Answer Key Midterm Exam for Economics 122a. Part I. Maximiliano Appendino

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1 Answer ey Midterm Exam for Economics 122a I. Shorties (25 minutes total) A. Part I Maximiliano Aendino a. FALSE. The CPI is an uward biased measure of inflation because it is a Laseyres index. For examle, the CPI does not reflect the substitution away from goods whose rices increase the most because it weights changes in rices with the fixed weights calculated using quantities in the base year. b. UNCERTAIN. A higher interest rate will have two effects: substitution and income effect. The substitution effect will lead you to save more (moving you from oint A to oint B in the diagram below) but the income effect will lead you to save less because you are a saver (you receive income only in the first eriod) and the higher interest rate means higher income in the second eriod making you richer. This way, the statement would be true if the substitution effect dominated the income effect (as in oint C) but would be false in the oosite case (in oint D). C2 (1+r )Y1 C (1+r)Y1 B A D C1 Y1 c. FALSE. In the labor-dynamics model, the Beveridge curve is the inverse relation between the job vacancy rate and the unemloyment rate given the job destruction rate and the matching function. If the matching function imroves, the Beveridge curve will shift inwards because for each level of the job vacancy rate a lower unemloyment rate will be observed because more unemloyed workers will find their match more frequently. 1

2 B. (10 Minutes) In the classical model, we assume that national outut, Y, is roduced with a roduction function, (, L), which exhibits ositive marginal roduct, F(,L) > 0 and F(,L) > 0. This way, if the immigration wave increases the labor force from L to L > L, national outut will increase Y = (, L) < (, L ). We also assume that factor markets are cometitive, therefore, the initial wage rate is W = F(,L). Since the roduction function exhibits diminishing P < F(,L), then, the immigration wave decreases the wage rate. returns, F(,L ) This was exected because the immigrants are substitutes of the existing workers. w (, L) W W L L L Total rentals on caital are equal to the rental rate, r, multilied by the P caital stock,. The caital stock does not change after the immigration wave, then, total rentals move in the same direction as the rental rate. As in the case of the wage rate, the rental rate is R = F(,L). Given that we also P assumed that the roduction function exhibits constant returns to scale (CRTS) we have that (, L) = L L 1 L,1 F(,L). This way, L F L,1 = F L, 1, then, F(,L) = F L,1 > F L,1 = F L,1 = F(,L ) = L F L,1 = because of the diminishing returns on caital, then, both the rental rate and total rentals on caital increase after the wave of immigration. This result was also exected because caital is a comlementary factor to labor. 2

3 r (, L ) (, L) R R The initial share of labor in national income is W L W L Y and after the immigration wave it is. We know that Y L > L, W < W and Y < Y, so we cannot conclude in which direction the share of labor in national income will change. The more inelastic is the demand for labor, F(,L), the lower will be the wage rate after the immigration wave and, therefore, the more likely the reduction of the share of labor in national income. In the articular case of the Cobb-Douglas roduction function, the labor income increases roortionately to the increase in outut and, as we know, the share of labor in national income is unchanged and equal to the exonent of labor: W L Y = α L 1 α L α L 1 α = α (1 α)l α L α L 1 α = 1 α = α (1 α)l α L α L 1 α = α L 1 α L α L 1 α = W L Y. 3

4 Solution for Midterm Question II ECON 122a Fall 2012 October 8, Economic Analysis in Marginalia 2.1 Since the roduction function exhibits constant returns to scale, we can write F (, L) = LF L, 1. Thus, the er-worker roduction function is y = f (k) =Ak 1 3 (1) 2.2 The Solow diagram is shown in Figure 1. Figure 1: The Solow diagram 1

5 2.3 Since k = L, the growth rate of caital er worker is k k = = s Y L L L (sy ) = {z} L n n = s y k n n = s Y n which gives At the steady state, k = sf (k) k =0. Thus, we obtain ( + n) k sf (k )=( + n) k Using (1), we can solve for k and y as 2.4 k = sa + n y = sa A + n = A 2 1 s 2 + n (2) (3) We need to comute the elasticity of k and y with resect to s. The elasticity of k is E Using logs, we can also exress the elasticity as s k E k,s ln ln s Taking the log of (2) yields ln k = 3 (ln s +lna ln ln n) 2 Thus, the elasticity of k with resect to s is 3 2. Similarly, the elasticity of y with resect to s is 1 2. Thus, when the saving rate increases by 1%, the steady state caital er worker increases by 1.5% and outut er worker increases by 0.5%. 2.5 The grah is shown in Figure 2a. At t = T, y jums u due to the technology shock and c jums roortionally as well since c =(1 s) y. Since the new technology generates more outut and more savings, caital er worker starts growing. Both new steady state caital 2

6 (a) A favorable technology shock (b) A saving rate jum Figure 2: Time aths of k and c and consumtion er worker are greater than the original levels (k >k and c >c ). 2.6 The grah is shown in Figure 2b. At t = T, consumtion er worker dros since s has jumed. After that, k starts growing due to additional savings. The new steady state caital er worker is higher than the original level (k >k ). Whether c is higher or lower than c deends on the Golden Rule saving rate s golden.if s<s 0 <s golden,weknowthatc >c.ifs golden <s<s 0, then c <c.ifs<s golden <s 0, either case c can be higher or lower than c. 3

7 III. (25 minutes) Life cycle after Yale Jean, a enniless 20-year old Yale senior about to graduate, recently listened to an old recording of Andrew Carnegie in which he discusses the morality of wealth. Imressed and insired, Jean decides to send the next twenty years as a researcher in the clean energy industry, earning $60,000 er year (in constant 2012 dollars). At age 40, Jean uses the hard-earned knowledge to start a fuel cell comany, after which annual income jums to $3,000,000. Following in Carnegie s footstes, Jean decides to retire at age 60, and devote the rest of life to charitable causes. He dies at age 80, having given away the wealth. Assume Jean considers charitable donations as art of consumtion, and that the real interest and discount rates are zero. A) For Part A, solve Jean s life-cycle lan at age 20. 1) Assume Jean can borrow and lend without limit, and is enniless at age 20. Comute consumtion exenditures for each remaining year. Jean earns $60,000 er year over the first 20 years ($1,200,000), and $3,000,000 er year over the next 20 years ($60,000,000). In total, his lifetime income is $61,200,000. Since we ve assumed the real and discount rates are zero (and assuming concave utility), then we know Jean will consume an equal amount each year. He has (80-20) = 60 years remaining, so his consumtion exenditure each year is $61,200,000/60 = $1,020,000. 2) Grah Jean s income, consumtion, and savings as a function of age. Recall that Savings (S) = Income (Y) Consumtion (C). 1

8 3) Draw a lot of Jean s wealth as a function of age. Wealth is defined to be lifetime savings. Since Jean borrows for the first twenty years, his wealth slowly declines. At 40, his income level greatly increases and his wealth begins to increase until age 60, when he retires. His wealth decreases until age 80, at which oint he dies enniless. At age 20, his savings are S = Y-C = 60,000 1,020,000 = -960,000. This is the first oint. Age 39 is the last year he has negative savings, and at this oint he has had savings of -960,000 for twenty years, so his total wealth at age 39 is -960,000*20 = -19,200,000. At age 40, his savings are S = Y-C = 3,000,000 1,020,000 = 1,980,000. Thus, at age 40 his income begins to increase. It increases for twenty years, so his wealth at age 59 is -19,200, *(1,980,000) = -19,200, ,600,000 = 20,400,000. At age 60, Jean retires and has no income. He begins giving his wealth away, and his total exenditures are 1,020,000 er year. His wealth decreases by this amount each year, at age 79 he consumes the remainder of his wealth, and when he turns 80 he dies instantaneously. 2

9 B) Now assume that Jean suddenly learns about liquidity constrained and cannot borrow. 1) Reeat art A (3). If Jean is borrowing constrained, then he at most can consume $60,000 er year for the first 20 years. He is unable to borrow to smooth consumtion. At age 40 his income increases and he is no longer constrained. He smoothes consumtion over the remaining lifetime income, which means he consumes (3,000,000*20)/40 = 1,500,000 er year. During his first 20 years, his savings are zero each year, so wealth is zero until age 40, when he starts earning $3,000,000. He consumes $1,500,000 of this and saves the other half. His wealth steadily increases by $1,500,000 er year for 20 years. Age 59 is his last year working, and his wealth increases to 20*$1,500,000 = $30,000,000. He consumes from this wealth over the remaining 20 years, finally ending his life at 0 wealth. 3

10 C) (This question illustrates the issues raised by Ricardian equivalence and its imlications for using taxes for stabilizing the business cycle.) Jean is still liquidity constrained. Because of a recession, the government announces a lan that will give Jean a tax cut of $3000 in year t with an offsetting tax increase of $3000 in year (t+10). What would be the effect on Jean s consumtion and saving if year t is: 1) when Jean is 35 years old? Jean would like to smooth consumtion, but is unable to do so because he is liquidity constrained. The unexected tax ackage increases his income by $3,000 when he is 35 (liquidity constrained) and decreases his income by $3,000 when is is 45 (no longer liquidity constrained). To smooth income, he sreads the $3,000 over the next five years (ages 35-39, inclusive), that is, he increases consumtion by $3,000/5 = $600 from ages 35 to 39. This means that at age 35 he saves $3,000-$600 = $2,400, and over the next four years he consumes $600 out of his savings. The second art of the tax reduces his income by $3,000 at age 45, and he knows this at age 35. Thus, during the eriod of his life when he is no longer liquidity constrained (ages 40 to 80), his income has decreased by $3,000. To smooth consumtion over this eriod, he thus consumes $3,000/(80-40) = $75 less each eriod. Since S = Y-C, we know his savings are increased by $75 each eriod, excet at age 45 where savings have decreased by 4

11 $2,925 (because at 45, Y has decreased by $3,000 and C has increased by $75) 2) when Jean is 50 years old? When Jean is 50 years old, he is no longer liquidity constrained. The $3,000 increase is offset by the later $3,000 decrease, and his remaining lifetime income is unchanged. He smoothes consumtion over his remaining lifetime income, thus his consumtion is unchanged because remaining lifetime income is unchanged. Since C stays the same, we can see from S = Y C that he saves an additional $3,000 at age 50 and saves $3,000 less at age 60. 5