The Ramsey Model Reading - Fidrmuc

1 The Ramsey ModelReading:Romer, Chapter 2-A;Developed by Ramsey (1928), later developed further by Cass (1965) and Koopmans (1965).Similar to the Solow Model : labor and knowledge grow at exogenous difference: capital stock is determined by optimization decisions of households are many identical firms, each with the same production functionY F K,AL .Theproduction function displays the same properties as before. Firms hire labor and capital incompetitive markets. For simplicity, we assume there is no depreciation of capital ( 0).HouseholdsHidentical infinitely-lived households. The size of each household grows at raten.

2 Eachhousehold member supplies one unit of labor and rents its capital to maximizes its lifitime utility:U t 0 e tu C t L t Hdtwhereu C t is theinstantaneous utilityof each member of the household,L t His the numberof members of the household and is the discount relative risk aversion (CRRA)utility function:u C t C t 1 1 where 0and n 1 g :u C C 0u C C 1 0so that thecoefficient of relative risk aversionis : Cu C u C C C 1C .Behavior of Households and FirmsFirms are competitive and earn zero profits. Firms hire capital and labor and pay them theirmarginal products. The real interest rate then is:r t f k t.

3 The real wage isW t F K,AL ALf k andk K/AL:W t Af k Akf k A f k t k t f k t .The wage per unit of effective labor then isw t f k t k t f k t .Households Budget ConstraintHouseholds takerandwas given. The budget constrain stipulates that the present value ofconsumption cannot exceed the sum of initial wealth and present value of labor t 0tr d so that one unit of the good invested at time 0 is wortheR t units at timet;R t captures theeffect of continuously compounding interest over the period 0,t . Similarly, one unit of outputat some future timetis worthe R t units at time budget constraint then is: t 0 e R t C t L t Hdt K 0 H t 0 e R t W t L t HdtorK 0 H t 0 e R t W t C t L t Hdt integral can be rewritten as a limit:lims K 0 H t 0se R t W t L t H C t L t Hdt capital stock at any timesisK s H eR s K 0 H t 0seR s R t W t C t L t Hdt,that is, household wealth at any time equals to the interest-compounded value of its initialwealth and its savings (positive or negative).

4 This can be rewritten asK s H eR s K 0 H t 0se R t W t C t L t Hdtso that the household budget constrain becomeslims e R s K s H implies that households cannot follow a path of consumption and investment that wouldresult in negative net present value of wealth (no-Ponzi-game condition).Households Maximization ProblemHouseholds maximize their lifetime utility subject to the budget constraint. All households areidentical, therefore all will choose the same path of consumption and consumption per unit of effective laborc t so thatC t A t c t (each worker hasAunits of effective labor) andC t 1 1 A t c t 1 1 A 0 egt 1 c t 1 1 A 0 1 e 1 gtc t 1 1.

5 Household life-time utility function becomesU t 0 e tC t 1 1 L t Hdt t 0 e tA 0 1 e 1 gtc t 1 1 L 0 entHdt A 0 1 L 0 H t 0 e t 1 gt ntc t 1 1 dt B t 0 e tc t 1 1 dtwhereB A 0 1 L 0 Hand n 1 g. Note that we assumed to be budget constraint, t 0 e R t C t L t Hdt K 0 H t 0 e R t W t L t Hdtcan be rewritten in terms of capital, consumption and wage per effective labor: t 0 e R t c t A t L t Hdt k 0 A 0 L 0 H t 0 e R t w t A t L t Hdt t 0 e R t c t e n g tA 0 L 0 Hdt k 0 A 0 L 0 H t 0 e R t w t e n g tA 0 L 0 Hdt t 0 e R t e n g tc t dt k 0 t 0 e R t e n g tw t limit version of the budget constraint can be also rewrittenlims e R s K s H 0lims e R s e n g sk s A 0 L 0 H 0lims e R s e n g sk s 0 Households BehaviorHouseholds choose the path ofc t that maximizes their lifetime utility subject to the budgetconstraint.

6 Because additional consumption always increases utility,u C 0, the budgetconstraint will be met as Langrangean:B t 0 e tc t 1 1 dt k 0 t 0 e R t w t e n g tdt t 0 e R t e n g tc t dtThe household choosesc t at any point in time according to the following FOC:Be tc t e R t e n g tand according to the budget logs of the FOC:lnB t lnc t ln R t n g tlnB t lnc t ln 0tr d n g derivatives t c t r t n gor t c t r t n g .Substituting n 1 g t c t r t g r t gSinceC t A t c t , consumption per worker grows at the rate of growth ofc t plus the rateof growth of knowledge,g: t C t r t .Hence, consumption per worker grows if the interest rate exceeds the discount rate and fallsotherwise.

7 This equation is referred to as theEuler equation; it describes howc t evolves forany given value ofc 0 . The household choosesc 0 so as to satisfy the budget constraint: thepresent value of lifetime consumption must equal the initial wealth plus the present value Dynamics of the EconomyDynamics of c: The Euler equation can be rewritten usingr t f k t t c t f k t g . t 0 whenf k t g. Denote the level ofkfor which this is the casek .Consumption is incresing for allk k and falling fork k .See Figure in the of k: Recall that, as we derived for the Solow Model ,k t sf k t n g k t . Here, assuming no depreciation and allowing savings to vary:k t f k t c t n g k t.

8 For anyk,k 0 whenc f k n g ,kwill remain constant if consumption equals the difference between output andbreak-even investment. The first term is increasing inkwith diminishing returns, the secondterm is linear ink. Therefore, the level of consuption that keepskconstant is hump-shaped inkand peaks at suchkfor whichf k n g(golden-rulelevel ofk). If consumption is lower,kis increasing, and vice Figure in the State:The value ofk is given byf k g. The golden-rulekis given byf kGR n n 1 g 0or g n ,k lines characterizing 0andk 0 can be combined in aphase diagram. For everyk 0 0, there is a unique level ofcthat is consistent with the household s intertemporaloptimization and will bring the economy to the steady state.

9 The set of all such combinationsofcandkis referred to as thesaddle Figures - in the Growth PathSolow and Ramsey models display similar properties in equilibrium:(1) Capital, output and consumption per unit of effective labor are constant. The savings rate,y cy, is also constant (because bothyandcare constant).(2)K,Yand total consumption growth at raten g.(3) Capital per worker, output per worker and consumption per worker grow at , the basic prediction of the Solow Model are reproduced also in the Ramsey Model : inthe steady state, the rate of growth of output per worker is determined entirely by , however, that the golden-rule level ofkwill not be attained in the Ramsey Model .

10 In theSolow Model , the savings rate is exogenous and therefore any level ofk, including thegolden-rule one, can constitute a steady state. In the Ramsey Model , the equilibrium is suchthatk the Ramsey Model , the savings rate is the outcome of households intertemporaloptimization rather than being exogenous. ChoosingkGRwould lead to higherc, but sincehouseholds discount future consumption, this is not in the Discount RateConsider an economy that is on the balanced growth path. Suppose the discount rate, , discount rate only affects the equation for consumption, t c t f k g .The steady-state capital is given byf k falls, this means that that the newk is higher than the original equilibrium.