The AK Model
Romer developed a Ramsey version of the AK model, in which the constant saving rate is replaced by intertemporal utility maximization by a representative individual with the assumption that individuals do not internalize the externalities associated with the growth of knowledge.
The Setup
Dynamic optimisation problem is given as
where k is the capital stock of the individual firm, y is its output, c = ct is the current consumption of its owner-worker, and A denotes aggregate productivity that is taken as given.
Having rational expectations, individuals correctly anticipate that all firms will employ the same capital in equilibrium so,
K = Nk
And therefore the preceding Euler condition can be written as
There are three cases to consider depending on the exponent α+η
Case 1. α+η < 1
In the case of decreasing returns, again growth will vanish asymptotically as in the neoclassical model without technological progress. Positive growth implies that the capital stock k will converge to infinity over time, implying that the right-hand side of equation 5 must converge to r, since the exponent α+η − 1 is negative, and in turn implying that the growth rate c /c (c dot upon c) will become negative, a result which contradicts our assumption of positive growth.
Case 2. α+η > 1
In the case of increasing returns to capital, then, as in the Frankel model, there will be explosive growth. This can be seen using the Euler equation 5. Specifically, if growth is positive in the long run, then the capital stock k converges to infinity over time. This result, together with the fact that α+η>1, implies that the right-hand side of equation 5 converges to negative infinity.
Case 3. α+η =1
In the AK case where there are constant social returns to capital, then, as in the Frankel model, the economy will sustain a strictly positive but finite growth rate g, in which diminishing private returns to capital are just offset by the external improvements in technology that they bring about .
In steady state, consumption and output will grow at the same rate. So this implies
we see that the higher the discount rate r (that is, the lower the propensity to save), or the lower the intertemporal elasticity of substitution measured by 1/e, the lower will be the steady-state growth rate g.
Welfare
Because individuals and individual firms do not internalize the effect of individual capital accumulation on knowledge, when optimizing on c and k, the equilibrium growth rate g is less than the socially optimal rate of growth.
The social planner who internalizes the knowledge externalities induced by individual capital accumulation would solve the dynamic program
With constant social return to capital (α+η=1), this gives the socially optimal rate of growth
Reference: Aghion & Howitt.
Romer developed a Ramsey version of the AK model, in which the constant saving rate is replaced by intertemporal utility maximization by a representative individual with the assumption that individuals do not internalize the externalities associated with the growth of knowledge.
The Setup
Dynamic optimisation problem is given as
where k is the capital stock of the individual firm, y is its output, c = ct is the current consumption of its owner-worker, and A denotes aggregate productivity that is taken as given.
Having rational expectations, individuals correctly anticipate that all firms will employ the same capital in equilibrium so,
K = Nk
And therefore the preceding Euler condition can be written as
There are three cases to consider depending on the exponent α+η
Case 1. α+η < 1
In the case of decreasing returns, again growth will vanish asymptotically as in the neoclassical model without technological progress. Positive growth implies that the capital stock k will converge to infinity over time, implying that the right-hand side of equation 5 must converge to r, since the exponent α+η − 1 is negative, and in turn implying that the growth rate c /c (c dot upon c) will become negative, a result which contradicts our assumption of positive growth.
Case 2. α+η > 1
In the case of increasing returns to capital, then, as in the Frankel model, there will be explosive growth. This can be seen using the Euler equation 5. Specifically, if growth is positive in the long run, then the capital stock k converges to infinity over time. This result, together with the fact that α+η>1, implies that the right-hand side of equation 5 converges to negative infinity.
Case 3. α+η =1
In the AK case where there are constant social returns to capital, then, as in the Frankel model, the economy will sustain a strictly positive but finite growth rate g, in which diminishing private returns to capital are just offset by the external improvements in technology that they bring about .
In steady state, consumption and output will grow at the same rate. So this implies
we see that the higher the discount rate r (that is, the lower the propensity to save), or the lower the intertemporal elasticity of substitution measured by 1/e, the lower will be the steady-state growth rate g.
Welfare
Because individuals and individual firms do not internalize the effect of individual capital accumulation on knowledge, when optimizing on c and k, the equilibrium growth rate g is less than the socially optimal rate of growth.
The social planner who internalizes the knowledge externalities induced by individual capital accumulation would solve the dynamic program
With constant social return to capital (α+η=1), this gives the socially optimal rate of growth
Reference: Aghion & Howitt.
Hi. I’m Vivek Sharma. I do write Blog on different issues and topics related to Economy, Exams and Political issues. Inspired to make things looks better.
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