14.02 Principles of Macroeconomics - MIT OpenCourseWare

(e) [2 points] Interpret the following two conditions that determine optimal tax collection.g 1 + g 21 + r2bt 1 = 2bt 2= t 1 + t 21 + rAnswer: The optimal pair must satisfy the budget constraint. Also, the marginalcollection cost has to be the same in each period (independent of r; sincethe tax revenue and the collection cost arise at the same point in time).(f) [4 points] Solve for the optimal tax path.Answer:The condition 0 (t 1) = 0 (t 2) together with the assumptions on imply that optimal taxes are constant in time t 1 = t 2 = t : Therefore,g 1 + g 2= t + t1 + r1 + r = 2 + r1 + r tt = 1 + r g 1 + g 22 + r 1 + r(g) [5 points] Use your results to justify how a government should …nance a temporary increasein government expenditures (e.g. wartime).Answer: The government should use debt to smooth the tax burden in timewhen there is a temporary increase in government expenditures.3 Open Economy (35 points)Consider a world economy with two countries. The home country is characterized by:C = c 0 + c 1 YI = c 2X = d ~ 1 YeM= d 2 Yewhere C; Y; ~ Y ; I; M; X; e denote aggregate consumption, domestic output, foreign output, investment,imports, exports and the real exchange rate, respectively. Assume c 0 ; c 2 > 0; 0 c 1 1; 0 d 1 1,1 c 1 + m > 0, 0 d 2 1.The foreign country is characterized byfM~e~C = ~c 0 + ~c 1~ Y~I = ~c 2= d 1~ Y~X = d 2Y~e4

Assume that parameters in the foreign economy satisfy assumptions analogous to the ones for thedomestic economy.We will assume throughout the question that the real exchange rate (e) is exogenously given.1. [5 points] Find the equilibrium level of output in the domestic goods market as a function ofe; G; ~ Y and parameters: Find the multiplier. Find equilibrium in foreign goods market as afunction of ~e; ~ G; Y and parameters.Answer:Y = c 0 + c 1 Y + c 2 + G + d 1 Y ~ =e d2 Y1hiY =c 0 + c 2 + d 1 Y ~ =e + G1 c 1 + d 2i= mhc 0 + c 2 + d 1 Y ~ =e + Gwhere M is the multiplier. Similarly,1h~Y =~c 0 + ~c 2 + d 2 Y=~e +1 ~c 1 + d ~ iGh 1= ~m ~c 0 + ~c 2 + d 2 Y=~e + G ~ i2. [3 points] How do net exports depend on the real exchange rate in this model?Answer: net exports are decreasing in the real exchange rate e:NX = d 1 ~ Yed 2 Y3. [3 points ] State a condition relating e and ~e.Answer:e = 1 ~e4. [6 points ] Solve for domestic output as a function of e; G; G ~ and parameters.Answer:iY = mhc 0 + c 2 + d 1 Y ~ =e + G"= m c 0 + c 2 + G + d 1 ~m ~c 0 + ~c 2 + d 2 Y e + G ~ #e"= m c 0 + c 2 + G + d 1 ~m ~c 0 + ~c 2 + G ~ #+ d 1 d 2 ~mYe5