A Note on Ramsey, Harrod-Domar, Solow, and a Closed Form

A Note on Ramsey, Harrod-Domar, Solow, and a Closed Form Saddle Path Halvor Mehlum Abstract Following up a 50 year old suggestion due to Solow, I show that by including a Ramsey consumer in the Harrod-Domar model, the knife-edge problem is solved More surprisingly, however, the derived saddle path happens to be a closed form expression The existence of a closed form expression greatly simplifies the analysis of how the parameters of the utility function affect investments and growth Keywords: Ramsey growth model JEL: D91, O41 Department of Economics, University of Oslo PO Box 1095, Blindern N-0317 Oslo, Norway E-mail: halvormehlum@econuiono 1

A note on Ramsey and a Closed Form Saddle Path 2 1 Introduction One of the main contributions to growth theory is the Solow model (1956) As a backdrop for his own modeling Solow expresses uneasiness with the knife-edge feature of the Harrod- Domar model and in the introduction to his article Solow writes he bulk of this paper is devoted to a model of long-run growth which accepts all the Harrod-Domar assumptions except that of fixed proportions Instead I suppose that the single composite commodity is produced by labor and capital under the standard neoclassical conditions [], to see if the Harrod instability appears (p 66) Indeed he finds that the knife-edge feature of the Harrod-Domar model goes away when using the neoclassical production function This was not the first time, however, that Solow expressed his uneasiness with the implications of the Harrod-Domar model Already in a note in 1953 Solow points out several problems in Mr Harrod s system Problems that eventually are handled in the 1956 article However, that the solution would lay in allowing for substitution in production is not pointed out in the 1953 note Here Solow s attention is rather on the assumption of a fixed savings rate In the concluding section of the 1953 note, when pointing out possible ways of building causal dynamics he writes: A mechanical firststepinthisdirection could be made by letting the choice between consumption and investment depend on the interest rate and price level in some arbitrary but simple way Another possibility would be to think of investors as Ramsey-type utility maximisers over time (p 79) In the present note I take up the tread from Solow s 50 year old note and replace Mr Harrod s assumption about a fixed savings rate with savings behavior a la Ramsey (1928) Paraphrasing Solow (1956): The bulk of this note is devoted to a model of long-run growth which accepts all the Harrod-Domar assumptions except that of fixed savings rate Instead I suppose that the savings decision is made by Ramsey-type utility

A note on Ramsey and a Closed Form Saddle Path 3 maximizers This note has two main results (1) Not surprisingly Solow was right: the Harrod instability disappears when the savings rate is flexible (2) Most surprisingly: in this problem, in contrast to most other cases, the Ramsey saddle path has a closed form solution The last result is both interesting in its own right and as a contribution to the literature on the Ramsey model An explicit saddle path makes it a lot easier to get to grips with the Ramsey framework in general - the essential feature of which is the optimizing consumer s behavior in a specific technological environment 2 The model Production per unit of labor is given as a function of capital per unit of labor, K, bythe fixed coefficient production function X = a min K, K, (1) where a is the output-capital ratio and K is the maximal capital stock that can be used productively K is determined by the productivity of labor relative to capital This production function is concave and has the essential properties needed for the Ramsey problem It can be shown that (1) is the limit of a constant elasticity of substitution production function as the elasticity of substitution between labor and capital goes to zero The dynamics of the economy depends on the capital accumulation All income is earned by a representative consumer who is the owner of the firms and the owner of capital The supply of capital accumulation is thus determined by this consumer s sav-

A note on Ramsey and a Closed Form Saddle Path 4 ings/investment decision The consumer maximizes a constant relative risk aversion utility function U = Z C 1 1 σ t=0 t 1 1 1 e θt dt, (2) σ where C is consumption, θ the rate of time preferences, and σ the intertemporal elasticity of substitution When including depreciation δ and using (1), investments (the time derivative of capital) is simply dk dt (a δ) K C when K< K = (3) a K δk C when K> K By using standard methods of dynamic optimization 1, maximizing (2) with respect to (3) and using from (1), the optimal consumptionpathischaracterizedby dc dt = Cσ [ X/ K δ θ] =Cσ [a δ θ]when K< K Cσ [δ + θ] when K> K (4) it further follows that the two steady state terminal conditions are K = K and C =(a δ) K C Throughout it is assumed that the consumer wants to hold some capital, hence a δ θ > 0 Then consumption grows exponentially with rate σ (a δ θ) aslongask < K When K> K, however, consumption will decline exponentially with the rate σ (δ + θ) Figure 2 shows the complete phase diagram when also incorporating the capital dynamics 1 In order to keep the note short and as I assume that the readers are familiar with the problem I avoid technicalities For a careful presentation of the Ramsey problem see for example Barro and Sala-I-Martin (1995)

A note on Ramsey and a Closed Form Saddle Path 5 C Figure 1: The saddle path dc dt =0 dk dt =0 K K from (3) The phase diagram shares all the essential features of the textbook exposition of the Ramsey growth model All paths except the indicated saddle path bumps into either zero capital or zero consumption constraints and are obviously not solutions Along the lower branch of the saddle path (K starting below K) both the capital stock and consumption grow over time until the steady state is reached Along the upper branch (K starting above K) both capital and consumption decline over time until the steady state is reached It should in passing be noted that the steady state condition K = K rules out the knife-edge problem Savings will adjust exactly so that over time there is neither idle capital nor labor As the consumption growth is exponential, the transition time is finite At the time of termination all capital is employed and the return to capital drops down to the level where the consumer has no incentive for further savings nor for dissavings Hence from the point of termination and onwards consumption is stable The two linear differential equations (3) and (4), in combination with the initial condition K = K 0 and the two terminal conditions C = C and K = K, aresufficient for solving for the two transition paths and for the transition time

A note on Ramsey and a Closed Form Saddle Path 6 21 The Saddle Path The saddle path is found by applying standard tools for differential equations The upper and the lower branch has to be treated separately Here I will concentrate on the lower branch, which, from a growth perspective, is the most interesting The solution for this branch is found by transforming the differential equations (3) and (4) to one differential equation between K and C This elimination of time from the system is possible since both K and C are strictly increasing in t LetK = K (C) indicate the function that describes the saddle path in the phase diagram 2 It follows from the rules for differentiation that dk dt K (C) dc = C dt K (C) C = dk dt /dc dt By inserting from (3) and (4) the result is a general linear first-order differential equation K (C) C (a δ) K (C) C = Cσ (a δ θ) (5) Hence, the solution to (5) is found by using standard formulas Following from the two terminal conditions C = C and K = K, the terminal steady state condition for K (C) is K C = K,which determines the constant of integration It can be confirmed by taking the derivative that the solution is simply K (C) = C 1 β C β βc (a δ)(1 β), β = a δ σ (a δ θ) (6) 2 IchoosetoexpressK as a function of C as this has a closed form solution while C as a function of K does not

A note on Ramsey and a Closed Form Saddle Path 7 This explicit expression for the saddle path is the main finding of this paper 3 It gives a closed form solution for the Ramsey saddle path In addition(4)can be solved to give the transition time T until the steady state is reached, where C = C Let the present consumption level be C then T is given by the equation C = Ce Tσ(a δ θ) Hence T (C) =ln µ C σ (a δ θ) (7) C The closed form solution provided in (6) and (7) gives anyone working with the Ramsey problem an accessible way to investigate the consequences of altering the parameters of the utility function One can employ standard methods from calculus and need not use numerical or approximate methods It is for example easy to find the saddle path by solving for K for different C Comparative statics shifts in the saddle path can be analyzed by taking the derivative of (6) with respect to the parameters The saddle path (6) and the transition time (7) give capital and transition time as functions of consumption In order to find the effect of parameter changes on C and T for a given K one has to approach (6) and (7) using implicit differentiation Figure 2 gives two examples of shifts in parameter values - easily and accurately plotted using a spreadsheet: When the intertemporal elasticity of substitution σ is high, consumeriswillingtoforegoconsumptiontoday in order to achieve faster transition to higher income Hence C is low relative to production and the growth rate is high On the other hand, if the rate of time preferences θ is high, the future does not matter that much and the consumer want to consume more today Hence, consumption is high relative to production and the growth is low 4 3 Inthecasewhereβ = 1 the solution becomes K (C) =C 1+ln C/C / (a δ) Along the upper branch of the saddle path both C and K are declining in t and the solution is found by the same method 4 In the numerical example I use the following parameter values: a δ =1/3, δ=005, K =3,σ=05,

A note on Ramsey and a Closed Form Saddle Path 8 C Figure 2: The saddle path 1 0 θ high σ high 1 2 3 dk dt =0 K 22 Transition Time The saddle path gives the relationship between K and C In order to find the consumption level for a given starting value of the capital stock one needs to solve the equation K 0 = K (C 0 ) Here one needs to revert to numeric methods, though of the less demanding sort which can be done even by hand Once C 0 is known (4) gives the exponential growth of C C (t) =C 0 e tσ(a δ θ) (8) Once C (t) is known it can be inserted into (6) to also get K as a function of time The economic growth over time is illustrated in Figure 3 When the intertemporal elasticity of substitution σ is high, investments are high and the growth is fast If the rate of time preferences θ is high, however, investments are low and the growth is slow This concludes the main analysis To summarize: The main result is the existence of the function (6) that gives the closed form solution for the saddle path Combined with (8) and the initial condition on the capital stock it is simple to calculate the time paths for consumption, capital and production θ =005, σ(high) = 1, θ(high) = 020, and in addition (for the next figure) K 0 =15

A note on Ramsey and a Closed Form Saddle Path 9 X Figure 3: Transition time 1 σ high θ high 0 t 0 5 10 15 3 Concluding Remarks Following up a 50 year old suggestion due to Solow, I have shown that by including a Ramsey consumer in the Harrod-Domar model, the knife-edge problem is solved More surprisingly, however, the derived saddle path happens to be a closed form expression Theliteraturecontainssomeotherexamplesof explicit solutions to the Ramsey problem, (see eg Benhabib and Rustichini 1994) These are all quite restrictive: Complete depreciation over one or two periods, crosscutting conditions on parameters in production and consumption et c The assumptions in the analysis above may also be characterized as restrictive - given the fixed coefficient production function The question is: restrictive compared to what? One quite general base-case for the Ramsey model, which satisfies all accepted neoclassical assumptions, is the combination of a general constant returns to scale CES production function, a general CRRA utility function, and depreciation/population growth/technical progress less than unity The present model is within this domain, though at the border, as the fixed coefficient production is the limit of the CES as the elasticity of substitution goes to zero Hence, compared to the base-case only one parameter is fixed - the elasticity of substitution- while all the other parameters are

A note on Ramsey and a Closed Form Saddle Path 10 free The cost of fixing this single parameter should be weighted against the benefit An explicit solution makes it a lot easier to get to grips with the essential feature of the Ramsey model: The optimizing consumer s behavior in a specific technological environment The present model allows the analysis of changes in the parameters of the utility function: the intertemporal elasticity of substitution and the rate of time preferences In addition, important aspects of technology can be analyzed, such as efficiency, a, capacity K and depreciation δ Obviously, all relevant questions cannot be addressed satisfactorily without substitution, eg questions related to factor prices The explicit function describing the saddle path K (C) in (6) follows from the linear differential equation (5) The essential condition for this result is that the marginal productivity of capital is constant during the transition; a property that follows from the fixed coefficient production function Given that this condition is satisfied the model s assumptions may be altered in a number of ways It can be shown that the following alterations can be done without the sacrifice of an explicit expression for the saddle path First, the utility function may be modified to include a minimum consumption, a la Stone-Geary, or be changed to the constant absolute risk aversion type Second, theproductionfunctionneednotstart out in the origin but at the constant b That is: X = b + a min K, K Readers who are unhappy with the lack of substitution may use this production function instead of (1), to approximate production functions with more curvature

A note on Ramsey and a Closed Form Saddle Path 11 References Barro, Robert and Xavier Sala-I-Martin (1995) Economic Growth, McGraw Hill, New York Benhabib, Jess and Aldo Rustichini (1994) A Note on a New Class of Solutions to Dynamic Programming Problems Arising in Economic Growth, Journal of Economic Dynamics and Control, 18, 808-813 Ramsey, Frank (1928) A mathematical theory of saving, Economic Journal, 38, 543-559 Solow, Robert (1953) A Note on the Price Level and Interest Rate in a Growth Model, The Review of Economic Studies, 21(1), 74-79 Solow, Robert (1956) A Contribution to the theory of economic growth, Quarterly Journal of Economics, 70, 65-94