PROBLEM SET FOUR SOLUTIONSProblem 1.a. PDV of your lifetime wealth.Note to graders: please be kind if exponents are off by only one yearLabor market income:$50,000[1+1.05/1.03+…+(1.05/1.03)44]= $50,000(1-1.01941745)/(1-1.019417)= $50,000(73.23911)= $3,543,156The exponent is the first line is 44 because there must be 45 terms in the series.Social security income:$75,000[1+1/1.03+…+1/(1.03)19]/(1.03)45= $75,000[(1-0.97087420)/(1-0.970874)]/(1.03)45= $303,915The exponent in the first line is 19 because there must be 20 terms in the series,while the exponent in the denominator is 45 as the stream of Social securitybenefits is discounted over 45 years.Lifetime wealth:W = $3,847,071b. Smooth consumptionC* = W/65 = $3,847,071/65 = $59,185c. Borrowing and liquidity constraintsSavingS20 = Y20 – C20 = Y20 – C* = $50,000 - $59,185 = -$9,185You must borrow this amount in the first year to perfectly smooth consumptionover your expected lifetime.When can you smoothNote W20 = Y20 +W21/(1.03) so W21 = (1.03)*(W20-Y20).More generally Wt+k = (1.03)*(Wt+k-1-Yt+k-1)Liquidity constraints in the first five years require Ct+k = Yt+k, so the abovesequence captures the evolution of lifetime wealth subject to our consumptionplan (which is just the path of income over the next five years). The series forwealth is generated in the second column of the table below. The last column isthe smooth consumption consistent with this wealth, simply constructed asWt+k/(65-k). This is simply the level of consumption consistent with remainingwealth. Note smooth consumption is less than income in each of the first fiveyears, so we will not be able to begin to smooth consumption until after our credithistory is long enough for us to borrow at age 25.Time YtWtc*t20$ 50,000.00 $ 3,847,071.80 $ 59,185.7221$ 52,500.00 $ 3,910,983.95 $ 61,109.1222$ 55,125.00 $ 3,974,238.47 $ 63,083.1523$ 57,881.25 $ 4,036,686.87 $ 65,107.8524$ 60,775.31 $ 4,098,169.79 $ 67,183.1125$ 63,814.08 $ 4,158,516.31 $ 69,308.61Note to graders: again be kind here, as I’m sure many people got lost in doingsome complicated math.d. Tax cutsA tax cut of $10,000 will be smoothed over the next 65 years of your life soconsumption today will rise by $10,000/65 = $154.The effectiveness of one-time changes in taxes on output is diminished whenagents make consumption decisions based on lifetime wealth.e. Tax cuts under liquidity constraintsIf you are liquidity constrained, you will be more likely to consume a largefraction of your one-time tax cut instead of spread it evenly over your lifetime.This implies that the presence of liquidity constraints implies that even if agentsare forward-looking in their consumption decisions, one-time changes in thestance of fiscal policy can have significant effects on output.Problem 2.a.E = Kb(AN)1-b-(r+N)K-wNfoc(K): bKb-1(AN)1-b – (r+N) = 0bKb(AN)1-b/K = (r+N)bY/K = (r+N)K* = bY/(r+N) = S2Y where S2 = b/(r+N)Y* = [bY*/(r+N)]b(AN)1-bY* = AN[b/(r+N)]b/(1-b) = AN*S1 where S1 = [b/(r+N)]b/(1-b)b.Irept = NKt-1 = NK*t-1 = NbYt-1/(r+N)Inett = AK*t = bAYt/(r+N)It = Irept+Inett = NbYt-1/(r+N)+bAYt/(r+N)c.Y* = 1*100*[0.6/(0.05+0.10)]0.6/(1-0.6) = 800K* = 0.6*800/(0.05+0.10) = 3200d.Note that the following is true after taking natural logs and time derivatives on yourequations from part a.gy* = gN+gA+gS1gk* = gy*+gS2Divide your investment equation by the capital stock from the previous period as follows:(It/Kt-1) = N+AK*t/Kt-1 = N+AK*t/K*t-1 = N+gk*Note finally gN = gS1 = gS2 = 0. We have gy* = 3% and (It/Kt-1) = 13%.e.From the above it follows that gy* = 5% while (It/Kt-1) = 15%. These are permanentchanges in the growth rate of output and investment as the change in technologicalprogress is permanent.f.Now we need to take into account gS1 = (S1,t+1-S1,t)/S1,t and gS2 = (S2,t+1-S2,t)/S2,t.We have S1,t = [0.6/(0.05+0.10)]0.6/(1-0.6) = 8 and S1,t+1 = [0.6/(0.055+0.10)]0.6/(1-0.6) = 7.6.We have S2,t = 0.6/(0.05+0.10) = 4 and S2,t = 0.6/(0.055+0.10) = 3.87These results imply gS1 = -4.7% and gS2 = -3.2%, which together imply thatgy,t = 3%-3.2% = -0.2% and (It/Kt-1) = 13%-7.9% = 5.1%. These changes in the growthrate of output and investment rate are temporary as there is a one-time permanent changein interest rates, so as there are no changes in the future there are no more changes in S1or S2.Note to graders: it is sufficient to say that the growth rate of output and investment rateincrease, and that these increases are temporary if students note the reason (one-timechanges in the theta’s).g.Firms will on average have more than the optimal amount of capital if they follow thispolicy. The policy simply tells them to compare the current capital stock with theoptimal capital stock. If the capital stock is less than optimal, invest until you reach theoptimal level. If the capital stock is more than optimal, irreversibility constrains you todo nothing. So part of the time you have optimal capital and part of the time you havemore than optimal, so on average you have too much.A better policy for the firm to follow would be when capital is less than optimal capital,invest more slowly (for example, invest only one-half of the way to the level of optimalcapital). Firms are still taking advantage of time being good, but reduce their downsiderisk if times turn bad again in the future.h.Irreversibility in investment implies that firms will be reluctant to invest when timesbecome good, especially if it looks like good times are temporary. In part e, the changein a permanent one, so firms would likely behave in a manner similar to your answer inthe case without irreversibility. This might be less so if it takes times for firms to realizethe change in technological progress is permanent. In part f the change is an increase inthe interest rate, which reduces optimal capital and temporarily reduces output growth,and investment rates. The firm will react less to this interest rate shock than in part esimply because of the irreversibility constraint.Note to graders: for the second part, I’m sure not many people realized that the shockwas negative, so it is fine to talk about a positive shock. Answer in this case as follows:A decrease in interest rates increases optimal capital and temporarily increases outputgrowth and the investment rate. Shocks to the interest rate are easily reversed by thecentral bank, so this may be perceived as more of a temporary shock, in which case
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