Consumption

• : consumption at .
• : discount rate
• : relative risk aversion coefficient
• assumed that every representative individual has infinite lifetime

• : initial asset
• : stream of the income is known at .
• : interest rate is assumed to be a constant

Note that we have

Lagrangian is F.O.C. thus we have i.e.

From the previous result of , we have if , meaning that the interest rate equals to the discount rate. Now denote the steady state as implying perfect consumption smoothing.

Remark that, for , we have (in quarterly), thus in annual, the interest rate is assumed to be .

Then how to drive ? As the budget constraint hold in equality by , we have and the left hand side is therefore, we have where denotes the permanent income. Finally, we have This implies that the changes in (instantaneous income) has very marginal effect on the current consumption, unless it is a permanent change.

Lifetime Utility function is and the budget constraint is thus when the equality holds, we have assuming , we have Lagrangian is F.O.C. Thus the Euler equation is

Assume then from the Euler equation, we have Plugging back to the budget constraint, we have thus Finally, we have this implies that does not depend on , since the properties of the log utility function shows that its income effect and the substitution effect are canceled each other.

Now assume Then, the Euler equation is Plugging back to the budget constraint, we have and Denoting the saving rate, we have