% 4.376149 5 1.48% 3.00% 4.130102 7 1.84% 3.73% 3.912402 10 2.05% 4.31% 3.574024 4.44925 4.44925 4.44925 4.44925 4.44925 Using uncovered interest parity, we can write expectations of future spot rates as T StET (1 it ,T ) St F 1 it ,T 3. Tax Wedge and Deadweight Loss. Assume that a firm has a quadratic production function in capital. GDP 5 K K 2 This implies that the faces a linear marginal product of capital schedule. MPK 5 2K Assume that the price of goods is always equal to the price of capital (i.e. pK = 1) and the depreciation rate is 10% (i.e. δ = .10). The real interest rate is equal to 10% (i.e. r = .1) so the real cost of capital is r+δ. a. Calculate, the profit maximizing level of capital when the tax wedge is zero and the level of output that could be produced with that amount of capital. Calculate the level of profit (Hint: The profit is the level of output minus costs. Costs are equal to the product of the cost of capital and the capital used, Profit = GDP – (r+δ) ․K* ). The profit maximizing level of capital sets the marginal product of capital equal to the cost of capital which in this case is (r+δ) = .2 = (r+ ) = MPK 5 2 K K * 4.8 2.4 2 GDP 5K K 2 12 5.76 = 6.24 Profit = GDP-(.2*K) = 6.24 - 0.48 = 5.76 b. Assume that government imposes a tax on capital with each firm having to pay taxes equal to a fraction of their capital stock, TaxesτK. Calculate the real cost of capital when the tax wedge is equal to 10% (i.e. τ=.1). What is the profit maximizing level of capital and output? What is the after-tax profit level (Profit = GDP – {r+δ+τ} ․K* ) and how much tax is collected? 4.7 2.35 2 GDP 5K K 2 11.75 5.5225 = 6.2275 Profit = GDP-(.3*K)=6.2275-0.705 = 5.5225 .3 = (r+ tw) = MPK 5 2 K K * (1.3) The loss in profits is about 5.76-5.5225 ≈ .2375 and the taxes gained are .235 so social costs are .0025. c. Calculate the social cost of the tax distortion which is the loss in profits that does not go to the government in taxes. Social cost is the profits from part a. minus the profits and tax revenues in part b. Draw a graph of these losses. The loss in profits is about 5.88-5.64 ≈ .24 and the taxes gained are .2375 so social costs are .0025. Profits r+δ+tw w r+δ Taxes Social Cossts 4. Compare output per capita in East Asia. Sometimes we would like to compare levels of GDP across countries. However, GDP is measured in national currencies. To compare, we have to convert into the same units. There are two methods, exchange rate conversion and PPP conversion. Exchange rate conversion simply multiplies by the nominal exchange rates, the number of foreign currency units per domestic currency units. This calculates the number of foreign currency units necessary to buy the domestic currency necessary to buy domestic GDP. The price of domestic currency relative to foreign currency. PPP conversion PF , the price of foreign goods relative to the price of P domestic goods. PPP conversion divides spending on domestic goods by the price level of domestic goods, then re-multiplies by the price of foreign goods. This gives the cost of domestic goods produced if their prices were the same as in the foreign country. multiplies by XP Table 2 gives some data for a number of East Asian economies in 2000 on GDP per capita measured in domestic currency, nominal exchange rates (# of domestic currency units per US$) and prices of goods in the US relative to domestic prices. Treating the US as the foreign economy, calculate GDP per capita for each economy in US dollars using exchange rate conversion and PPP conversion. Exchange Rates Hong Kong China Japan Macau Singapore Philippines Indonesia GDP per Capita (US$ per in Domestic Currecy LCU) 192,776 6,423 3,920,922 132,230 45,193 35,433 4,732,761 0.128 0.121 0.009 0.125 0.580 0.023 0.022 Foreign Prices relative to Domestic Prices XP 0.150 0.522 0.006 0.203 0.724 0.091 0.130 GDP per Capita in US Dollars Exchange PPP Rate Conversion Conversion US$24,742.84 US$28,845.64 US$775.84 US$3,353.14 US$36,384.01 US$25,121.99 US$16,475.35 US$26,870.64 US$26,214.53 US$32,720.06 US$801.80 US$3,219.59 US$105,309.11 US$616,783.49 5. Technology and the Current Account. Assume that a countries savings level is fixed and equal to S = 100. In every period, the countries capital depreciates 100% (i.e. δ = 1). In this economy, the real price of capital is constant and equal to pK = 1. Further, the technology of this economy is described with a Cobb-Douglas production function. 12 GDP 24 K MPKt 1 Kt 1 Therefore, the real cost of capital is rck = 1+r where r is the real interest rate. The level of investment today is next period’s capital stock, Kt+1 = It. a. Calculate the level of investment when the real interest rate is 10% (r = .10). What is the current account (Hint: The current account is S - I)? The level of investment is the level of capital that maximizes profits 2 12 12 MPK t 1 1.1 K * I 119.008 K t 1 1.1 (1.4) CurrentAccount S I 100 119 19 b. Calculate the autarky interest rate. The autarky interest rate sets current account equal to zero or savings equal to investment. 2 12 100 r A .2 A 1 r (1.5) c. Technology increases so that next period’s marginal product of capital is 12.5 GDP 25 K MPKt 1 Kt 1 . What is the current account after the increase of technology? Draw a graph of the effect of the increase in technology on the current account. 2 12.5 1.1 K 129.1322314 * 1.1 K S-I = -29.13 12.5 * (1.6) S 1+rA 1+r=1.1 .1 I’ I