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NU ES_APPM 411 - Rough sketch of FINAL EXAM answers

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Christiano411, Fall, 2007Rough sketch of FINAL EXAM answersAllocate your time to the following four questions in proportion to thenumber of points a vailable. If a question seems ambiguous, state wh y,sharpen it up and answer the revised question. Good luck!Ask stuff about the factor-price frontier.1. A representative, competitive firm produces a homogeneous final good,Y, using the following production function:Y =∙Z10yλidi¸1λ, 0 <λ<1,where yiis the quantity of the ithintermediate good. The final goodproducer is competitive in the market for yi, where it faces a price, Pi.Each intermediate good is produced by a single monopolist which setsprice, Pi, to maximize profits. The first order necessary condition forprofit maximization by the representative final good producer is theone associated with:maxyiY −Z10Piyidi,and the first order condition is:ÃYyi!1−λ= Pi. (1)The ithintermediate good producer tak e s (1) as its demand curve, withY being an exogenous variable.The ith,i∈ (0, 1) intermediate good is produced by a monopolist usingthe production function:yi= AkαiN1−αi, 0 <α<1,where A is a parameter of technology, and ki,Nidenote capital andemployment, respectively. Suppose that each m onopolist is a price1taker in the factor markets, where r and w are the rental rate on capitaland wage rate on labor, respectively.Monopolists, i ∈ (0,γ)aresubjecttoa‘financial friction’: they mustborrow the funds to pay rkiand wNiat the beginning of the period.So, at the end of the period these firms owe the bank principle plusinterest equal to Rrki+ RwNi, where R is a gross rate of interest (i.e.,anumberlike1.08 for ‘8 percent interest’). Monopolists with i ∈ (γ,1)face no financial friction and simply pa y rkiand wNiat the end of theperiod for its factor payments.With the Cobb-Douglas production function, marginal cost for a firmthat pays rental rate ˜r and wage rate ˜w is:MC(˜r, ˜w)=ξ˜rα˜w1−α,where ξ is a constant. The end of period profits for intermediate goodproduces arePiyi− MC(r, w)yi,fori ∈ (γ,1)Piyi− R × MC(r, w)yi,fori ∈ (0,γ) .To solve the monopolist maximization problem, substitute the demandcurve out of the profit function:(Y )1−λyλi− Ciyi,where Cidenotes marginal cost:Ci=(R × MC(r, w) i ∈ (0,γ)MC(r, w) i ∈ (γ,1),which is beyond the con t rol by the monopolist. The first order condi-tion is:λ (Y )1−λyλ−1i= Ci,or,Pi=1λCi,2for all i ∈ (0, 1) . Solving,λ (Y )1−λCi= y1−λi,or,yλi=ÃλCi!λ1−λYλ.Let i0∈ (0,γ) ,i∈ (γ,1) , soyi0yi=⎡⎢⎢⎣³λR×MC(r,w)´λ1−λYλ³λMC(r,w)´λ1−λYλ⎤⎥⎥⎦1λ=µ1R¶11−λ,or,yi= yi0R11−λ,so the unconstrained producers, i, produce more than the unconstrainedones. Substituting,Y =∙Z10yλidi¸1λ=h(1 − γ) yλi+ γyλi0i1λ=h(1 − γ) yλi0Rλ1−λ+ γyλi0i1λ=h(1 − γ) Rλ1−λ+ γi1λyi0=h(1 − γ) Rλ1−λ+ γi1λAkαi0N1−αi0=h(1 − γ) Rλ1−λ+ γi1λAµNi0ki0¶1−αki0To understand how producers use the factor inputs, consider the costminimization problem for a producer which pay s ˜r for capital and ˜wfor labor:min ˜rki+˜wN + υhyi− AkαiN1−αii.3The first order conditions associated with this minimization problemare:˜r = υαµNiki¶1−α˜w =(1− α) υµNiki¶−αand, the ratio impliesrw=α(1 − α)Niki,for all i ∈ (0, 1) . So,Niki=Nk,i∈ (0, 1) ,where N and k are the aggregate quantities of capital and labor. Fromthis we see that producers use the same capital to labor ratio. As aresult,ki= ki0R11−λNi= Ni0R11−λ,sok = γki0+(1− γ) ki=hγ +(1− γ) R11−λiki0N =hγ +(1− γ) R11−λiNi0Concludeh(1 − γ) Rλ1−λ+ γi1λAµNk¶1−αhγ +(1− γ) R11−λi−1kY =h(1 − γ) Rλ1−λ+ γi1λh(1 − γ) R11−λ+ γiAµNk¶1−αk,or,Y =h(1 − γ) Rλ1−λ+ γi1λh(1 − γ) R11−λ+ γiAkαN1−α.4From this, we see that estimated multifactor productivit y isYkαN1−α=h(1 − γ) Rλ1−λ+ γi1λh(1 − γ) R11−λ+ γiA.This can mo ve around in part because of movements in tec hnology, A.However, it can also move around with movements in the endogenousvariable, R. To see this, consider the case γ =0.5,λ= .9andR in therange 0.5-2:0.51 1.520.920.930.940.950.960.970.980.991Note how the function is maximized at R = 1 where resources aredistributed equally (and, efficiently) among all producers. It is also thecase that the inefficiency is quite small. For R =1.05,TFP is about0.9975, so that there is only a loss of about 0.25 percent in output.2. Suppose there is a given total amount of some finite resource, X. Thisresource can be allocated among a range of inputs, x (i) ,i∈ (0,M)subject to the resource constraint:ZM0x (i) di ≤ X.5Suppose the inputs can be converted into output according to the fol-lowing technology:y = n1−αZM0x (i)αdi, 0 <α<1.Explain the sense in which the present environment is one in whichthere are gains from specialization.answer: if the total resource, X, is allocated over a greater variety ofinputs, x (i) ,i∈ (0,M) , then more output is possible. Thus, supposeeachinputisoperatedatintensityx (i)=XM,so that the resource constraint is satisfied. Then, output isy = n1−αµXM¶αM = n1−αXαM1−α,which is increasing in M, the range of inputs. As M increases, the rangeof inputs is greater, and this can be interpre ted as greater specialization.3. Consider the following competitive, two-period lived overlapping gen-erations economy. People work and save when young. Their timeendowment when young is unity. When, old households do not workand they pay for consumption out of the rent from capital accumu-lated while young. The population is constant, and the num ber ofyoungbornineachperiodisequaltothenumberofoldwhodieinthesameperiod. Letcttand ctt+1denote the period t and t +1con-sumptions, respectively, of agents born in period t. Let wtdenote theperiod t wage rate. Preferences of households are given by u³ctt,ctt+1´and these are increasing and concave. The young supply one unit oflabor inelastically. The budget constraint of the young and old are,respectiv ely,ctt+ kt+1≤ wtctt+1≤ rt+1kt+1+(1− δ) kt+1where rt+1denotes the rental rate of capital in period t +1 and wtdenotes the wage rate in period t. All quantities chosen by the household6are required to be non-negative. The initial generation of old peopleowns the initial stock of capital, k0, and simply consume the incomefrom this stock. A representative, competitive firm chooses capital


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