Econ 102 Mid Term 1 & 2 Solutions
Econ 102 Mid Term 1 & 2 Solutions Econ 102
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This 19 page Bundle was uploaded by Nick Zalevskiy on Sunday March 15, 2015. The Bundle belongs to Econ 102 at University of California - Los Angeles taught by Geerolf in Winter2015. Since its upload, it has received 90 views. For similar materials see Macroeconomic Theory in Economcs at University of California - Los Angeles.
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Date Created: 03/15/15
Macroeconomic Theory 102 Last Name Winter 2015 Francois Geerolf Midterm 1 First Name Wednesday January 28 2015 Time Limit 75 Minutes Teaching Assistant Student ID Number Signature Test A This exam contains 9 pages including this cover page You can earn 100 points 6 bonus points If you are stuck at some point don t forget to answer the easy questions worth many points there are many of them Instructions 1 Print your Last name First Name Teaching Assistant Name as a reminder teaching assistants are Flavien Moreau Keyyong Park Matias Vieyra and Gabriel Zaourak Student ID Number and Signature at the top of this page 2 The only items which should be on your desk are pencils and or pens NO other items are allowed Place any other item UNDER your desk Calculators are NOT allowed 3 Once the exam begins you are not allowed to leave the room until you hand in your exam Good luck Budget your time wisely skip the question or even the exercice if you get stuck Grade Table FOR TEACHER USE ONLY Question Points Score 1 10 2 14 3 6 4 20 5 20 6 36 Total 106 Macroeconomic Theory 102 Midterm 1 Page 2 of 9 Wednesday January 28 2015 Multiple Choice 10 points 1 10 points These are multiple choice questions Mark box if true a 2 points If per capita GDP in 2013 was 20000 and per capita GDP in 2015 is 20200 then how much approximately was the growth rate of per capita GDP every year Q 1 Q 10 05 2 points Over the past 50 years Brazil s population growth rate has averaged about 23 percent According to the rule of 70 Brazil s population will double in about 0 3 years 30 years 0 33 years 0 161 years 0 16 years 2 points The key difference between the Solow model and the production model 182 The Solow model endogenizes the process of capital accumula tion 0 The standard model endogenizes the process of capital accumulation Q The Solow model uses different values for the capital share 0 The Solow model does not contain a productivity measure 0 The Solow model exogenizes the process of capital accumulation 2 points In the Solow model if we assume that capital depreciation rates are the same across all countries differences in per capita output can be explained by O the steady state capital stock 0 the initial capital stock and saving rates differences in productivity and saving rates 0 the labor stock and saving rates 0 None of these answers are correct 2 points An implication of the Solow model is that once an economy reaches the steady state per capita consumption is constant 0 per capita output is constant but per capita capital is not 0 per capita capital is variable 0 per capita consumption continues to grow 0 per capita consumption is growing Macroeconomic Theory 102 Midterm 1 Page 3 of 9 Wednesday January 28 2015 Two Questions directly from the course 20 points 2 14 points Assume that production is given by the following production function Y 2 AthLS and that At A Give short answers you won t be penalized if you do not write sentences Go quickly to the next sections a 2 points What is the name of this production function Solution This is a Cobb Douglas production function b 2 points What is the relationship between a and b in this production function if returns to scale are constant with respect to capital and labor Solution a b 1 c 2 points Why do macroeconomists use this production function a power function of the quantity of capital and the quantity of labor What statistical regularity makes them con dent to use this function Solution The fact that the share of labor in GDP and the share of capital in GDP are constants For example the marginal product of capital shows that the share of capital is constant Y TK t gt t tza 730 Kt Y2 Similarly the marginal product of labor is given by Y L thb t gt wt t b Lt Y2 d 2 points What are a and 9 equal to typically Solution a 13 and b 23 e 2 points Why are they equal to these numbers Solution The share of capital in value added is approximately 13 and that of labor is approximately 2 3 f 2 points Write an example of a production function with the same inputs tech nology At capital Kt and labor Lt but decreasing returns to scale with respect to capital and labor Macroeconomic Theory 102 Midterm 1 Page 4 of 9 Wednesday January 28 2015 Solution There are many options One is Y Athl4Ll4 g 2 points Write an example of a production function with the same inputs but increasing returns to scale again with respect to capital and labor Solution Again many possibilities One is Y 14th 3L 3 but you could use your imagination 3 6 points Quantity indexes a 3 points De ne the Laspeyres index and the Paasche index Solution The Laspeyres index calculated changes in real GDP using the initial prices The Paasche index in contrast calculates changes in real GDP using the nal year prices b 3 points What is the Fischer or chain weighted index Solution It is an average of the Laspeyres and the Paasche indexes Exercice 1 20 points 4 20 points Consider the following model of the labor market with labor supply L5 and labor demand L0l respectively given as a function of the wage by L5 711 Ld f w a 2 points In the following box represent this model graphically as well as its equilibrium L Represent an increase in EL on the graph using a dotted line Represent the effect of this increase in EL on the wage and the effect on employment Solution See the TA section b 2 points Again represent the labor market model in the following box but this time consider a decrease in f Represent the effect on the wage and the effect on employment Solution See the TA section Macroeconomic Theory 102 Midterm 1 Page 5 of 9 Wednesday January 28 2015 c 4 points One last time represent the labor market model in the following box but consider the introduction of a proportional taX 739 on wages such that when w is paid by employers employees actually only receive w1 739 Show the triangle corresponding to deadweight loss also known as Harberger s 1924 triangle Solution See the TA section d 4 points Solve analytically for the equilibrium values of L and w in this model with no taxes Solution Using a little algebra one can show that w L 1a 1a e 4 points Show analytically the effect on the L and w of the two previous exper iments increase in EL decrease in f Solution When EL increases 71 decreases You can see that through the deriva tiveor just because EL is in the denominator and the wage has to be positive so f l The comparative statics with respect to EL are slightly more complicated 8L f l 85 1 5 Because the wage is positive in equilibrium this derivative also is positive f 4 points Show analytically the effect on the equilibrium wage w and employment L of the tax 739 as a function of 739 Solution Labor demand is given by what the employers pay wage 71 but employees receive only w1 739 so that L5ELw1 739 l Ld f 71 Note that it is as if EL had switched to EL1 739 Therefore M f z Za l THf 1EL1 739 1EL1 739 Of course you could as well do the same calculations again Therefore the tax 739 increases the wage paid by the employer by the following amount of course what the employee gets w1 739 decreases 9711 f l a 9739 1 EL1 7392 gt0 Macroeconomic Theory 102 Midterm 1 Page 6 of 9 Wednesday January 28 2015 The tax leads to the destruction of jobs an 87 1 1 72 Exercice 2 20 points 5 20 points Consider the following Solow model Output Y is produced using capital Kt and labor Lt according to the production function Y AXE11 note the exponents The amount of labor is xed Lt E and a fraction of s of output is saved by agents in this economy so that investment is It 2 sY Capital depreciates every period at a rate d a 4 points Write the laW of motion for capital Solve for the steady state value of capital K and the steady state value of Y in this economy Solution AKt l l Kt1 Kt Y2 CZKt gAK34E34 3K Steady state obtains With AKtH 0 so that Y 2 JK The production function gives a second equation Y A K14 E34 Again this is a two equations two variables system Whose solution is A 43 13 43 K7 L 13 A b 4 points Comment the dependence of Y With 1 Compare With the production model Why is there a difference Solution There is an exponent 4 3 The production function has linearity in total factor productivity The reason is that capital accumulation ampli es productivity differences because more productive countries accumulate more capital Which increases their production further Macroeconomic Theory 102 Midterm 1 Page 7 of 9 Wednesday January 28 2015 c 4 points Imagine that the government can enact policies aimed at targeting peo ple s savings rate g Which savings rate would maximize steady state production What would steady state consumption then be equal to Solution Y is maximized when savings are maximum so the savings rate maximizing steady state production would be s 1 However people would be starving in this economy All its output would be used towards increased capital accumulation d 4 points Using that the steady state value for consumption is equal to C 1 Y show that the savings rate s has two opposing effects on consumption Give an intuition Solution 1 g decreases in while Y increases in g On the one hand steady state consumption increases when savings increase because there is then more production But on the other consumption decreases because a higher fraction of production is then saved rather than consumed e 4 points Adam Smith once wrote Consumption is the sole end and purpose of all production Calculate the steady state consumption maximizing level of the savings rate s in the Solow model Solution You need to maximize the following g 13 43 Y 1 s A d The rst order condition gives note you actually just need to maximize 1 13 8Y 3 s 1 1 0 gt 13 231 0 gt Z25 Exercice 3 30 points 6 bonus points 6 36 points We consider an economy with the following production function Y 14K 3L 3Nt where the amount of land Nt is also entering as an input in the pro duction function rms need land to produce for example because they need of ces Assume that the number of people working is xed and given by 1 Capital depreciates at rate d and the savings rate is constant equal to g It is also assumed that there is a xed quantity of land in the economy given by N a 6 points Write the law of motion for capital as a function of Y and Kt Macroeconomic Theory 102 Midterm 1 Page 8 of 9 Wednesday January 28 2015 Solution AKt l l Kt1 Kt Y2 CZKt b 6 points Replace Y in the previous expression to derive the law of motion for the capital stock where only the capital stock and exogenous variables appear Solution AK1 AZ23NK13 JKt c 6 points Explain using a graph as well as words why the level of capital converges to a steady state Take the example of an initially lower than steady state amount of capital K0 lt K note that the introduction of land does not fundamentally change the capital accumulation process as land is assumed to be in constant supply Solution See the TA section Solution Same as Solow replacing the constant 14 11 123 by 14 11 12 3N d 6 points What is the marginal product of land Solution The marginal product of land is given by m AE23K13 8N t e 6 points Imagine the economy starts with an initially too low level of capital relative to the steady state level How does the marginal product of land varies over time Why Solution As the capital stock increases the marginal product of land increases from the previous equation This is intuitive land being in xed supply its scarcity is becoming all the more limiting as the economy produces a lot This helps explain why the price of land in practice covaries with GDP in booms house prices are high while they are low in recessions It also helps explain why China has had soaring real estate prices recently as its economy was accumu lating more capital or as it was becoming more wealthy in general Macroeconomic Theory 102 Midterm 1 Page 9 of 9 Wednesday January 28 2015 f 6 points Bonus extend this model to one Where there is a research sector a production sector and ideas accumulate over time Go as far as you can in that investigation if needed use the back of the sheet Solution Again see thenotes posted on the Website except you replace the constant AL23 by AL23N Macroeconomic Theory 102 Last Name Winter 2015 Francois Geerolf Midterm 2 First Name Wednesday February 25 2015 Time Limit 75 Minutes Teaching Assistant Student ID Number Signature Test A This exam contains 10 pages including this cover page You can earn 100 points Instructions 1 Print your Last name First Name Teaching Assistant Name as a reminder teaching assistants are Flavien Moreau Keyyong Park Matias Vieyra and Gabriel Zaourak Student ID Number and Signature at the top of this page 2 The only items which should be on your desk are pencils and or pens NO other items are allowed Place any other item UNDER your desk Calculators are NOT allowed 3 Once the exam begins you are not allowed to leave the room until you hand in your exam Good luck Budget your time wisely skip the question or even the exercice if you get stuck Grade Table FOR TEACHER USE ONLY Question Points Score 1 30 2 15 3 5 4 50 Total 100 Macroeconomic Theory 102 Midterm 2 Page 2 of 10 Wednesday February 25 2015 Multiple Choice 30 points 1 30 points Mark box if true each multiple choice question has only one right answer a 2 points If Pt is the price level in time t then in ation is calculated as 0 1Pt 0 Pt1Pt O Pt1 Pt 0 Pt Pt1 Pt1 PtPt b 2 points In the United States money is backed by 0 oil 0 gold 0 silver no physical commodity 0 None of these answers are correct c 2 points According to the quantity theory of money the price level is Q Exogenous O Determined by the money supply only Determined by the ratio of the effective quantity of money to the volume of goods 0 Indeterminate in the long run 0 Determined by the volume of goods produced d 2 points Net worth is equal to a bank s 0 investments minus deposits 0 cash plus reserves 0 deposits plus loans 0 loans minus capital total assets minus total liabilities e 2 points Using the IS curve Z 2 EL 5Rt 77 in the long run EL and so that 0 equals one Rt 2 77 the economy is in recession 0 is greater than one Rt gt 77 the economy is at its long run equilibrium equals zero Rt 2 77 the economy is at its longrun equilibrium 0 equals one 5 2 EL the economy is expanding 0 equals one Rt 2 1 the economy is in recession f 2 points Consider the consumption function CtYt 2 Etc If 5 05 a 2 percent demand shock Macroeconomic Theory 102 Midterm 2 Page 3 of 10 Wednesday February 25 2015 O raises short run output by 1 percent 0 raises short run output by 05 percent raises shortrun output by 4 percent 0 reduces short run output by 4 percent 0 has no impact on short run output g 2 points With adaptive expectations the Phillips curve can be written as Am 2 BY 0 A70 2 7Tt 1 l WHYt 0 7 t 7Tt1 17 0 Am 2 But 0 7Tt 7Tt 1 h 2 points Which of the following best describes why the aggregate demand curve slopes downward Q If the central bank observes a low rate of in ation the monetary policy rule dictates an increase in the real interest rate The high interest rate reduces output by reducing investment demand in the economy 0 If the central bank observes a high rate of in ation the monetary policy rule dictates a decrease in the real interest rate The low interest rate increases output by reducing investment demand in the economy If the central bank observes a high rate of in ation the mon etary policy rule dictates an increase in the real interest rate The high interest rate reduced output by reducing investment demand in the economy 0 If the central bank observes a low rate of in ation the monetary policy rule dictates a decrease in the real interest rate The low interest rate reduces output by reducing investment demand in the economy 0 None of these answers is correct i 2 points The adjustment process back to the steady state in the short run model hinges on the 0 rate of unemployment O immediate reaction to a change in the in ation rate Q consumers response to in ation shocks 0 government s response to in ation shocks slow adjustment of in ation re ected in the aggregate supply curve j 2 points Which of the following represents the AD curve with a nancial friction za bf BmQrt Macroeconomic Theory 102 Midterm 2 Page 4 of 10 Wednesday February 25 2015 k 2 points The Fisher equation is given by O W 7Tb 12Yt O p Q Am 2 1717t 5 0 it Rt 7Q it Rt 71 l 2 points When the central bank announces expansionary monetary policy and all other economic agents build this into their decision making as a consequence with no economic bene t this is called the problem 0 output rises policy lag Q unemployment rises time inconsistency O expectations rise adaptive expectations in ation rises time inconsistency Q in ation rises discretionary m 2 points In the presence of rational expectations the central banks willingness to battle in ation 0 causes future in ation 0 becomes a determinant of past in ation O undermines the ability to ght in ation becomes a determinant of expected in ation O weakens the central government n 2 points If the government gives rms a temporary investment tax credit 0 rms will invest now rather than in the future 0 it will increase EL 0 it will increase EL All of these answers are correct 0 None of these answers are correct 0 2 points Suppose we assume that initially it 0 B 05 Rt 2 7 5 if Etc rises 2 percent and the real interest rate falls 2 percent short run output 0 falls 2 percent 0 rises 1 percent rises 3 percent 0 falls 1 percent 0 does not change Macroeconomic Theory 102 Midterm 2 Page 5 of 10 Wednesday February 25 2015 Exercice 1 15 points 2 15 points Consider the following balance sheets for three hypothetical nancial insti tutions bank A bank B and bank C Bank A s Balance Sheet Assets Liabilities Cash 600 Deposits 1400 Loan to bank B 500 Mortgage Backed Securities 400 Total assets Total Liabilities Equity net worth Bank B s Balance Sheet Assets Liabilities Cash 1000 Deposits 900 Loan to bank C 500 Loan from Bank A 500 Total assets Total Liabilities Equity net worth Bank C s Balance Sheet Mortgage Backed Securities 800 Deposits 200 Loan from Bank B 500 Total assets Total liabilities Equity net worth a 25 points Fill in the missing entries in the balance sheets denoted Solution 15001400100 2 and 800700100 b 25 points What is the leverage ratio in each bank Solution The leverage ratio of the rst bank and second bank Bank A and B are given by 14001002141 That of Bank C is given by 700100271 c 10 points Suppose housing prices fall sharply and the mortgage backed securities as a consequence fall in value by 50 Assume that banks rst make good on their deposits before actually repaying other banks Calculate bank A s new net worth be careful two banks hold Mortgage Backed Securities Solution Bank A s net worth falls to 300 200 is lost directly through MBS 200 is lost through Bank C s MBS Cushion of 200 of net worth in between Macroeconomic Theory 102 Midterm 2 Page 6 of 10 Wednesday February 25 2015 Exercice 2 5 points 3 5 points According to the life cycle permanent income hypothesis consumption de pends on the present discounted value of income An increase in the real interest rate will make future income worth less thereby reducing the present discounted value and reduc ing consumption To incorporate this channel into the model suppose the consumption equation is given by C ath bcRt Yt Derive the IS curve for this new speci cation Solution As always start with the de nition of GDP and divide both sides by Z EQEE EX 1M YY Y Y Y Z39 Plug the equations in Table 111 given in the slides to replace each one of these components and get short run output Y 4 1zac bcRt fa bRt rfagaem am 1 t zacaagaem am 1 ECBRt rf y nzi lza bc kaRt f Yt So the new IS curve is Y1t a bc l bRt f Macroeconomic Theory 102 Midterm 2 Page 7 of 10 Wednesday February 25 2015 Exercice 3 50 points 4 50 points Take the usual AS AD model ruling out Aggregate Demand shocks so With EL 2 0 but assuming a one time unexpected oil price shock 50 gt 0 One time means that the oil price shock lasts only for one period in period t 0 and that at 0 for all subsequent 75 E 1 2 Unexpected means that the economy was originally in steady state and in particular that 710 7 Unless otherwise noted agents have adaptive expectations about in ation The economy is described by an AS AD model In particular the AS curve is given by be careful about the convention on the timing of the oil shock 5t1 l 7Tt 716 171 5t1 The AD curve is the standard one used throughout the course a 5 points What are the values of 711 and 171 in terms of the parameters of the model in particular the size of the oil price shock 50 Solution The AS AD equations are Wy i 171 Bm7r1 7 r Using the second equation to plug in the rst one gets 17 5 gt 7 5m m 1 0 O 1 1 0 1 1 9m 0 Using this to replace in one or the other equation one gets 50 7T17T 1Emz739 b 5 points Show analytically1 the effect of a more agressive monetary policy on in ation and short run output in period 1 do in ation and short run output increase or decrease With a more agressive monetary policy Solution 871391 517 TOO 8m 1 mu2 8171 517 1That is in mathematical terms Macroeconomic Theory 102 Midterm 2 Page 8 of 10 Wednesday February 25 2015 Both decrease Since in ation was increasing initially this means there is a more muted response of in ation To the contrary since output was already decreasing this means that the response of short run output is actually more important With a more agressive monetary policy the bulk of the adjustment goes through unemployment and a decrease in short run output Note of course adaptive expectations neglect the fact that if monetary policy was ex pected to be agressive then people may anticipate that in ation will be lower in future periods which on the contrary mitigates the needed adjustment of short run output c 5 points Illustrate this on two graphs with the AS AD curves show one AS AD diagram with a soft monetary policy and next to it another AS AD diagram with an agressive monetary policy Show 7T1 and 171 as well as the long run values of in ation and short run output on these graphs Solution See TA section d 5 points Eliminating 17 from the AS AD model nd a difference equation2 for 71 for t 6 23 Solution Because people have adaptive expectations he have the following equations 7Tt 7Tt 1 171 5t 1 5m7rt 7r Solving this system of two equations and two unknowns where the unknowns are Ema it is easy to show that in ation follows the following difference equation 1 Ema 1 7r 1bmz7 t1 7T 7T 0 t 1bmz7 1bmz7t 1 Moreover we have that 5t1 0 for all t E 2 3 because the shock is a one time shock So nally Ema Vt E 2737 7ft W 73939 1bmz7 t 1 e 5 points Substracting 7 on both sides in the difference equation for 71 show that 2A difference equation is an expression of an economic quantity as a function of its previous lagged values generally the value in the previous period For example 7Tt expressed as a function of 7Tt1 is a difference equation for 7Tt Macroeconomic Theory 102 Midterm 2 Page 9 of 10 Wednesday February 257 2015 7Q 7 satis es a simpler difference equation than 7rt3 Solve for this difference equation This should give you an expression for 7Q as a function of 71391 Then replace 7T1 With the value found in question a7 to get 7Q as a function of time and the parameters of the model Solution We have 1 Ema 1 Wt 1 Ema 1 977 217 1 Bmz77rt1 7T This difference equation iterates just as those in the Romer model through 1 t l 7ft 7 1quot T 7T1 1 bmz Using the expression for 7T1 in the previous question 50 1 5727 1 t 271 5 t 1bmz7 0 f 5 points Use the AD curve to then calculate 12 as a function of time and the parameters of the model 7T1 This gives Solution We use the AD curve which is Y1t bm7rt 7r With the espression found in question f 1 t Y2 4977 T 50 1 bmz g 5 points Numerical Application Suppose the parameters of the AS and AD curves take the following values 50 2 EL 2 07 5 127 772 127 17 127 and 7 2 Solve for the value of short run output and the in ation rate for the rst 2 years after the shock express your result as a single fraction7 since you do not have a calculator 3Simpler in the sense that you can solve for it For example a simple difference equation is one of the form at put1 Whose solution is at pt1u1 Macroeconomic Theory 102 Midterm 2 Page 10 of 10 Wednesday February 25 2015 Solution Therefore numerically 1 8 9 1 Ema 39 Time In ation 7Q ShortRun Output Z 0 2 0 1 2 89gtIlt2349 49 2 2 892 gtIlt 2 29081 3281 h 5 points Calculate how much realized in ation differs from expected in ation or 7Q 75 for any 75 Z 1 in this model Simplify the expression so that its sign appears clearly4 Solution We have that 6 7rt 7rfz7rt 7rt1 1 t 1 75 1 1 67 00 1 Ema 00 gm 5 lt 0 1 Emmt l 0 i 5 points Why are these adaptive expectations where 7 7rt1 considered as non rational What is irrational about them expectations they do not Solution Agents consistently overestimate what in ation will be next period They don t understand that in ation tomorrow will be determined by the re action of the central bank which will tighten monetary policy in the face of too high in ation and therefore lower in ation That is irrational Moreover this is a mistake that agents could learn not to make over time as they are always making the same mistake in the same direction But under adaptive j 5 points What would 7Q 7r be equal to under rational expectations Solution Under rational expectations we would have 7rt 7rf0 4That is it should be clear from your expression of 7Tt 7139 whether it is positive or negative
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