Ramsey’s Conjecture in a Stochastically Growing Economy

Abstract

Becker (1980) confirms Ramsey’s conjecture (Ramsey 1928) in a discrete-time model by proving that the most patient individuals own the entire capital of the economy, while others consume their wage income in the long-run.

Appendix

Applying Ito’s lemma to the maximization of $V_{i} (t)$ in (4.4), we get the following stochastic Bellman equation:

$$\begin{aligned} 0= & {} \max \limits _{c_{i}}\{ [(1-\gamma _i )/(1-1/\varepsilon _i)]c_i^{1-1/\varepsilon _i} -\rho _if([1-\gamma _i]J_i(k_i))\nonumber \\+ & {} (1-\gamma _i)f'([1-\gamma _i]J_i(k_i)) [ J_i'(k_i)(Ak_i-c_i)+(1/2)J_i"(k_i)(\sigma A k_i)^2 ]\}. \end{aligned}$$

(4.32)

From (4.32), the first-order condition for $c_{i}$is

$$\begin{aligned} c_{i}^{-1/\varepsilon _{i}}=f'([1-\gamma _{i}]J_{i}(k_{i}))J_{i}'(k_{i}). \end{aligned}$$

(4.33)

Equation (4.4) suggests that $J_{i} (k_{i} )$ takes the following form:

$$\begin{aligned} J_i(k_i)=(b_ik_i)^{1-\gamma _i}/(1-\gamma _i), \end{aligned}$$

(4.34)

where $b_{i}$ is a positive constant to be determined. Now, (4.33) becomes

$$\begin{aligned} c_{i} =u_{i} k_{i}~~with~~ u_{i} =b_{i}^{1-\varepsilon _{i} }. \end{aligned}$$

(4.35)

Substitution of (4.34) and (4.35) into (4.32) gives

$$\begin{aligned} u_{i}=A-\varepsilon _{i}(A-\rho _{i})-\frac{(1-\varepsilon _{i} )\gamma _{i} (\sigma _{i} A)^{2}}{2}, \end{aligned}$$

(4.36)

and therefore,

$$\begin{aligned} b_{i}=\left[ A-\varepsilon _i(A-\rho _i)-\frac{(1-\varepsilon _i)\gamma _{i} (\sigma _{i} A)^2}{2}\right] ^{1/(1-\varepsilon _i)}. \end{aligned}$$

(4.37)

Substitution of (4.37) into (4.35) gives the following consumption function:

$$\begin{aligned} c_{i} =u_{i}k_{i}=\left[ A-\varepsilon _{i}(A_{i} -\rho _{i})-\frac{(1-\varepsilon _{i} )\gamma _{i}(\sigma _{i} A)^{2}}{2}\right] k_{i} . \end{aligned}$$

(4.38)