Economics Honors Exam 2008 Solutions Question 5

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1 Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I + G = c 0 + c (Y T ) + I + G = c 0 + c (Y 0 Y ) + I + G ( c + c )Y = c 0 c 0 + I + G. Therefore he equilibrium oupu is, Y = c 0 c 0 + I + G c + c. 3 poins: for realizing Y = C + I + G 2 poins: for subsiuing in C = c 0 + c Y D 2 poins: for subsiuing in T = 0 + Y 2 poins: for subsiuing in Y D = Y T 3 poins: for geing he final answer correc (b) (2 poins) The muliplier is <. C + c c The economy responds less o changes in auonomous spending when is posiive. Afer a posiive change in auonomous spending, he increase in oal axes (because of he increase in income) reduces consumpion and ends o lessen he increase in oupu. poins: for geing he muliplier correc poins: for realizing ha he economy responds less when is posiive 0- poins: depending on he qualiy of he explanaion (c) (6 poins) Because of he auomaic effec of axes on he economy, he economy responds less o changes in auonomous spending han in he case

2 where axes are independen of income. So oupu ends o vary less and fiscal policy is called an auomaic sabilizer. 0-6 poins: depending on he qualiy of he explanaion 2

3 Economics Honors Exam 2008 Soluions Quesion 6 (a) (0 poins) Since he rae of growh of E is 0, E is consan. Leing A E α, we can rewrie he aggregae producion funcion as Y = E α K α L α = A K α L α. Thus income per worker, y, can be wrien as. In seady sae, we have y = Y L ( K = A L = A k α. ) α k = sy (δ + n)k. Subsiuing in he expression for k, we can ge s Ak α (δ + n)k = 0 k α = sa ( k = Subsiuing ino he expression for y, we ge δ+n sa δ+n ) α. ( y = A A α α s ( = A α s δ + n δ + n ) α α ) α α. poin: for realizing ha E is consan 2 poins: for geing he expression y = A k α correc poin: for realizing ha in seady sae, k = 0 2 poins: for geing he expression sy (δ + n)k = 0 correc 2 poins: for geing he seady-sae k correc 2 poins: for geing he seady-sae y correc 3

4 (b) (3 poins) Consumpion per worker is c = ( s) y = ( s) A α ( ) α s α. δ + n 2 poins: for geing c = ( s) y correc poin: for geing final answer correc (c) (3 poins) We know ha y = Ak α, while A E α. Therefore if E increases, oupu per worker would increase as well. poin: for realizing ha y increase on he day of he change 0-2 poins: depending on he qualiy of he reasoning (d) (7 poins) An increase in E is equivalen o improved efficiency in he producion funcion: As could be seen from he graph, in he new seady sae, boh capial per worker (k) and oupu/income per worker (y) are higher, herefore he ransiion pah is illusraed overleaf: 2 poins: for realizing ha income per worker keeps on increasing over he ransiion pah

5 0-2 poins: depending on he qualiy of he reasoning (does no necessarily have o draw he firs graph) poin: for drawing a discree jump a 0 in he second graph poin: for showing ha y increases over ime afer 0 in he second graph poin: for showing ha y converges o he new seady sae value in he second graph (e) (7 poins) Immediaely afer he shock, here are wo compeing effecs: E increases bu capial sock is desroyed, hence he efficiency gain is offse by he capial loss. The ne effec on iniial oupu per worker is ambiguous. If he drop in capial sock dominaes he increase in E, oupu per worker would acually drop on he day of he change. Oherwise, oupu per worker would sill jump up on he day of he change, hough o a lesser exen han in par (c). Over ime, oupu per worker is going o be higher since k is higher in he new seady sae. The new graphs are shown below: poin: for realizing ha he immediae effec of desrucion of capial sock is o reduce y 2 poins: for realizing ha he ne effec on income per worker a 0 is ambiguous poin: for realizing ha y is going o be higher in he new seady sae (does no necessarily have o draw he firs graph) poin: for drawing a leas wo curves in he second graph, wih a discree jump up or down respecively a 0 poin: for showing ha y increases over ime afer 0 in he second graph poin: for showing ha y converges o he new seady sae value in he second graph 5

6 6

7 Economics Honors Exam 2008 Soluions Quesion 7 (a) (6 poins) Consumpion is: C = Y G K + = G 2 K G K + Differeniaing his wih respec o G gives: Seing his equal o zero gives: dc dg = 2 G 2 K G = K 2 2 poins: for geing he expression for consumpion C correc poin: for differeniaing C wih respec o G and seing his o zero 2 poins: for solving he differeniaion problem correcly poin: for geing he final answer G = K 2 correc (b) (2 poins) The household s budge consrain is: C = G 2 K G K + Subsiuing his ino he uiliy funcion of he represenaive agen gives: U = =0 β u(g 2 K G K + ) The consumer does no ake governmen spending as given, herefore G mus be subsiued ou of his expression, i.e.: U = = =0 =0 β u( 2 K K K + K 2 ) β u( K 2 K + ) Taking firs order condiions wih respec o K gives: 0 = u ( K 2 K ) + β 8 K 2 u ( K 2 K + )

8 Therefore he firs order condiion gives: = β 8 K 2 K = ( 8 β ) 2 2 poins: for subsiuing he expression for C ino he lifeime uiliy funcion 2 poins: for subsiuing G = K 2 in 2 poins: for geing U = =0 β u( K 2 K + ) correc 2 poins: for aking firs order condiion wih respec o K 2 poins: for solving he differeniaion problem correcly 2 poins: for geing he final answer K = ( 8 β ) 2 correc (c) (9 poins) The household s budge consrain is: C = G 2 K G + K + Subsiuing his ino he uiliy funcion of he represenaive agen gives: U = =0 β u(g 2 K G + K + ) Taking firs order condiions wih respec o G gives: 0 = u (G 2 K G K ) + β 2 G 2 K u (G 2 K G + K + ) In seady sae he argumens are he same, herefore: G 2 = β 2 K G = β2 K 2 2 poins: for geing he new expression for C correc (noe he subscrips of G) poin: for subsiuing C ino he lifeime uiliy funcion 2 poins: for aking firs order condiion wih respec o G 2 poins: for solving he differeniaion problem correcly poin: for realizing ha in seady sae, G is consan poin: for geing he final answer G = β2 K 2 correc 2

9 (d) (3 poins) This is less han he answer from par (a). I is because here is an exra cos o governmen spending now, in ha i mus be from savings. Because agens are impaien, his means ha i is less desirable. poin: for realizing ha he new governmen spending level is lower 0-2 poins: depending on he qualiy of he reasoning 3

10 Economics Honors Exam 2008 Soluions Quesion 8 (a) (6 poins) K = I δk = sy δk = sa()(k()) α (L()) α δk() k k = K K L L = sa()(k()) α δ n poin: for realizing ha K = I δk 2 poins: for geing he expression K = sa()(k()) α (L()) α δk() correc poin: for realizing ha k k = K K L L 2 poins: for geing he final answer correc (b) (6 poins) y = A()k() α In seady sae, i mus be ha y and k grow a he same rae, call i g. Therefore i mus be ha: A ( α)g = A i.e. he growh of A mus be consan. The growh of A is given by: A() A() = y() A() = k()α This mus be consan, i.e. k() mus be consan. However, i is no because if i were hen k k would be growing consanly over ime, which can be seen from par (a). poin: for acknowledging ha his model does have such a seady sae poin: for geing y = A()k() α correc poin: for realizing ha y and k mus grow a he same rae in seady sae

11 poin: for geing ( α)g = Ȧ A correc A() poin: for geing A() = k()α correc poin: for realizing ha he growh rae of A is consan (c) (9 poins) A() ( α)g = A() = k()β A() Therefore if he las par is consan hen k() grows a a rae β A() rae of A, which is consisen wih A() = ( α)g if ( α) = β. A() A() = k()β A() 2 poins: for geing correc 2 poins: for equaing his o ( α)g he growh A() A() is consan if k() grows a a rae β he 3 poins: for realizing ha growh rae of A 2 poins: for realizing ha his is saisfied if ( α) = β (d) (6 poins) Here, savings affec growh. I does no in he oher case. The more paien people are, he higher he opimal s will be. The usual Golden Rule does no depend on he ime preference. 2 poins: for realizing ha savings affec growh here, while i does no in he oher case 2 poins: for realizing ha he more paien people are, he higher he opimal s will be 2 poins: for realizing ha he usual Golden Rule does no depend on he ime preference (e) (3 poins) Learning by doing. 5

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