Intermediate Macro In-Class Problems

Transcription

1 Inermediae Macro In-Class Problems Exploring Romer Model June 14, 016 Today we will explore he mechanisms of he simply Romer model by exploring how economies described by his model would reac o exogenous changes. Firs, recall his summary of he Romer model: Producion : Y = A L y Technological Dynamics : A +1 = za L a Resource Consrain : L y + L a = L Labor Allocaion Rule : L a = l L And he model soluion: 1 Labor Force L a = l L L y = 1 l ) L A = 1 + z l L ) A0 Y = 1 l ) 1 + z l L ) A0 L Analyze he effecs of a governmen policy encouraging immigraion in he conex of he Romer model. Remember, any proper analysis of policy in a macro model will evaluae he impac of he policy on every aspec of he model ypically, he model parameers, bu occasionally also he funcional forms hemselves a higherdimensional parameer). Idenifying Model Parameers Wha are he parameers of he Romer model? Wha are he quaniies ha are aken as given, and which are allowed o change as a resul of agens behavior? The parameers are z, l, L, and A 0. 1

2 Evaluaing he Policy Wha is he poenial effec of he immigraion policy on each of he parameers you idenified above? z : No reason o believe his will change wihou furher informaion hough argumens can be made in eiher direcion). l : idem. A 0 : idem. L : Obviously a pro-immigraion policy will increase L. Overall, his will lead o an increase in he growh rae of he economy from z l L o z l L, wih he same increase in he growh rae of knowledge. Reallocaion of Labor Analyze he effecs of a governmen policy which subsidizes research in he conex of he Romer model. Wha is he poenial effec of he immigraion policy on each of he parameers you idenified above? z : No reason o believe his will change wihou furher informaion hough argumens can be made in eiher direcion). l : Such a policy is likely o resul in reallocaion of labor o he research secor from he producion secor, which means l will increase. A 0 : No reason o believe his will change wihou furher informaion hough argumens can be made in eiher direcion). L : idem. Overall, his will lead o an increase in he growh rae of he economy from z l L o z l L, wih he same increase in he growh rae of knowledge. 3 Texbook Exercises 6.1 Explain wheher he following goods are rivalrous or nonrivalrous: 1. Beehoven s Fifh Symphony. An ipod 3. Mone s paining Waer Lilies

3 4. The mehod of public key crypography RSA) 5. Fish in he ocean 1. Nonrivalrous he noes can be played by an infinie number of people hrough he end of ime.. Rivalrous only one person can own a paricular ipod a a given ime. 3. Rivalrous hough replicas can be made, only one will be MONET s paining. 4. Nonrivalrous RSA is jus an algorihm; any compuer can and does) reproduce i. 5. Rivalrous hough heir quaniy is vas, i s ulimaely finie; each fish in paricular can only be enjoyed by one person or oher fish). 6. Suppose a new piece of compuer sofware say a word processor wih perfec speech recogniion can be creaed for a oneime cos of $100 million. Suppose ha once i s creaed, copies of he sofware can be disribued a a cos of $1 each. 1. If Y denoes he number of copies of he compuer program produced and X denoes he amoun spen on producion, wha is he producion funcion; ha is, he relaion beween Y and X?. Make a graph of his producion funcion. Does i exhibi increasing reurns? Why or why no? 3. Suppose he firm charges a price equal o marginal cos $1) and sells a million copies of he sofware. Wha are is profis? 4. Suppose he firm charges a price of $0. How many copies does i have o sell in order o break even? Wha if he price is $100 per copy? 5. Why does he scale of he marke he number of copies he firm could sell maer? 1. Y = { 0 X < 100 = X X) 1 [X X] X 100 X 100 Where X = 100, 000, 000, defined for conciseness. The laer form is more convenien/concise for wha will follow below. The funcion 1 [ ] is he indicaor funcion, aking he value 1 if is argumen is rue and 0 oher wise. For example, 1 [3 > 4] = 0 while 1 [ z 0 ] = 1 for all real values of z). 3

4 . Simple coninuous piecewise funcion fla hrough X = 100, 000, 000, hen wih slope 1 hereafer. Reurns o scale are weakly) increasing if we double X from 100, 000, 001 o 00, 000, 00, we increase oupu from 1 o 100, 000, 001, which is far more han double. Mahemaically, Y X) = X X) 1 [X X] X X) 1 [X X] X X) 1 [X X] = Y X) The firs inequaliy holds because clearly X > X, and subracing somehing larger makes he objec smaller; he inequaliy is no sric because he indicaor may be 0, negaing he sric size difference of he firs produc. The second inequaliy holds because he se of X for which X X is a sric subse of he se of X for which X X. Tha is, if X X, surely X X, which means ha whenever he former akes he value 1, he laer cerainly does as well. 3. Π = revenue cos = 1, 000, , 000, , 000, 000 = 100, 000, 000 Revenue is 1, 000, 000 one million copies a one dollar a pop); variable coss are he same since he marginal cos of producion is also $1). The fixed cos of research mus sill be considered, however, leading o massive losses. This is he oucome ha would arise under perfec compeiion, where price equals marginal cos. 4. If he firm charges $0 per copy, hey make profis of $19 per copy having subraced ou he marginal cos from he price). To break even, hey need o sell 100,000, , 63, 158 copies. Similarly, wih a price of $100, per-uni profi is $99, so hey need o sell 100,000, , 010, 101 copies. 5. The larger he marke, he lower he price can be which allows breaking even. 6.8: A variaion on he Romer model Consider he following variaion: 4

5 Y L y A +1 = za L a L y + L a = L L a = l L There is only a single difference: we ve changed he exponen on A in he producion of he oupu good so ha here is now a diminishing marginal produc o ideas in ha secor. 1. Provide an economic inerpreaion for each equaion.. Wha is he growh rae of knowledge in his economy? 3. Wha is he growh rae of oupu per person in his economy? 4. Solve for he level of oupu per person a each poin in ime. 1. The firs equaion is producion. This producion funcion sill exhibis he increasing reurns o scale ha are he hallmark of echnological innovaion, bu now exhibis diminishing reurns o echnological innovaion. The second equaion is he rajecory of echnological innovaion. The economy grows is produciviy by assigning par of is workforce o doing research. The hird equaion is he labor resource consrain. Toal labor used in boh secors of he economy mus be equal o he oal supply of labor. The final equaion is sociey s rule for allocaing labor. I says ha, in each period, a fixed proporion of he workforce will be assigned o each secor.. The growh rae of knowledge is given by A+1 A = zl a = z l L, jus as before. 3. The growh rae of oupu is given by 5

6 Y +1 Y = A 1 +1 Ly +1 A 1 L y A 1 L y A 1 A A 1 = = A+1 A 1 l) L A 1 A 1 1 l) L ) A ) A = 1 + z l L ) l) L 4. Since we again have A = 1 + z l L ) A0, oupu per person is simply: Y y = L a + L y L y L a + L y = 1 l ) 1 + z l L ) A 1 0 6