Overlapping Generations

Size: px

Start display at page:

Download "Overlapping Generations"

Jasmin Cox
5 years ago
Views:

1 Eco. 53a all 996 C. Sims. troductio Overlappig Geeratios We wat to study how asset markets allow idividuals, motivated by the eed to provide icome for their retiremet years, to fiace capital accumulatio for ecoomic growth. We cosider a highly simplified model, i which people live two periods. The periods therefore have to be thought of as about 30 years log. At ay oe date t there are N t youg people bor at that date ad N t old people who were bor at the previous date. The youg people are bor without ay wealth, but are each edowed with oe uit of labor, which they sell to provide themselves with the produced good. f they wat to cosume i their old age (secod period of life), they caot cosume all of their wage earigs, but istead must whe they are youg put some of their earigs aside to fiace their cosumptio whe old.. Cosumer Behavior We assume that a idividual bor at t maximizes a utility fuctio that has the special form b g. () U C ( t ), C ( t + ) C ( t ) C ( t + ) Here C is cosumptio by this youg perso i his or her youth, i.e. at time t, while C( t + ) is cosumptio i old age, the secod period of life, i.e. at time t +. The cosumer faces two costraits, ad C ( t ) + B ( t ) W ( t ). () b g. (3) C ( t + ) + r B Here B is savigs ad r is the retur o savigs. We use the letter B because the cosumer is treatig savigs like a bod that pays iterest rate r, though i this versio of the model there is o oe issuig the bod. t is just output put aside for the future. W is both total wage icome ad the price of oe uit of labor. Sice idividuals each have oe uit of labor ad do ot gai utility from leisure, they supply their labor ielastically (i.e., they work for oe uit of time, regardless of the wage). f we solve () ad (3) to elimiate B, we obtai a sigle budget costrait i which ( + r ) is the price of C i terms of C : C( t + ) C + W. (5) + r This produces a diagram of the covetioal cosumptio-theory type, as show below.

2 Budget Costrait ad differece Curve, W, r.5.5 C C Note that the iterest rate of 50% i the diagram is ot urealistically high, because the time period of about 30 years makes a 50% rate correspod to a aual rate of.4%. As the iterest rate rises, the budget costrait at a give value of W swigs outward, rotatig aroud the fixed poit C 0, C W. A rise i r must icrease C, but i geeral, because of offsettig icome ad substitutio effects, may icrease or decrease C. With our particular, Cobb-Douglas, form for the utility fuctio, it turs out that C is a fixed fractio,, of W, regardless of the value of r. To see this, we solve the costraied optimizatio problem of maximizig () subject to () ad (3). The Lagragia for this problem is b g b g. (6) C C ( t + ) λ C + B W µ C ( t + ) ( + r) B Because we are lookig for a competitive equilibrium solutio, cosumers are assumed to treat the prices W ad r as beyod their cotrol. Choice variables for a cosumer bor at t are C ( t ), C ( t + ), ad B. The first-order coditios (OC s) with respect to these three variables are C : C C t λ ( + ) KJ C : ( ) µ C ( t + ) C ( t + ) KJ B : λ + r µ (7) (8) b g (9) t is ot hard to verify that these three equatios ca be solved to elimiate λ ad µ, deliverig

3 C ( t + ) ( )( + r). (0) C Combiig (0) with the budget costrait (5) writte i terms of C s aloe gives us the coclusio we already aouced above, C Wbtg. () rom () ad the costraits () ad (3), we ca fid the values of the other two choice variables as. Productio B ( ) W C ( t + ) + r W, b gb g. () We assume that at each date t there are M idetical firms, each of which hires labor from youg people ad purchases capital held over from the previous period by old people, combiig them to produce ew output. The firm aims to maximize profits, which are give by AK L + ( δ ) K W L Q K. (3) Here K is capital, L is labor, ad Q is the market price of capital. The first term i (3) is ew product, give by a Cobb-Douglas productio fuctio with costat returs to scale. The secod term is the proportio of the capital stock that is ot used up i productio ad that ca therefore be sold as output alog with ew product. The last two terms are the cost of iputs. Maximizig (3) with respect to the firm s two choice variables K ad L produces as OC s K : AK L + ( δ ) Q (4) L : ( ) K L W ( t ). (5) Though (4) ad (5) costitute two equatios i the two ukow decisio variables K ad L, it turs out that they caot actually be solved for K ad L. The reaso is that both ivolve K ad L oly as the ratio k K L. Sice either equatio ca be solved for k, together they require a certai relatio betwee W ad Q. Usig (5) to get rid of k i (4) we arrive at Q t A W ( ( ) t ) δ + KJ. (6) This is a stadard situatio whe the techology is costat returs to scale. actor prices determie the ratios of iputs, but the absolute levels of them are idetermiate. Also factor prices have to be such as to leave profits zero. f profits are positive, scalig productio up always icreases profit, without boud. f profits are egative, the maximum profit of zero is obtaied by scalig productio back to zero. Ad if profits are zero, they are zero at every scale 3

4 of productio. This result, that with costat returs to scale ad competitive equilibrium profits are zero, is sometimes summarized as factor paymets exhaust the product. Note that we have ot said who ows these firms. Presumably the firms motivatio for maximizig profits is that there are owers receivig profits as icome who istruct the firms to do this. Yet our discussio of the cosumer made o metio of icome from owership of the firm. We ca igore owership ad profits i the cosumer problem because, with our costatreturs-to-scale setup, profits are zero i competitive equilibrium. Owership of the zero-profit firm has o effect o the cosumer s maximizatio problem. V. Equilibrium We ow must add equatios eforcig cosistecy betwee choices of firms ad choices of cosumers. Purchases of iputs by firms must match sales of them by cosumers: b g ( ) ( ) ( ) M t K t N t B t (7) M L N. (8) The umber of people per geeratio is assumed to grow at the rate per period, so i every period. Takig the ratio of (8) to (7), we get Usig (5) ad the first equatio i (), we obtai N ( + ) N( t ) (9) B( t ) k. (0) + b gb g. () B Ak Q is the same thig as + r( t ) ; we used separate otatio just to emphasize the iterpretatio of it as a iterest rate. Recogizig this, we ca use (4) ad () to rewrite (0) etirely i terms of k: k b gb g () Ak( t ) + The steady-state level of k, which we call k, is obtaied by settig k k( t ) ad solvig (), k b gb g A + KJ (3) Note that is the rate of savig out of labor icome (ot exactly the s of the Solow model, which is the rate of savig out of total gross product icludig depreciatio). [A good exercise to check uderstadig: compare this formula for the steady state to that for the Solow model. Which parameters correspod? Do similar parameters have similar effects o steady state i each case? The two that clearly do t are δ ad. ] 4

5 V. Golde Rule ad Dyamic Efficiecy Just as i the case of the Solow model, we ca ask here what is the highest level of utility that ca be sustaied forever. Whe we ask this questio, we take the viewpoit of a plaer that ca choose C ( t ), C ad B ( t ) directly at every date. We assume the solutio will make these variables the same at every date. (This is ot completely obvious. f you wat a really challegig mathematical problem, try to prove rigorously that i this model the highest feasible costat level of utility is obtaied with C ( t ), C ad B ( t ) all costat.) The plaer s problem is subject to C max C C C, C, B C + + B A B + + s (4) K J + B ( δ ). (5) + [Where does this equatio come from? Why the + divisor for C? Why the + divisor for B i some places but ot others?] troducig λ as the Lagrage multiplier for the costrait (5), we obtai OC s : C C λ (6) C : C λ gc C + b K J + + B : δ A B + B. (8) (7) Equatio (8) implies that, if this allocatio were cosistet with a competitive equilibrium, the iterest rate would satisfy r, (9) a result similar to that for the Solow model. [You should be able to prove this. Usig (3), you should be able to show that the competitive equilibrium steady state is ot i geeral a golde rule steady state, ad ideed that it ca be iefficiet, i that it ca make r <.] {} 5

Estimating Proportions with Confidence

Estimating Proportions with Confidence Aoucemets: Discussio today is review for midterm, o credit. You may atted more tha oe discussio sectio. Brig sheets of otes ad calculator to midterm. We will provide Scatro form. Homework: (Due Wed Chapter