Macroeconomics Exam & Solution, UPF/BGSE MSc Economics Professor Antonio Ciccone

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1 Macroeconomics Exam & Solution, UPF/BGSE MSc Economics Professor Antonio Ciccone Question : Consider two Solow economies that have identical savings rates, population growth rates, and depreciation rates; the rate of technological change is equal to zero. The two economies also start out with the same amount of capital per worker. The aggregate production function in the two economies is given by Y = K ( EL) where E is the level of technology in labor-augmenting form. The only difference between the two economies is that economy starts out with a higher level of E than economy 2. a) Show that the economy with higher E starts out with higher real interest rates and higher real wages rates. Calculate the MPL and the MPK using the CD production function given above, which are equal to w and r+delta in a competitive equilibrium. From the expressions for MPL and the MPK it is easy to see that, as K/L is the same in the two countries, it is the country with higher E that has higher wages and interest rates. b) Show that the two economies have the same real interest rates in the steady state. Show this analytically as well as graphically. Explain intuitively (very briefly). Recall that the MPK depends on the capital per efficiency worker (ktilde) only in the Solow model (this can be also be shown immediately for the CD production function given above which implies MPK K ( EL) k = = % ). Hence, the two countries will have the same MPK and interest rate in the SS if they have the same ktilde (defined as K/(EL)). That the two countries have the same ktilde follows from the fact that the have same s, n, delta, and production function. You can see that graphically by putting down the very first of the Solow diagrams. Analytically it follows from the capital accumulation equation of the Solow model when the growth of ktilde is equal to 0 (which is the case in the SS). c) Show that the economy with higher E has a higher real wage rate in the steady state. Explain intuitively (very briefly). Is the percentage difference in real wages between the two economies higher in the beginning or in the steady state?

2 With the CD production function, wages are proportional to income per worker. Income per worker is higher in the country with higher E in the SS because both countries will have the same ktilde, and income per worker is y = Ey% = Ek % (the last equality is using the CD production function, but that would not be necessary to make this point). The difference in wages is larger in the SS than the beginning. Intuitively this is because in the beginning countries differ in E but have the same K/L. In the SS, the country with higher E will also have a higher K/L. hence, workers (wages) will not just benefit from high E but also from high K/L. Formally, w = ( ) y = ( ) Ey% = ( ) Ek%. Hence, as the ktilde are the same in SS, SS differences in wages are proportional to differences in E. This is not true in the beginning however where wages differ by less that the E because the K/L are identical for the two countries (just calculate the MPL as in part (a)). d) Suppose that these economies liberalize their financial markets and that capital therefore flows towards the country with the higher MPK. Will capital flow from the country with low total factor productivity to the country with high total factor productivity? In the SS, the MPK and real interest are the same and so capital does not flow. In the beginning capital would flow from high TFP to low TFP country because high TFP country has higher real interest rate (recall that K/L is the same in both countries in the beginning) e) Explain very briefly what should happen to wages and output per worker if the countries had allowed free cross-border flows of workers but not capital? Workers should go to high MPL (w) country which is the one with high TFP. Eventually migration should equalize wages and, because w= ( ) y with CD production function, output per worker. 2

3 Question 2: Consider the Ramsey-Cass-Koopmans model without technological change and without population growth. The capital depreciation rate is δ and the aggregate production function is given by Y = AL+ BK L. Suppose the economy starts in steady state. a) Show graphically the adjustment path for consumption and capital if there is an unanticipated and temporary fall in A. Explain intuitively. (You do not need to derive the Euler/consumption profile equation.) This corresponds exactly to the downward shift in the k-isocline that we analyzed in class (SLIDE SET 4; SLIDES -6). The key is to see that a change in A does not affect the MPK for a given K/L, which means that the c-isocline stays unchanged. Hence, what happens in the usual RCK graph is that the hill shifts temporarily down but then moves but again. The details are in SLIDE SET 4; SLIDES -6. b) What if instead there is an unanticipated and temporary fall in B. Explain intuitively the differences with the scenario in (a) above. The difference with (a) above is that now the hill (the k-isocline) shifts down and, at the same time, the vertical c-isocline shifts left. This is because a decrease in B lowers output for a given K/L and, at the same time, lower the MPK for a given K/L. STARTING POINT c Temporary, anticipated fall in B c-isocline: NO CONSUMPTION GROWTH k-isocline: NO CAPITAL GROWTH 0 k* k BGSE/UPF Macroeconomics, SLIDE SET 4 SLIDE 3

4 TEMPORARY FALL IN B Temporary, anticipated in B: PART II c NEW c-isocline: NO CONSUMPTION GROWTH NEW k-isocline: NO CAPITAL GROWTH 0 k* k BGSE/UPF Macroeconomics, SLIDE SET 4 SLIDE 2 RETURN OF B TO ORIGINAL VALUE Temporary, anticipated in B: PART II c OLD c-isocline: NO CONSUMPTION GROWTH SADDLE PATH OLD k-isocline: NO CAPITAL GROWTH 0 k* k BGSE/UPF Macroeconomics, SLIDE SET 4 SLIDE 3 It is also possible that the green path hits the saddle path before crossing the new c- isocline. In this case the path gose down and left until it hits the saddle path and then goes up and right. 4

5 Question 3: Consider the Ramsey-Cass-Koopmans model without technological change, without population growth, and without capital depreciation. The aggregate production function is given by Y = K L. Suppose that the government puts a tax τ on the capital income of households. a) What is the relationship between the real interest rate earned by households after taxes and the marginal product of capital? In equilibrium, the interest rate paid by firms to banks will still be the MPK. But households will receive the after tax MPK only: N G r = ( τ) r = ( τ) MPK = ( τ) k Where N stand for net and G for gross; the last equality makes use of the CD production function. b) Calculate the steady-state capital intensity in the economy as a function of the tax on capital and the rate of time preferences of households. Graph the economy s steady-state capital intensity as a function of the tax rate. (You do not need to derive the Euler/consumption profile equation.) In SS, the Euler equation yields ρ = ( τ) k And this can be solved for SS k k SS ( τ ) = ρ c) Find the expression for total steady-state tax revenues from the capital income tax. Graph government tax revenue as a function of the tax rate. Explain why government tax revenues will eventually fall as the capital income tax increases. Tax revenue will be G r k = kmpk = k k τ τ τ 5

6 which simplifies to τ k Now we just need to plug in the expression for the SS k to get τ k ( τ ) = τ ρ this expression is hum-shaped in tau. Tax revenue can be a decreasing function of the tax rate because investment and ultimately the SS capital stock fall as taxes go up (a Laffer curve argument for those of you who have done some taxation theory). d) Find the capital income tax that maximizes the government s tax revenues in steady state. Here you had to maximize τ k order condition is ( τ ) = τ ρ ( τ ) ( τ ) = 0 ρ ρ ρ which simplifies to with respect to tau. The first ( τ) = ρ ρ or τ = or τ = 6

7 Question 4: Consider two Solow economies that have identical savings rates, population growth rates, and depreciation rates; the rate of technological change is equal to zero. The two economies also have the same number of workers and start out with the same amount of capital per worker. Workers work 8 hours per day 250 days per year. Output per year is given by Y = K L where K is the capital stock in the economy and L the total number of workers. The only difference between the two economies is that in economy workers work in shifts. Half of the workers work 8 hours during the day shift and the other half use the same capital stock for 8 hours during the night shifts. a) What is total output in economy? (Start by writing down output per year produced by workers working the day shift and output per worker produced by workers working the night shift.) Which economy produced more output per worker and why? Calculate the percentage difference in output per worker /2 assuming that = /3. (One of the following results may be useful: 2.4; /3 2 =.25; /4 2 =.2; /2 3 =.7; /3 3 =.45.) Yearly output for a give capital stock and number of workers is equal to Y = K L in the economy where people do not work shifts. Shift work implies that the day shift produces Yday = K ( L /2) as only half of the workers work this shift. The night shift produced with the same capital stock and yields Ynight = K ( L /2). The sum of these is yearly output in the economy working shifts. Y = 2 K ( L/2) = 2 K L Hence it is as if the shift economy has higher TFP (TFP is 2 in the shift economy versus in the non-shift economy). This expression yields immediately that output per worker in the shift economyc is /3 2 =.25 times output per worker in the non-shift economy (for alpha=/3). I.e. 25% higher! You could also think in terms of labor-augmenting efficiency. In this case the shift economy ouput should be written as /( ) Y = 2 K ( L/2) = K (2 L) 7

8 Hence it is as if labor-augmenting technology in the shift economy was versus in the non-shift economy. /( ) 2 b) What is the steady-state difference in output per worker between the two /2 economies? (One of the following results may be useful: 2.4; /3 /4 /2 /3 2 =.25; 2 =.2; 3 =.7; 3 =.45.) The two economies are identical in everything except (labor-augmenting technology). Recall that in the SS y = Ey% Where E is labor-augmenting technology. As the two economies are identical in s,n, delta, they have the same ytilde. So SS output per worker only differ because of the /( ) E. What is the difference in the E? Recall from above that E = 2 in the shift economy and in the non-shift economy. Hence, in the SS, output per worker in the (/3)/(2/3) shift economyc is 2 =.4 times output per worker in the non-shift economy (for alpha=/3). I.e. 40% higher! Why is th edifference bigger in the SS? Because of the usual readon: capital accumulation response to the higher efficiency and the shift economy will therefore have more capital per worker in the SS. 8