ECN101: Intermediate Macroeconomic Theory TA Section (jwjung@ucdavis.edu) Department of Economics, UC Davis November 4, 2014 Slides revised: November 4, 2014
Outline 1 2 Fall 2012 Winter 2012
Midterm: this Thursday (Nov. 6) in class Format: 10 MC + 2 SQ + 1 LQ for 80 minutes from 1:40 p.m. Bring a pencil or a pen a calculator (smartphone is not allowed) Scantron forms (UCD 2000 BLUE) no need to bring bluebooks
Suggestions/Tips Don t be late. You will need full 80 minutes for the exam. Also time management is enormously important in the exam. Don t spend too much time for just one question. You don t need to memorize basic equations such as 5 equations of the Solow model that will be provided on the exam. But you should know how to use them for questions. Some key concepts even from Chapter 1/2/3 can be asked in MC questions. Be familiar with them. You can get a hunch from past year exams. It is really important to go over past year exams ON YOUR OWN (i.e., without answer keys).
Suggestions/Tips During the exam, follow the instruction given in questions. Distinguish what is asked/what you need to answer first. Sometimes, you could miss what is asked, specifically when one question asks several things. Show your work. Even though some questions ask to get just simple numbers, it would be better to show your logic with sufficient verbal explanations and formulas. It will give you at least partial credit even if your answer is incorrect. When graphing please remember to label all axes, lines, curves, equilibrium points. Remember to try to provide an enough explanation of your answer. Try not to assume that TAs know what you re talking about when grading.
Question 1(4 points) In Germany the real GDP (in billions of Euros) in 2011 was 2,451, in 2012 the real GDP was 2,473. In 2011 the German nominal GDP was 2,495, while in 2012 the nominal GDP was 2,601. The inflation rate in Germany in 2012 was approximately equal to: (a) 0.9 % (b) 4.25 % (c) 3.35 % (d) 5.15% The answer is (c). NGDP = P RGDP % NGDP = % P + % RGDP g(x t) % X t = X t X 0 X 0 2601 2495 = π + 2473 2451 2495 2451
Question 2(4 points) In the Solow model, the economy s ( ) is equal to its ( ) (a) output; consumption (b) investment; consumption (c) fixed capital stock; investment (d) None of the above The answer is (d). (a): Y = C + I (b): I = sy (c): no K in the Solow model
Question 3(4 points) Consider the variables x, y. Suppose that x t = (1.01) t and y t = (1.03) t. What is the growth rate of z, where z = x 1/2 y 1/3? (a) 0.5% (b) -0.5% (c) 4% (d) 1.5% The answer is (b). ( ) 1/t x 0 = (1.01) 0 = 1, g(x) = xt x 1 = 0.01 0 g(y) = 0.03 g(z) = 1 g(x) 1 g(y) = 1 0.01 1 0.03 = 0.005 = 0.5% 2 3 2 3
Question 4(4 points) Consider the production function Y t = A 1/2 t K 1/3 t L 6/9 t, and recall the definitions of objects and ideas from Chapter 6. Which of the following statements is correct? (a) The production function exhibits CRS with respect to objects. (b) The production function exhibits CRS with respect to ideas. (c) The production function exhibits IRS with respect to objects and ideas jointly. (d) All of the above are correct. The answer is (a). Basic rule: replace the variable with doubled up value of the variable then check Y. (a): A 1/2 t (2K t) 1/3 (2L t) 6/9 = A 1/2 t 2 1/3 K 1/3 t 2 6/9 L 6/9 t = 2 1/3+6/9 A 1/2 t K 1/3 t L 6/9 t = 2Y t: CRS w.r.t. objects. (b): (2A t) 1/2 K 1/3 t L 6/9 t = 2 1/2 A 1/2 t K 1/3 t L 6/9 t = 2 1/2 Y < 2Y : DRS w.r.t. ideas. (c): 1 + 1 + 6 = 3 > 1, IRS w.r.t both objects and ideas jointly 2 3 9 2
Question 5(4 points) Consider two countries with fixed populations given by L 1 (country 1), and L 2 (country 2), and assume that L 1 < L 2. According to the Romer growth model, if everything else (except the populations given above) in these countries is exactly the same, then: (a) Per capita output growth will be higher in country 1. (b) Per capita output growth will be higher in country 2. (c) Per capita output growth will be the same in both countries, since per capita means that we divide by the country s population. (d) None of above is correct. The answer is (b). Basic Romer model: no capital, g(a) = g(y ) = g(y) = ḡ (balanced growth path) ḡ = z l L. g 1 < g 2 because L 1 < L 2
Question 6(4 points) In the production model of Chapter 4, the demand for labor is derived by (a) the worker s optimal consumption-saving choice. (b) the worker s optimal work-leisure choice. (c) the firms optimal choice of capital. (d) the firms optimal choice of labor. Answer is (d). (a): supply of K. Here it is exogenous (b): supply of L, Chapter 7. Here it is exogenous (c): r = MPK, demand for capital (d): w = MPL, demand for labor
Question 7(4 points) Consider the Solow model. Assume that the economy is at the steady-state. Which of the following will have the different effect on the capital stock? (a) The depreciation rate falls. (b) The savings rate goes up. (c) The level of TFP rises. (d) People become less patient towards future consumption. The answer is (d). ( 3/2 K = s d Ā) L (d): less patient = lower s
Question 8(4 points) Which of the following statements is false? (a) The Romer model has diminishing returns to ideas. (b) The Romer model does not exhibit transition dynamics. (c) The Romer model produces the desired long-run economic growth. (d) The Romer model focuses on the distinction between ideas and objects. The answer is (a) (a) and (d): IRS due to no depreciation of ideas (b): no transition. When anything happens, the growth rate of A suddenly changes. See Q1 in PS #3. (c): the reason why we need this model, compared to the Solow model.
Question 9(4 points) In 2012, a used-car dealer buys a 2007 model for $12,000, and sells it to a family for $14,000. By how much does 2012 GDP rise? By how much does current GDP rise? (a) $12,000 (b) $14,000 (c) $2,000 (d) $4,000 The answer is (c). The difference between $12,000 and $14,000 is a commission, which becomes a part of GDP.
Question 10(4 points) If the population of Bulgaria was about 24.5 million in 1960 and the average population growth rate is 0.7 percent per year, then Bulgaria s population would have been approximately ( ) in 2000. (a) 35.66 millions. (b) 32.38 millions. (c) 31.33 millions. (d) 34.18 millions. The answer is (b). X 2000 = X 1960 (1 + g) 2000 1960 = 24.5 (1 + 0.007) 40 = 32.38
Short Question 1(15 points) Consider a model of production similar to the one in Chapter 4. Final good (Y ) is produced using capital (K ) and labor (L) according to: Y = ĀK 1/3 L 2/3 (1) The supply of capital, labor, and land are given exogenously by K and L, respectively. Assume that K = 100 and L = 1, 000.
Short Question 1(15 points) Compared to the model that we discussed in class, we will make one important change: we will assume that the firms of this economy do not only own the technology described by the production function above, but they also own the capital (recall that in class we assumed that the capital stock was owned by the workers). Assume that all the markets where the agents of this economy interact are perfectly competitive (that is, in all market transactions agents are price takers).
Short Question 1(15 points) (a) Carefully state all the endogenous variables and the parameters of the model. (4 points) endogenous variables (what variables are you solving for?) : Y, K, L, w. No r because of no market for K parameters (what numbers/constants are given?) : Ā, K, L (b) If Ā = 100, calculate the equilibrium quantities Y, K, L. (4 points) : K = K = 100 and L = L = 1000 as given. Y = ĀK 1/3 L 2/3 = 100 100 1/3 1000 2/3 = 46415.89 (c) If Ā = 100, calculate the equilibrium price of all the inputs traded in this economy. (4 points) From the profit maximization problem of the firm max Π = Y wl Π L = 0 = w = MPL = 2 3 ĀK 1/3 L 1/3 = 2 Y 3 L = 30.94
Short Question 1(15 points) (d) In this economy, what is the share of output that is paid to labor? (for full credit show all your work, including the definition of the labor share of output) (3 points) : labor share of output (income) = W L From (c), w = 2 Y 3 L W L = 2 Y 3 Y
Short Question 2(15 points) Consider the Romer model of Chapter 6. Final good is produced using ideas (or technology), A t, and labor which is specifically assigned to final good production, L y t, according to the following formula: Y t = A t L yt (2) The stock of new ideas is given by A t+1 A t = za t L at (3) where L at is the amount of labor employed at the labs trying to come up with new technologies, and z = 0.0001. Also, assume that in the initial period, Ā0 = 100.
Short Question 2(15 points) The total supply of labor in this economy is fixed and given by L = 1000. Finally, this economy allocates 80% of its labor force to the production of final good. (a) In class, we discussed that ideas are non-rival, while objects are rival. Is this discussion somehow depicted in the formulas described above? (4 points) A t is used in both sectors. cf. L t = L yt + L at
Short Question 2(15 points) (b) What is the growth rate of output per person in this economy? What is the per capita output and the stock of ideas in t = 50 (i.e., y 50, A 50 )? (6 points) : g(a) = z l L, where l is the fraction of labor employed in the research sector l = 1 0.8 = 0.2. We also know that z = 0.0001 and L = 1000. So g(a) = 0.02. For given production function, g(y ) = g(a) + g(l y) = g(a) = 0.02 since L y is a constant. ( Y ) g(y) = g = g(y ) g( L) = g(y ) = g(a) = 0.02 L A 50 = (1 + g) 50 A 0 = (1 + 0.02) 50 100 = 269.16 y 0 = Y 0 = A 0L y0 L = 80 L y 50 = y 0 (1 + g) 50 = 215.33 or y 50 = A 50 (1 l) = 215.33
Short Question 2(15 points) (c) Suppose that starting in t = 51, the fraction of the labor force allocated to the production of new ideas becomes 0.3. What is the per capita output and the stock of ideas in t = 51 (i.e. y 51, A 51 )? Is there a time period after t = 50 in which per capita output equals y 50? If yes, what is that time period? (5 points) y 51 = Y 51 L = A 51L y,51 = A 51 (1 0.3) = 194.06 L y 51 < y 50. more labor for A and less labor for Y y t 51 time
Short Question 2(15 points) y 50 = y 51+t = (1 + 0.03) t y 51 = y 50 y 51 = (1 + 0.03) t = log ( y50 y 51 ( ) ) log y50 y 51 = t log(1.03) = t = log(1.03) = 3.52 y 54 < y 50 = y 51+3.52 < y 55, 4 years after t = 50
Long Question(30 points) This question is based on the Solow growth model of Chapter 5. Consider an economy where the period production function is given by Y t = K 1/3 t L 2/3 t, where Y t, K t, L t are output, capital, and labor in period t, respectively. In this economy, there is a fixed supply of labor equal to L, and output can be used for one of two purposes, consumption or investment: Y t = C t + I t (4)
Long Question(30+5 points) Also, in our economy capital depreciates at rate d per period. Therefore, the capital in period t + 1 can be described fully if one knows the capital in period t: K t+1 = K t + I t dk t (5) Assume that the capital stock of the initial period t = 0 is known and denote it by K 0. Finally, the agents in this economy invest a fraction s of the yearly output and consumes the rest. This implies that I t = sy t (6) Assume that L = 1, 000, s = 0.2, and d = 0.05.
Long Question(30 points) (a) Suppose K 0 = 10, 000. Draw a Solow diagram and explain what will happen to the capital stock of this economy in the short run and in the long run, having as starting point period t = 0. (6 points) : You need to draw a typical Solow diagram like, for example, Figure 5.1. The initial point is K 0 = 10, 000. In the long run we will reach the steady state, which can be found by solving sy t = dk t. Using the production function, the fact that L t = L in every period, and a little algebra, we find that K = ( s Ā ) 3/2 L = 8000 d The capital in our economy will converge to that level in the long run. In the short run, since we start with a level of capital bigger than K, we know that capital will decrease. In the beginning these downward changes will be big, but as time passes they will get smaller and smaller. i.e., the growth rate of K decreases.
Figure : Solow Diagram
Long Question(30 points) (b) What is the long run value of consumption per capita, c? (6 points) c = C L Y = K 1/3 L2/3 = 2000 (1 s)y 2000 (1 0.2) = = = 1.6 L 1000
Long Question(30 points) Suppose that this economy has been resting at the steady state for many years up to, and including, period, t = 1000, but then a preference shock takes place, so that starting in period t = 1001, the value of s rises to s = 0.3. The other parameters remain the same as above. (c) What will happen to the capital stock, K, of this economy, many many years after t = 1000 (in the long run)?. (6 points) : Just use the same numbers as before, except now s = 0.3 in the steady state K formula, ( ) ( ) 3/2 K = s d 3/2 L = 0.3 0.05 1000 = 14696.94. K > K
Long Question(30 points) (d) As a result of the change in s described above, what will happen to consumption in the short run and in the long run? More precisely, what is the value of consumption in period t = 1001 and how does it compare to the new steady state level of consumption (many years after t = 1000)? What is the intuition behind your results? (6 points) In the long run, C = (1 s )Y = 1714.64 > C In the short run, C t = (1 s)y t. C t can be lower than C K 1001 = K 1000 + s Y 1000 dk 1000 = 8200 L 1001 = L = 1000 Y 1001 = (K 1001 ) 1/3 (L 1001 ) 2/3 = 2016.53 C 1001 = (1 s )Y 1001 = 1411.57
Long Question(30 points) (e) Suppose that the saving rate is not a parameter ( s), but a variable (call it s) that a hypothetical decision maker could choose in order to maximize the total steady-state consumption of the economy. Express the objective function (that is, the total steady-state consumption) of this economy as a function of the saving rate s, and describe it graphically. What is the optimal choice s of the decision maker and how does it depend on the model s parameters L, d? (6 points) want to maximize C instead of max. Y or K with s C = (1 s)y = (1 s)(k ) 1/3 ( L) 2/3 ( ( ) ) 3/2 1/3 s = (1 s) L ( L) d 2/3 ( ) 1/2 = (1 s) L s d
Long Question(30 points) Figure : Consumption over s
Long Question(30 points) Find s subject to C s = 0 L d 1/2 [ s 1/2 + 1 2 (1 s)s 1/2 ] = 0 It is independent of L and d. = 1 2 (1 s) = s = s = 1 3
Fall 2012 Question 7(4 points) The income and expenditure approaches for measuring GDP: (a) Are the same if the income approach considers value added. (b) Are different measures. (c) Are the same if the expenditure approach excludes net factor payments. (d) Yield the same result for GDP no matter what. The answer is (d). All the approaches for measuring GDP yield the same result. income (Y = rk + wl) = expenditure (C + I + G + NX) = output (Y )
Fall 2012 Question 8(4 points) The president of the World Bank has asked you to calculate the average per capita GDP growth in Bulgaria from 1970 to 2000. In 1970, per capita GDP was about $3,600 and in 2010 it was about $13,500. Your answer would be: (a) 2.45 percent (b) 4.55 percent (c) 3.35 percent (d) 5.15 percent The answer is (c). ( ) 1/t g(y ) = Yt Y 1 from Yt = Y 0 (1 + g) t 0 ( ) 1/(2010 1970) Y2010 Y 1 = 0.03359. 1970
Fall 2012 Short Question 1(15 points) Consider a model of production similar to the one in Chapter 4. Final good is produced using capital (K ), labor (L), and land (X), according to the following production function: Y = ĀK 1/3 L 1/3 X 1/3 (7) The supply of capital, labor, and land are given exogenously by K, L, and X, respectively. Assume that K = 100, L = 1000, and X = 10. The demand for these inputs comes from firms that have access to the technology described above (the production function), and try to maximize their profits taking the prices of all factors as given (in other words, the markets for those factors are perfectly competitive). Let w denote the price of one unit of labor, r the rental rate of one unit of capital, and p the rental rate of one unit of land.
Fall 2012 Short Question 1(15 points) (b) Assume again that Ā = 100. Calculate the equilibrium values of the prices of the various inputs, w, r, p. (7 points) w = MP L = 1 3 ĀK 1/3 L 2/3 X 1/3 = 1 Y 3 L = 3.33 Y r = MP K = 1 3 ĀK 2/3 L 1/3 X 1/3 = 1 3 K = 33.33 Y p = MP X = 1 3 ĀK 1/3 L 1/3 X 2/3 = 1 3 X = 333.33
Fall 2012 Short Question 1(15 points) (c) In equation (1), the labor is raised to the power of 1/3. Do you think this is a good idea in light of data regarding labor income in the US in the last 60 years? Explain. (4 points) : From the model with the Cobb-Douglas production function, the exponent power for each factor, is the share of that factor s payment out of the total output. See (b). wl = 2. Y 3 From the US data that in the last (at least) 60 years, the share of labor has been roughly 2/3. Therefore, the choice of raising labor in the C-D function to the power 1/3 is not a good one.
Fall 2012 Short Question 2(15 points) Consider the Romer model of Chapter 6. Final good is produced using ideas (or technology), A t, and labor which is specifically assigned to final good production, L y t, according to the following formula: Y t = A t L yt (8) The stock of new ideas is given by A t+1 A t = za t L at (9) where L at is the amount of labor employed at the labs trying to come up with new technologies, and z = 0.002. Also, assume that in the initial period, A 0 = 1000.
Fall 2012 Short Question 2(15 points) The total supply of labor in this economy is fixed and given by L = 100. Finally, this economy allocates 75% of its labor force to the production of final good, that is: L yt = 0.75 L. (b) What is the value of per person output in the economy in t = 20? How many years (from t = 0) will it take for output per person to double?(5 points) : y 20 = (1 + g(y)) 20 y 0 = (1 + 0.05) 20 y 0 = 1989.97. Use the rule of 70. For any variable that grows with an annual (and constant) rate of g percent, the number of years it takes this variable to double is 70/g = 70/7.5 = 9.33 years.
Winter 2012 Question 9(4 points) In the equation Y = F (K, L) = Ā K 1/3 L 2/3, the lack of a bar over L means that it is (a) an exogenous variable (b) a constant (c) a parameter (d) an endogenous variable The answer is (d). bar means the variable with bar is constant and a given value. So now L should be determined in the model: an endogenous variable