Homework 1: Warm-up Exercise Solutions Eco 603 Fall 2016

Transcription

1 Homework : Warm-up Exercise Soluions Eco 603 Fall 206 Quesion : Normalizaion a. The producion funcion is consan reurns o scale in K and L. Therefore, we can use he procedure given in he class noes o remove. The nex hing o noe is ha he producion funcion is no consan reurns o scale in K and A, herefore dividing hrough by A does no eliminae A from he problem. To use he same rick, we need o conver A ino somehing ha is consan reurns o scale wih K, ie. somehing o he power. This is done by noing ha: C +K + = K γ ( ) +( δ)k () So now A augmens he produciviy of labor. From here: C + K + = K γ ( ) γ +( δ) K (2) C + Le c = C A K + + +, ec. Then: ++ = ( K ) γ +( δ) K (3) c +(+η)(+θ) k+ = k γ +( δ)k (4) So we have eliminaed A and L from he resource consrain. Now we have o do he same for he objecive funcion. This is again easy for L, since we maximize per capia uiliy. Bu we also have o divide and muliply by A: max c,k + β σ ( C A ) () + A (5)

2 = max c,k + [ β σ c ] + β A (6) = σ 0 L 0 max c,k + ) [ (β(+η)(+θ) c ] +a 0 (7) Here a 0 is a consan (acually an infinie sum of consan erms which converges o a finie sum). Since he opimal decision is no affeced by adding a consan or muliplying a consan, we can ignore hem. Assuming σ is no one, he discoun rae is also changed. Thus we can solve he following problem (which has neiher A nor L): = max c,k + Subjec o: [ c ] ˆβ c +(+η)(+θ) k+ = k γ +( δ)k (9) Here ˆβ = β(+η)(+θ). b. We can inerpre he sae variable as he per capia capial sock, bu where persons are weighed by heir labor produciviy. Thus, we have capial per produciviy adjused worker. c. For finie welfare, we need he discoun rae o be less han one: ˆβ = β(+η)(+θ) < (0) We canno have he echnology growh rae be so large ha high echnology capial in he far fuure becomes so aracive ha households consume zero in order o obain infinie uiliy from capial laer. d. In he seady sae, y approaches a consan ȳ. Here y is oupu per produciviy adjused person. For oupu per person, we have: (8) ȳ = y = Y = Y, () 2

3 Y = ȳ, (2) In he seady sae, he growh rae of he righ and lef hand sides mus be equal, so: = 0 (+θ), (3) gy L = (+θ). (4) Similarly, for oal GDP we have: g Y = (+η)(+θ). (5) Quesion 2. For he limiing case, noe ha: lim σ u(c) = 0 0. (6) Since he limi is indeerminae, we can use L Hopial s rule. Taking he derivaive of he numeraor and he denominaor, we see ha: log(c)c limu(c) = lim σ σ = log(c). (7) Noe: oakehederivaiveofc wihrespecoσ,firsrewrieusing: c = exp(()log(c)). Quesion 3. a. Wih he increase in produciviy we have wo effecs: The reurn o savings rises: nex period, savings becomes capial ha is more producive and hus generaes more oupu. This effec causes savings o rise. This is a subsiiuion effec (consumpion o savings). Consumpion becomes less smooh: holding savings consan, consumpion becomes less smooh as consumpion will rise in he fuure (higher produciviy means more producion and herefore more consumpion, holding savings consan). Since households prefer smooh consumpion, hey will reduce savings and increase consumpion oday o ry o smooh consumpion. This is an income effec. b. As he uiliy funcion becomes more concave, households have sronger preferences 3

4 for smooh consumpion. The uiliy from smooh consumpion becomes increasingly beer han he uiliy from non-smooh consumpion: u(c φ ) >> φu(c )+( φ)u(c + ), c φ = φc +( φ)c +, 0 < φ <. (8) This increases he income effec. c. The uiliy funcion becomes more concave as σ rises. u u σ=4 σ= u c c c + c Figure : Preference for smooh consumpion ( C) increases wih σ. For σ = 0, uiliy is linear. As σ rises, he difference beween he lef and righ hand sides of equaion (8) increases. d. Savings will fall as σ rises due o he sronger income effec. e. Less savings is equivalen o discouning he fuure more (β closer o zero). As σ increases, he discoun facor falls, reflecing he sronger income effec. Increasing θ magnifies boh he income and subsiuion effecs. Therefore, he effec of θ on he discoun rae depends on σ, which deermines which effec is sronger. f. Under log uiliy, he income and subsiuion effecs exacly cancel, and so he discoun rae is unaffeced by echnological change. Quesion 4 4

5 a. We have one sae variable which is k. So v is a funcion of he one sae variable. Afer subsiuing for c using he resource consrain, we have a single conrol variable which is k. Thus: v(k) = max k { [ } k γ +( δ)k (+η)(+θ) k ] + ˆβv(k ) (9) b. The firs order condiion is: c σ = ˆβ Γ v k(k ), (20) Γ (+η)(+θ). (2) The lef hand side is he marginal uiliy of consumpion and he righ hand side is he marginal uiliy of invesmen. Wih one more uni of resources, we should be indifferen beween consuming and saving. If consuming gave more uiliy, hen we should go back and reallocae more resources from saving (which gives low uiliy) owards consumpion (which gives higher uiliy). Therefore, we mus be indifferen wih he las resource uni. We divide by he growh raes because he savings mus be spli among a larger produciviy adjused populaion. Quesion 5 a. Firs, we can build he resource consrain by seing resources equal o allocaions. Resources are producion and capial afer depreciaion, as in quesion. For allocaions, we have consumpion as in quesion, and invesmen. Since any invesmen in period becomes capial in wo periods, invesmen is X 2. So he resource consrain is: C +X 2 = K γ ( ) +( δ)k (22) Noe ha we also have a consrain ha invesmen oday is capial under consrucion in he nex period: X 2 = X,+. (23) Finally, capial nex period is equal o capial under consrucion oday plus capial which has no depreciaed: K + = X, +( δ)k. (24) 5

6 Normalizing as in quesion resuls in: c +x 2 = k γ +( δ)k, (25) x 2 = Γx,+. (26) Γk + = x, +( δ)k. (27) b. Saevariablesaregivenfromoday sperspecive, buchangeover ime. Clearlyk and capial under consrucion x are given oday. The conrol variables are invesmen x 2 and consumpion c. So we can wrie he value funcion as: v(k,x ) = max x 2 { [k γ +( δ)k x 2 ] + ˆβv(k,x ) }. (28) Noice ha we have wo variables no defined as eiher saes or conrols, x (a conrol) and k (a sae). We also have wo remaining consrains ha need o be subsiued in. Hence: { [ ] ( v(k,x ) = max k γ +( δ)k x 2 + x 2 ˆβv x +( δ)k, x ) } 2 (29) Γ Γ 6