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 by
where 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 to
where 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
which 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 is
which 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 
if 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:
is 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)
is 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
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 by
where 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 towhere 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
which 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 iswhich 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 if 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:is 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)is 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
Mahalanobis - am 2005-09-13 09:54 - Rubrik: EconoSchool
