(1 ) = 1 for some 2 (0; 1); (1 + ) = 0 for some > 0:

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1 Answers, na. Economics 4 Fa, Christiano.. The typica househod can engage in two types of activities producing current output and studying at home. Athough time spent on studying at home sacrices current production, it augments future output by increasing the househod's future stock of human capita, k t+ The househod has one unit of time avaiabe to spit between home study and current production. Any given amount of human capita accumuation, k t+ =k t ; eaves an amount of time, h t ; eft over for producing current output, where h t = (k t+ =k t ). Here, is stricty decreasing, stricty concave, and continuousy dierentiabe, with ( ) = for some 2 (0; ); ( + ) = 0 for some > 0 The variabe, h t, must satisfy 0 h t A househod's eective abor input into production is the product of its time and human capita h t k t Tota output is reated to eective abor input by f(h t k t ) = (h t k t ) ; 2 (0; ) The resource constraint for this economy is c t f(h t k t ); and the initia eve of human capita, k 0 ; is given. The utiity vaue of a given sequence of consumption, c t ; is given by X t u(c t ); where u(c t ) = c t =; < 0 a, b, c (0) Express the panning probem for this economy as a sequence probem (SP). Write out the associated functiona equation (FE).

2 subject to so, X v(k 0 ) = max t c t ; c t = f(h t k t ); h t = (k t+ =k t ) where But, so, v(k 0 ) = max fk t+ g X v(k 0 ) = max k f tg 0 t ((k t+=k t )k t ) t = k t+ =k t k t = k 0 t j=0 j ; = k 0 max f t+g = Ak 0 ; = max f t;k t+ g X t k t X h t t j=0 i ( t ) j X h t t j=0 i ( t ) j ( t ) ; where X A = h t t j=0 i ( t ) j To estabish < A < ; proceed as foows. That A > foows from the fact that the objective is non-negative. Now, we estabish that A < Then, X A = max h t t f t+g j=0 i ( t ) ( X j < max h i ) ( t t f t+g j=0 j max f t+g t= ( ) X ( ) = f ( + ) g t < ; given the boundedness condition. 2

3 Writing out the expression for A carefuy, we nd A = max Dene the foowing mapping, T (w) + A T (w) = max + w To verify that this mapping has a unique soution, verify that Backwe's sucient conditions are satised. Monotonicity requires T (w) T (v) ; if w < v To verify this, et so that w arg max T (w) = ( w) + w; + w w by w<v z} { < ( w) by optimaity z} { T (v) + w v Discounting requires that where " 2 (0; ) and g 0 is a scaar. T (w + g) = max T (w + g) T (w) + "g; max + [w + g] " # + w g = T (w) + max + max g 3

4 Reca that t + The soution to the above maximization probem is the argest possibe vaue of which is + In this case, T (w + g) T (w) + ( + ) g But, 0 < ( + ) <. Discounting is estabished by setting " = ( + ) Thus, T is a contraction and it has a unique xed point, arrived at by A = im j! T j w; for any initia w The optima vaue of ; ; is = arg f max +g + A But, is dened as k t+ =k t ; so that the soution is a poicy rue, g (k t ) = k t f. The graph of = against is downward soped. The graph of A against is positivey soped. Assuming the ecient growth rate is interior, these two curves intersect in the interior of the set that restricts Ony the positivey soped graph is a function of That graph shifts to the right with an increase in ; and so the intersection of the two curves (assuming the equiibrium is interior) shifts to the right, to a higher vaue of 2 The optimization probem is with rst order conditions max Y Z 0 p j y j dj Margina cost for the j th intermediate good rm is Y y j = p j r w s = 4

5 This is obtained by studying its cost minimization probem min j ;k j w j + rk j + s [y j f (k j ; j ) ] This probem eads to the foowing rst order necessary condition for an interior optimum w = s ( ) r = s k j j y j + = f (k j ; j ) Rearrange the rst two conditions and mutipy! k j j!! ( ) w = s k j j! r ( ) = s k j ; j w r s = The prot maximization probem of an intermediate good rm is (apart from a constant having to do with xed costs) max p j y j sy j ; p j ;y j After substituting out the demand curve y s y p j According to this demand curve, the vaue of quantity demanded is aways a constant. As a resut, with higher prices revenues are constant, but of course costs a ower because quantity sod is ess. There 5

6 is no soution to the monopoy probem because for whatever p j the monopoist contempates, a higher price aways brings in more prot. If everyone has the same technoogy, then if a monopoist attempted to make positive prots, no matter how sma, an entrant woud come in and charge a sighty ower price to take a the business away from the monopoist. Thus, the monopoist who actuay produces must make zero prots. Zero prots by the monopoist impies p j y j s (y j + ) = 0; or, substituting in the production function, f j = y j p j y j = sf(k j ; j ) The rm markup is j into account = p j =s Dividing by p j and taking the atter y j = j f(k j ; j ); as requested. Tota costs break down into a part, s; associated with the xed cost and a part, sy j ; that is associated with the scae of operation. If the rm set p j = s; then its revenues woud match the part of its costs not reated to xed costs. To make zero prots, the rm must set price higher than s, so that revenues are enough to cover a its costs. This is why j > If the scae of production is high, then the markup wi be ow because the xed cost is reativey sma. One can see this, by substituting out for y j in terms of f in the above expression f(k j ; j ) = j f(k j ; j ); or, after rearranging, j = f(k j; j ) f(k j ; j ) = =f(k j ; j ) Note that if f is high, then j is ow. 6

7 Since each rm's probem is symmetric, it wi set the same price and hence it wi have the same markup,, output, y; and inputs, k and. Substituting this into the na good production function Z Y = exp [og y j ] dj = exp [og y] = y = f(k; ) (2) 0 The share of income going to capita and abor may be computed from the eciency conditions associated with cost minimization Thus, w = s ( ) k r = s k! w + rk = s ( ) k! = s [( ) + ] k = sf(k; ) = sy;! + s k! k by (2). Now, = p=s; where p denotes the price of the intermediate good producer. According to, p = so that we can concude w + rk = Y; and income going to capita and abor is precisey equa to tota na output. Intuitivey, this is no surprise since there are zero prots. Consider abor's share, w = s ( ) k! = s ( ) f (k; ) = s ( ) Y = ( ) Y Simiary, capita's share is rk = Y Conventiona measures of TFP take tota output and divide by capita, k; and abor, ; each raised to a power that corresponds to its share 7

8 of income. In this case, that's just f (k; ) So, TFP is = If output responds to shocks outside the rm sector, then wi fa when output is high and rise when output is ow, i.e., it wi be countercycica. But, this means that estimated TFP is procycica. An econometrician might be tempted to concude that RBC theory is vindicated, in think ing that he/she has uncovered the shock that drives the business cyce. In this case, that woud be a mistake because TFP is just responding endogenousy to other things, and is not itsef causa in this exampe. 3 A sequence of market equiibrium is a sequence of quantities, n o c t t; c t t+; k t+ ; c and prices, fr t ; w t ; r k;t g ; such that each period's househod and rm probems are satised and abor and capita markets cear. The period t househod probem is h i h max og(c t t)+ og(c t t+)+ t wt c t c t t ;ct t+ ;k t k t+ +2t rt+ k t+ ct+i t t+ The rst order conditions are Combining these, we obtain c t t c t t+ = t = 2t t + 2t r t+ = 0 0 ; c t t = r c t t+ ; t+ or, in the steady state equiibrium we consider c o = r (3) cy w = c y + k rk = c o (4) 8

9 The eciency conditions of the rms impy The resource constraint impies r = k (5) w = ( ) k (6) c y + c 0 + k = k (7) There are 7 equations, (3)-(7), = and 6 unknowns r; w; c y ; c o ; k and There is one redundancy in these equations because the househod budget constraints add up to the resource constraint (Waras' aw) after imposing (5) and (6). To see this, so that c y + k = ( ) k c o = k ; c y + k + c o = ( ) k + k = k Thus, we may drop one of the set of three equations the three budget constraints and the resource constraint. In addition, from here on we impose = We drop the resource constraint. We can substitute out capita from the two househod budget constraints, to obtain a singe ifetime budget constraint w = c y + co r Imposing the intertempora euer equation, so that w = c y + rcy r! w = c y ( + ) ; c y w = + c o w = r + k = w + 9

10 These are just a rewrite of the househods three equations. The equiibrium suppies two additiona equations, (5) and (6) r = k (8) w = ( ) k (9) Using this to substitute out for the wage in the househod's capita decision k = ( ) k ; + which we can sove for capita k = " ( ) + # This aows us to compute the return on capita r = k = + ; as required. Note that the return on capita is zero when = 0 This is because capita is worthess in this case. Aso, if is zero, the return on capita is innite because in this case, capita won't be accumuated and its margina product wi be innite. The object, r; is the rate of return on capita because a rate of return is the ratio of the tota payo on that asset to its price. The tota payo on capita in this mode is just its renta rate because it competey depreciates in one period. The price of capita in this mode is pinned down at unity by the technoogy. For any parameters in which we wi have r < > + ; Note from (3) that when r < consumption of the od is reativey ow. Note from (7) that it is feasibe to reaocate consumption from c y to c o if it is done one-for-one. That is, suppose we increase consumption 0

11 of the od by and reduce consumption of the young by the same amount. Consider what such a reaocation does to utiity f = u (c y ; c o + ) = og (c y ) + og c 0 + We now ask what the sope of f is with respect to, when the sope is evauated at the equiibrium aocations. Dierentiating, f 0 = c y + c 0 + At = 0 and r = c o =c y f 0 (0) = c o c o c + = y c [ r + ] = [ r] > 0 o co when r < Thus, the reaocation increases the utiity of each generation. This is because the terms of the intergenerationa transfer (one-for-one) are better than those oered by the market. Note that this transfer increases the utiity of each agent born in period 0,, 2,..., as we as the utiity of the current od. Their utiity obviousy rises because they simpy receive a transfer without making any payment. If there were a ast date for the economy, then the ast generation of initia young (who do not survive into od age) woud be worse o under the transfer and so the weath transfer is not obviousy wefare improving. Of course, it might be.