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From a chronological viewpoint, the starting point for modern growth theory is Ramsey's article on the optimal social savings behaviour entitled A Mathematical Theory of Saving. Since Ramsey applied variational calculus* to answer the question how an infinitely lived representative agent should allocate his resource over time, his work--regarded by Keynes as "one of the most remarkable contributions to mathematical economics ever made"(obituary notice of Ramsey)--was "terribly difficult reading for an economist" (Keynes, ibidem). In case you have troubles following Ramsey's arguments, I highly recommend reading "The Ramsey Exercise" over at the History of Economic Thought website. Unfortunately, the site fails to mention that in Ramsey's paper the Euler equations are introduced before the optimization problem is specified, i.e. they can also be seen as a stand-alone marginalist result.

Ramsey's growth model was refined by Cass (1965) and Koopmans (1965). Here is a short sketch (borrowing heavily on Robert Lucas' Lectures on Economic Growth): Consider a closed economy with competitive markets, with identical rational agents and a constant returns technology. At date t there are L(t) persons, the exogeneously given rate of growth of L(t) is n. Real, per capita consumption is a stream c(t) t ≥ 0, of units of a single good. Preferences over consumption streams are given bycass01where the discount rate ρ and the coefficient of relativ risk aversion θ are both positive. The elasticity of substitution for this utility funtion is the constant σ = 1/θ. Hence, this form is either called the constant-relative-risk-aversion utility (CRRA) function or the constant intertemporal elasticity of substitution (CIES) utility funtion. Production per capita of the one good is divided into consumption c(t) and capital accumulation. If we let K(t) denote the total stock of capital, then total output is L(t)c(t) + dK(t)/dt. Production is assumed to depend on the level of capital and labor inputs and on the level A(t) of the "technology" according tocass02where 0 <β< 1 and where the exogenously given rate of technological growth is μ>0.

The ressource allocation problem faced by this economy is simply to choose a time path c(t) for per capita consumption. Given a path c(t) and an initial capital stock K(0), the technology (2) then implies a time path K(t) for capital. One way to think about this allocation problem is to think of choosing c(t) at each date, given the values of K(t), A(t), and L(t) that have been attained by that date. Evidently, it will not be optimal to choose c(t) to maximize current-period utility, [1/(1 - θ)][c(t) - 1]^(1-θ)L(t), for the choice that achieves this is to set net investment d(K(t)/dt equal to zero: One needs to set some value or price on increments to capital. A central construct in the study of optimal allocations, allocations that maximize utility (1) subject to the technology (2), is the current-value Hamiltonian H defined by cass03which is just the sum of current-period utility and (from (2)) the rate of increase of capital, the latter valued at the (shadow) "price" λ(t). An optimal allocation must maximize the expression H at each date t, provided the price λ(t) is correctly chosen. The first-order condition for maximizing H with respect to c iscass04which is to say that goods must be so allocated at each date as to be equally valuable, on the margin, used either as consumption or as investment. It is known that the price λ(t) must satisfy at each date t cass05if the solution c(t) is to yield an optimal path. Now if (4) is used to express c(t) as a funtion of λ(t), and this function is substituted in place of c(t) in (2) and (5), these two equations are a pair of first-order differential equations in K(t) and its "price" λ(t). Solving this system, there will be a one-parameter family of paths (K(t), λ(t))), satisfying the given initial condition on K(0). The unique member of this family that satisfies the transversality condition:cass06is the optimal path! The next step is to construct from (2), (4), and (5) the system's balanced growth path: the particular solution (K(t), λ(t),c(t)) such that the rates of growth of each of these variables is constant. To cut a long story short: The interesting finding is that along a balanced path, the rate of growth of per capita magnitudes (κ, same for capital and consumption)cass07is simply proportional to the given rate of technical change, μ, where the constant of proportionality is the inverse of labor's share, 1 - β. The rate of time preference ρ and the degree of risk aversion θ have no bearing on this long-run growth rate. Low time preference ρ and low risk aversion θ induce a high savings rate s, and high savings is, in turn, associated with relatively high output levels on a balanced path. A thrifty society will, in the long run, be wealthier than an impatient one, but it will not grow faster!

*AFAIK Evans (1924) was the first to use variational calculus to address an economic issue.

related items: Growth Theory: Stylized Facts
Mike Linksvayer (guest) meinte am 13. Sep, 21:10:
more wealth, same growth?
I don't follow the math, (so?) the following seems paradoxical:

"A thrifty society will, in the long run, be wealthier than an impatient one, but it will not grow faster!"

In the long run how can a society be wealthier without growing faster? Output is higher in the initial state, so the same rate of growth results in more absolute wealth? 
Mahalanobis antwortete am 14. Sep, 01:09:
Balanced Growth Path
We are looking at the steady state, so transitional dynamics don't matter. As the economy develops and approaches the steady state many forces act upon the savings rate. On one hand, agents will save less when they are poor due to the form of the utility function (declining marginal utility, consumption smoothing), on the other hand, the return on saving is higher when the capital stock is low - which actually should lead to a higher savings rate in poor economies. How the saving rate (and therefore output growth) develops over time for different countries depends on quite a lot of paramters.

However, the savings rate will be constant in the long-run (but not necessarily the same for different countries) and so will be output growth.

Maybe you have heard of the Solow model (you will find lots of information on the internet) where the savings rate is exogeneously given. Here it makes more sense to ask what happens if the savings rate is incrased by x%. The answer is that output growth would temporarily increase but after a while output growth would be the same as before. So an incrase in the savings rate only has a level effect. 
Daniel Lam (guest) antwortete am 14. Sep, 04:08:
Risk Aversion?
Since there is no uncertainty in this model, I'm not sure that it adds anything to interpret theta as the coefficient of relative risk aversion. Its sole effect is that of (the reciprocal of) the elasticity of intertemporal substitution. 
Mahalanobis antwortete am 14. Sep, 05:45:
Elasticity of Marginal Utility
Lucas notes in a footnote* that "the coefficient of risk aversion is sometimes called the intertemporal elasticity of substitution. Since all the models considered in this chapter are deterministic, this latter terminology might be more suitable."

When thinking about the curvature of the utility function (u(c) → ln(c) as θ → 1 and u(c) → c -1 as θ → 0) I immediately categorize in risk averse, risk neutral, and risk loving. Maybe that's why I also tend to denote θ the coefficient of (relative) risk aversion. I do not spend more time thinking about the form of the utility function. Btw, it has never happend to me that I met a student who could correctly explain the Elasticity of Substitution.

*after denoting θ (actually σ - he uses a different notation) the coefficient of (relative) risk aversion 
kimcil (guest) antwortete am 7. Apr, 12:17:
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When's Part II being released? I learn more on this blog than I do watching PBS. 
Mahalanobis antwortete am 19. Sep, 22:58:
This is a weekly (oops) semimonthly series ;-D. 
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radek (guest) meinte am 21. Sep, 02:13:
The fact that the same coefficient captures both elasticity of substitution and risk aversion causes some problems in growth-style models with uncertainty (like capital asset pricing models and Ramsey-based RBC models). And just intuitively, there is no reason to think that a person who is say risk averse, MUST regard consumption in different periods in a particular way. Some of the newer behavioral economics has sought to disentangle the two. I can't remember any papers specifically off the top of my head, but I believe Thaler has some stuff on it.

BTW, even though you say you never met a student who could explain elasticity of substitution, you link to the wrong thing. That page is about substitution between capital and labor. Here we're talking about the substitution between consumption today and consumption tomorrow. 
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