Ramsey Model - Cashier - Coopmans

The Ramsey – Kass – Koopmans model ( the Ramsey model ) is a neoclassical model of equilibrium economic growth in which the “trajectory” of consumption and savings is determined on the basis of solving the problem of optimizing households and firms in the conditions of perfect competition .

Content

1 History of creation
2 Model Description
- 2.1 the challenge of consumer choice
- 2.2 The task of the company
- 2.3 General economic balance
- 2.4 Modified Golden Rule
3 Analysis of the model
4 See also
5 notes

Creation History

In 1928, F. P. Ramsey 's work “The Mathematical Theory of Savings” ^{[1] was} published, in which the optimality condition and the intertemporal utility function were stated. in his 1965 paper “Optimal Growth in the Aggregated Capital Accumulation Model” ^[2] and Tjalling Kupmans in his 1963 paper “On the Concept of Optimal Economic Growth” ^[3] introduced the optimal consumption method into the neoclassical growth model, defining an endogenous definition savings rates, forming in general the Ramsey – Kass – Coopmans model ^[4] .

Model Description

The model uses a representative household with an infinite lifespan. The dynamics of its consumer and savings behavior is analyzed. A closed economy with real variables (in units of goods and services) is assumed in conditions of perfect competition.

The Choice of Consumer Choice

A representative household is considered. It is conditionally assumed that the decisions of this household are equivalent to the decisions of an infinitely living individual who takes into account current and future welfare and resources. The utility function of this individual, representing the entire population, has the form:

U=\int _{0}^{\infty }u(c_{t})e^{-\rho t}dt

{\ displaystyle U = \ int _ {0} ^ {\ infty} u (c_ {t}) e ^ {- \ rho t} dt}

,

Where $c_{t}$ ${\ displaystyle c_ {t}}$ - per capita consumption at time $t$ ${\ displaystyle t}$ ; $\rho$ ${\ displaystyle \ rho}$ - a positive discount factor reflecting the intertemporal preferences of the individual.

Utility function $u(c_{t})$ ${\ displaystyle u (c_ {t})}$ is separable, that is, it depends only on consumption at that moment. In addition, it is assumed that marginal utility (derivative $u'(c)$ ${\ displaystyle u '(c)}$ ) is a positive and decreasing function and the Inada conditions are satisfied - when consumption tends to zero, marginal utility tends to infinity, and when consumption tends to infinity, marginal utility tends to zero.

The income of the “individual” is composed of wages $w_{t}$ ${\ displaystyle w_ {t}}$ and income $r_{t}a_{t}$ ${\ displaystyle r_ {t} a_ {t}}$ from assets $a_{t}$ ${\ displaystyle a_ {t}}$ belonging to him and having profitability $r_{t}$ ${\ displaystyle r_ {t}}$ (assets can also be negative, which reflects the situation of net debt, and the rate on borrowed funds is assumed to be the same with the return on positive assets). At the same time, income is spent either on consumption or on increasing assets (savings). Thus, an increase in assets per unit time is $w+ra-c$ ${\ displaystyle w + ra-c}$ . You must also consider that the population is growing $n$ ${\ displaystyle n}$ , therefore, assets per person are reduced at the same rate, that is, the rate of change of assets at each point in time is reduced by $n a$ ${\ displaystyle na}$ . Thus, the final budgetary restriction of an individual has the form:

{\dot {a}}=w+ra-c-na

{\ displaystyle {\ dot {a}} = w + ra-c-na}

.

The goal of optimizing consumer behavior is to maximize $U$ ${\ displaystyle U}$ under this restriction. Using the Pontryagin maximum principle , the Hamilton function is constructed:

H=u(c)e^{-\rho t}+\lambda (w+ra-c-na)

{\ displaystyle H = u (c) e ^ {- \ rho t} + \ lambda (w + ra-c-na)}

and maximum conditions:

\lambda =u'(c)e^{-\rho t}

{\ displaystyle \ lambda = u '(c) e ^ {- \ rho t}}

and

{\dot {\lambda }}=-(r-n)\lambda

{\ displaystyle {\ dot {\ lambda}} = - (rn) \ lambda}

.

From here the equation of dynamics of consumption is derived:

{\frac {\dot {c}}{c}}={\frac {1}{\theta }}(r-n-\rho )

{\ displaystyle {\ frac {\ dot {c}} {c}} = {\ frac {1} {\ theta}} (rn- \ rho)}

,

Where $\theta =-{\frac {u''(c)}{u'(c)}}c$ ${\ displaystyle \ theta = - {\ frac {u '' (c)} {u '(c)}} c}$ - elasticity of marginal utility for consumption. This value is positive due to the positiveness of marginal utility and the negativity of the second derivative of utility (decreasing marginal utility).

For a stationary state to exist, it is necessary that $\theta$ ${\ displaystyle \ theta}$ asymptotically tended to a constant value, therefore, as a utility function $u(c)$ ${\ displaystyle u (c)}$ use a function of the following form:

u(c)={\frac {c^{1-\theta }-1}{1-\theta }}

{\ displaystyle u (c) = {\ frac {c ^ {1- \ theta} -1} {1- \ theta}}}

,

Where $\theta$ ${\ displaystyle \ theta}$ - constant.

Company Objective

A representative firm is considered whose production function describes the aggregate supply. The production function is a neoclassical and similar production function in the Solow model : $Y_{t}=F(K_{t},L_{t}E_{t})$ ${\ displaystyle Y_ {t} = F (K_ {t}, L_ {t} E_ {t})}$ where $K$ ${\ displaystyle K}$ - capital $L$ ${\ displaystyle L}$ - labor $E$ ${\ displaystyle E}$ - labor efficiency. It is assumed that labor efficiency is growing at a constant rate $g$ ${\ displaystyle g}$ .

Due to the homogeneity of the production function, we can write $Y=LEf(k)$ ${\ displaystyle Y = LEf (k)}$ where $k=K/LE$ ${\ displaystyle k = K / LE}$ - capital-labor ratio with constant efficiency. Then:

Y'_{K}=LEf'(k)/LE=f'(k)

{\ displaystyle Y '_ {K} = LEf' (k) / LE = f '(k)}

,

Y'_{L}=[f(k)-kf'(k)]E_{t}=[f(k)-kf'(k)]e^{gt}

{\ displaystyle Y '_ {L} = [f (k) -kf' (k)] E_ {t} = [f (k) -kf '(k)] e ^ {gt}}

,

where, in order to simplify, it is assumed that labor efficiency at zero time is equal to unity, therefore, the dynamics of labor efficiency is described as $E_{t}=e^{gt}$ ${\ displaystyle E_ {t} = e ^ {gt}}$ .

In conditions of perfect competition, marginal productivity by factors of production is equal to the prices of these factors. The price of labor is equal to wages $w$ ${\ displaystyle w}$ , and the price of capital is $r+\delta$ ${\ displaystyle r + \ delta}$ where $\delta$ ${\ displaystyle \ delta}$ - the rate of depreciation of capital. Then:

f'(k)=r+\delta

{\ displaystyle f '(k) = r + \ delta}

,

[f(k)-kf'(k)]e^{gt}=w

{\ displaystyle [f (k) -kf '(k)] e ^ {gt} = w}

.

General economic balance

Since a closed economy is considered, capital is owned by residents and unit capital $K/L$ ${\ displaystyle K / L}$ per employee equals assets $a$ ${\ displaystyle a}$ .

By analogy with the Solow model, we can write the equation of the dynamics of capital-labor ratio of labor with constant efficiency:

{\dot {k}}=f(k)-{\hat {c}}-(\delta +n+g)k

{\ displaystyle {\ dot {k}} = f (k) - {\ hat {c}} - (\ delta + n + g) k}

,

Where ${\hat {c}}=c/LE$ ${\ displaystyle {\ hat {c}} = c / LE}$ - consumption per unit of labor with constant efficiency.

Given that ${\dot {\hat {c}}}/{\hat {c}}={\dot {c}}/c-g$ ${\ displaystyle {\ dot {\ hat {c}}} / {\ hat {c}} = {\ dot {c}} / cg}$ Based on the solution of the consumer problem, the following equation can be written:

{\dot {\hat {c}}}/{\hat {c}}={\frac {1}{\theta }}(r-n-\rho -g\theta )

{\ displaystyle {\ dot {\ hat {c}}} / {\ hat {c}} = {\ frac {1} {\ theta}} (rn- \ rho -g \ theta)}

,

or taking into account equality $f'(k)=r+\delta$ ${\ displaystyle f '(k) = r + \ delta}$ substituting expression $r$ ${\ displaystyle r}$ across $k$ ${\ displaystyle k}$ and $n$ ${\ displaystyle n}$ :

{\dot {\hat {c}}}/{\hat {c}}={\frac {1}{\theta }}(f'(k)-\delta -n-\rho -\theta g)

{\ displaystyle {\ dot {\ hat {c}}} / {\ hat {c}} = {\ frac {1} {\ theta}} (f '(k) - \ delta -n- \ rho - \ theta g)}

.

This differential equation together with the differential equation for capital-labor ratio $k$ ${\ displaystyle k}$ and determine the economic dynamics in the framework of this model.

Modified Golden Rule

In the Solow model, a golden savings rule is established to maximize consumption. In the Ramsey model, this rule is modified and has the form:

f'(k)=\delta +n+\rho +\theta g

{\ displaystyle f '(k) = \ delta + n + \ rho + \ theta g}

,

which corresponds to constant consumption per unit of labor with constant efficiency (or growth in consumption per person at a rate $g$ ${\ displaystyle g}$ )

Model Analysis

The model explains the basic macroeconomic patterns, but does not explain the reasons for global economic growth. The overall long-term growth in this model provides a parameter of labor efficiency, which is not explained in the model, but is given exogenously .

Notes

↑ Ramsey FP A mathematical theory of saving // The Economic Journal. - December 1928. - P. 543–559.
↑ Optimum Growth in an Aggregative Model of Capital Accumulation // The Review of Economic Studies. - July, 1965. - P. 233-240.
↑ Koopmans TC On the concept of optimal economic growth // Cowles Foundation for Research in Economics, Yale University. - December, 1963.
↑ Barro R. J. , Sala-i-Martin H. Economic Growth. - M .: BINOM. Knowledge laboratory. - 2010 .-- S. 27-30,115-184. - ISBN 978-5-94774-790-4 .

[:1-1] Ramsey FP A mathematical theory of saving // The Economic Journal. - December 1928. - P. 543–559.

[:2-2] Optimum Growth in an Aggregative Model of Capital Accumulation // The Review of Economic Studies. - July, 1965. - P. 233-240.

[:3-3] Koopmans TC On the concept of optimal economic growth // Cowles Foundation for Research in Economics, Yale University. - December, 1963.

[:4-4] Barro R. J. , Sala-i-Martin H. Economic Growth. - M .: BINOM. Knowledge laboratory. - 2010 .-- S. 27-30,115-184. - ISBN 978-5-94774-790-4 .