Q2: Consumption-saving decision

Max c1, c2, s [(c1^(1-))/(1-)+ *(c2^(1-))/(1-)]

s.t.c₁ + s = () y₁

c₂ = () (1 + r)s + y₂

where c1 and c2 are consumption in periods 1 and 2 respectively; y1 and y2 are income in periods 1 and 2 respectively; s is savings; r is the interest rate on savings. is a parameter controlling the concavity of the utility function, and will determine intertemporal substitution of consumption.4 We assume that > 1; so utility is increasing and concave in consumption.

(a)Use the method of substitution to solve for optimal savings s*

(b)Use the method of the Lagrangian to solve for optimal savings s*

(c)Find the solutions for optimal consumption in periods 1 and 2, ie. c*1 and c*2 .

(d)For = 1 look at how consumption in period 1 depends on the interest rate r (you can just copy the expression for c*1 from the lecture slides). Describe how monetary stimulus (a decrease in the interest rate) works to increase demand (consumption) today. (Whether taking derivatives of the optimal choices with respect to the interest rate, or just by describing it's effect, is up to you.)