Tufts University Department of Economics EC162 International Finance Prof. George Alogoskoufis Fall Semester 2016-17 Home Assignment 1 Financial Openness, the Current Account and Economic Welfare Consider an economy lasting for two periods, t=1,2. Output Υ is produced according to a production function of the form,! Y t = A t L (1) where L is total employment and Αt the productivity of labor in period t. In this economy there are N identical households, and each household supplies one unit of labor in every period. The productivity of labor is higher in period 2 than in period 1, according to,! A 2 = (1+ g)a 1 (2) where g>0. The representative household selects its consumption path so as to maximize the inter-temporal utility function,! U = u(c 1 ) + 1 (3) 1+ ρ u(c ) 2 under the constraint,! c 1 + c 2 (4) 1+ r = y + y 2 1 1+ r where ct=ct/n is consumption per household, yt=yt/n is income per household, t=1,2, u is a concave periodic utility function, ρ>0 is the pure rate of time preference and r is the real interest rate at which households can borrow and lend between periods 1 and 2. 1. Derive and interpret the first order conditions for the maximization of the utility of the representative household. 2. Analyze diagrammatically the macroeconomic equilibrium of this economy under financial autarky.
Professor George Alogoskoufis, International Macroeconomics 3. Analyze diagrammatically the macroeconomic equilibrium of this economy under financial openness, assuming households can borrow and lend freely from the rest of the world, at an exogenous international real interest rate r*. Analyze and discuss the path of the trade balance and the current account. 4. Discuss the differences between a financially autarkic and a financially open economy. 5. Assume that the per period utility function of the representative household takes the form, u(c)=ln(c). Calculate the path of output, consumption and the real interest rate under conditions of financial autarky, assuming the following parameter values: Α1=1, g=0.5, N=100, and ρ=0.5. 6. Under the same assumptions as in question 5, calculate the path of output, consumption and the current account, if the economy is financially open, and can borrow and lend freely at a real interest rate r*=0.75. 7. Explain the differences in the inter-temporal utility of the representative household between autarky and openness. 8. Assume now a world economy, consisting of two economies, similar to the economy we have described so far. These two economies are the Home Economy (H) and the Foreign Economy (F). The two economies differ only with respect to the initial productivity of labor A, and the rate of growth of productivity g. For the Home Economy, A1H=1 and gh=0.5, and for the Foreign Economy, A1F=1.1 και gf=0.2. (a). Calculate the path of output, consumption and the real interest for each economy, under financial autarky, assuming logarithmic preferences, and N=100, ρ=0.5 for each economy. (b). Under the same assumptions, calculate the path of output, consumption and the current account for each economy, as well as the international real interest rate, under financial openness, when the two economies can borrow and lend freely between them. 9. How do you explain the differences in output, consumption and economic welfare in the two economies between financial autarky and financial openness?!2
Professor George Alogoskoufis, International Macroeconomics Answers 1. Under financial autarky, the representative household maximizes,! U = u(c 1 ) + 1 (3) 1+ ρ u(c ) 2 subject to,! c 1 + c 2 (4) 1+ r = y + y 2 1 1+ r One could form the Lagrangean, derive the first order conditions and then eliminate the Lagrange multiplier in order to derive the Euler equation for consumption. Alternatively, one could solve (4) for c2, substitute for c2 in (3), and maximize with respect to c1, since y1, y2 and r are exogenous for the household. From (4), in follows that,! c 2 = (1+ r)(y 1 c 1 ) + y 2 (5) Substituting (5) in (3) we get,! U = u(c 1 ) + 1 (6) 1+ ρ u ( (1+ r)(y c ) + y 1 1 2 ) Taking the first order conditions for the maximization of (6) with respect to c1, we get,! u (c 1 ) 1+ r (7) 1+ ρ u ( (1+ r)(y c ) + y 1 1 2 ) = u (c 1 ) 1+ r 1+ ρ u (c ) = 0 2 (7) can be rearranged as, u (c! 1 ) (8) u (c 2 ) = 1+ r 1+ ρ (8) is known as the Euler equation for consumption, and can be re-written as, 1 u (c! 2 ) (9) 1+ ρ u (c 1 ) = 1 1+ r The left hand side of (9) is the marginal rate of substitution between present and future consumption. At the optimum this is equated to the right hand side, which is the relative price of future and current!3
!! Professor George Alogoskoufis, International Macroeconomics consumption. The representative household cannot thus improve her lifetime utility by further substituting present for future consumption when (9) is satisfied. From (8), it is clear that the ratio of the marginal utilities of consumption, and thus the ratio of period 1 and period 2 consumption depends on the relation between the pure rate of time preference of the representative household and the real interest rate. If the real interest rate is higher than the pure rate of time preference, then consumption in the second period is greater than consumption in the first period (the marginal utility is smaller), as the rate of return on savings exceeds the pure rate of time preference of the household. From (8) it follows that, r > ρ u (c 2 ) < u (c 1 ) c 2 > c 1 The opposite happens if the real interest rate is lower than the pure rate of time preference. Then, second period consumption is lower than first period consumption, i.e, r < ρ u (c 2 ) > u (c 1 ) c 2 < c 1 If the real interest rate is equal to the pure rate of time preference, then, it follows that consumption in the two periods is the same. Thus, it follows that,! r = ρ u (c 2 ) = u (c 1 ) c 1 = c 2 = (1+ r)y + y 1 2 (10) 2 + r Thus, in the special case where r=ρ, we end up with absolute consumption smoothing between the two periods. 2. Since in the economy that we analyze y1 < y2, and all households are the same, no households can increase their utility by saving for the future and lending their savings to other households under financial autarky. Hence savings will be equal to zero, and consumption in each period will be equal to current income. The interest rate under autarky, which in this case will be a shadow interest rate, as no borrowing and lending takes place at this interest rate, will be determined by (9), if we substitute current income for consumption. Thus, the autarky interest rate will be given by, u (y! 1 ) (11) u (y 2 ) = u (A ) 1 u (A 2 ) = 1+ r A 1+ ρ where ra denotes the autarky real interest rate. This is the one that satisfies (8) for consumption equal to current income. The equilibrium is depicted diagrammatically in Figure 1. The autarky real interest rate is determined by the slope of the indifference curve at point A, in which consumption is equal to income in each period. At this interest rate, no household has an incentive to change its consumption relative to current income in any of the two periods. Thus, the representative!4
Professor George Alogoskoufis, International Macroeconomics household neither saves nor borrows, and no borrowing and lending takes place in this economy at the autarky real interest rate. Figure 1 The Determination of the Autarky Real Interest Rate 3. The equilibrium under financial openness is depicted in Figure 2. In Figure 2 we have assumed that the world real interest rate r* is lower than the autarky real interest rate ra. Therefore, households consume more than their current income in period 1, and the economy runs a trade deficit c1-y1, and they consume less than their current income in period 2, and the economy runs a trade surplus y2-c2. Figure 2 The Determination of Consumption and the Current Account under Financial Openness 4. Consumption under financial openness is different than current income in both periods, and is determined at B, which corresponds to a higher indifference curve than the one corresponding to A under financial autarky. Thus, economic welfare is higher for the representative household under financial openness than under financial autarky. The reason is that financial openness gives domestic households more options to smooth consumption relative to the options that they have under financial autarky. 5. When the per period utility function is logarithmic, marginal utility is given by,!5
!!! Professor George Alogoskoufis, International Macroeconomics! u (c) = lnc (12) c = 1 c Substituting (12) in (8), it follows that, c! 2 = 1+ r (13) c 1 1+ ρ Given that under financial autarky c1=y1, c2=y2, it follows that the autarkic interest rate ra is given by, 1+ r A = y 2 y 1 (1+ ρ) = (1+ g)y 1 y 1 (1+ ρ) = (1+ g)(1+ ρ) = (1.5) 2 = 2.25 Therefore, the real interest rate under autarky is equal to 1.25. As for the rest of the variables, we have that, c1=y1=1, c2=y2=1.5, and C1=Y1=100, C2=Y2=150. 6. Under financial openness and a world real interest rate of 0.75, we have from the Euler equation and the inter-temporal budget constraint that, c 2 = 1+ r c 1 1+ ρ = 1.75 1.5 = 1.167 c 1 + c 2 1.75 = y + y 2 1.5 1 = 1+ 1.75 1.75 = 1.857 Solving those two equations for the two unknowns, c1 and c2, we get,! c 1 = 1.114,! c 2 = 1.300 The current accounts in the two periods are given by,! y 1 c 1 = 1 1.114 = 0.114,! y 2 + r(y 1 c 1 ) c 2 = 0.114 Thus, under financial openness our economy has a current account deficit equal to 0.114 per household in the first period, and a current account surplus equal to 0.114 per household in the second period. 7. Differences in inter-temporal utility between autarky and openness can be calculated by substituting for consumption in the two periods in the utility function of the representative household under autarky and openness. Under autarky inter-temporal utility is equal to 0.27, and under openness it is equal to 0.28. Thus, inter-temporal utility rises by 3.7% under financial openness. The reason behind the welfare gains is that under financial openness, domestic households can engage in an inter-temporal reallocation of their consumption which was not possible under autarky. As a result, they can achieve a higher level of welfare. The reason is that financial openness allows domestic households to choose their consumption on the basis!6
! Professor George Alogoskoufis, International Macroeconomics of their inter-temporal budget constraint and not under the constraint that consumption must be equal to their current income in each period. 8. The solution under autarky for the Home economy is as in the answer to question 5. For the Foreign economy, we get that, 1+ r FA = y F 2 y F1 (1+ ρ) = (1+ g F )y F1 y F1 (1+ ρ) = (1+ g F )(1+ ρ) = (1.2)(1.5) = 1.80 Therefore, the real interest rate for the Foreign economy under autarky is equal to 0.80. As for the rest of the variables, we have that, cf1=yf1=1.1, cf2=yf2=1.32, and CF1=YF1=110, CF2=YF2=132. Under autarky, the utility of the respresentative household in the Home economy is equal to 0.27 and the utility of the representative household in the Foreign economy is equal to 0.28. Under financial openness, households in the Home economy will seek to borrow from households in the Foreign economy, who have higher first period income. A world real interest rate will be established between the two economies. The world real interest rate will lie between the autarky real interest rates of Home and Foreign. Let us denote this world real interest rate by r*. In equilibrium we shall have, c! H 2 = c F 2 = 1+ r * (14) c H1 c F1 1+ ρ (14) denotes the Euler equations for the two economies! c H1 + c H 2 (15) 1+ r * = y + y H 2 H1 1+ r *! c F1 + c F 2 (16) 1+ r * = y + y F 2 F1 1+ r * (15) and (16) are the inter-temporal budget constraints for Home and Foreign respectively.! c H1 + c F1 = y H1 + y F1 (17)! c H 2 + c F 2 = y H 2 + y F 2 (18) (17) and (18) denote the resource constraints for the world economy, i.e. that in every period global consumption must be equal to global output. Solving these five equations for the five unknowns r*, ch1, ch2, cf1, cf2, using the given parameters for output and the pure rate of time preference, we get, r*=1.01, ch1=1.05, ch2=1.41, cf1=1.05, cf2=1.41!7
Professor George Alogoskoufis, International Macroeconomics Thus the Home economy will have a trade and current account deficit of 0.5 in the first period, and the Foreign economy will have a trade and current account surplus of 0.5. In the second period the situation is reversed. 9. The utility of the representative home consumer is equal to the utility of the representative foreign consumer, and is equal to 0.28. Thus, home consumers gain from inter-temporal trade, while foreign consumers do not lose out (in fact, if calculations are carried out to the fifth digit, foreign households have a small gain as well).!8