Question

Solve for c₁ and c₂ for the case of the two-period consumption and saving model with certainty, using the quadratic form u(c) = c - 0.5ac² for the instantaneous utility function and with the following income patterns {y₁, y₂} and initial wealth w₀.

Answer 1

Consider

$u(c) = c-0.5ac^2 \\ c_1+b_1=y_1 \\ c_2+b_2 = y_2 + (1+ω)b_1 \\ c_1 + \frac{c_2}{1+ω} = y_1 + \frac{y_2}{1+ω}$

Consider

$c_1 + \frac{c_2}{1+ω}=x$

Define

$u'(c_1)dc_1 + βu'(c_2)dc_2 = 0 \\ \frac{dc_2}{dc_1} = \frac{-u'(c_1)}{β(u')c_2} \\ u(c_1>c_2) = c_1 -0.5ac_1^2 \\ = 1 -ac_1$

Similary $(1-ac_2)$

Consider $c_1=c_2$

$\frac{c(1-a)}{c(1-a)}=1$

However

$\frac{dc_2}{dc_1} = \frac{-c_2}{βc_1} = \frac{1}{β}$