Question $2.$

Consider a household whose utility is determined by its consumption in periods $0$ and $1.$ Let

$c_0$ and $c_1$ denote the consumption in periods $0$ and $1,$ respectively. The utility of this household

can be represented by a utility function

$U(c_0,c_1)=u(c_0)+u(c_1).$

Assume further that the per$-$period utility $u(c)$ is given by $u(c)=logc,$ and the discount factor

$$ is given by $=\frac{9}{10}.$ In periods $0$ and $1,$ this household is endowed with incomes $y_0=190$ and

$y_1=380,$ respectively. Importantly, this household can save or borrow in period $0$ at the interest

rate $r=\frac{1}{9}.$

$($a$)$ Check if the per$-$period utility function $u(c)$ satisfies i$)u^{\text{'}}(c)>0$ and ii where $u^{\text{'}}(c)$

denotes $d\frac{u}{d}c$ and $u^{\text{'}\text{'}}(c)$ denotes $d^2\frac{u}{d}{c}^2.$ Describe the economic meaning of these conditions.

Show that when the per$-$period utility function $u(c)$ satisfies the two conditions above, house$-$

holds' total utility $U(c_0,c_1)$ satisfies i$)U_{c_0}>0,U_{c_1}>0$ and ii where $U_{c_0}$ and

$U_{c_1}$ denote del$\frac{U}{d}el{c}_0$ and del$\frac{U}{d}el{c}_1,$ respectively, and $U_{c_0,c_0}$ and $U_{c_1,{}_1}$ denote $de{l}^2\frac{U}{d}el{c}_0^2$ and $de{l}^2\frac{U}{d}el{c}_1^2,$

respectively.

$($b$)$ Write down this household's intertemporal optimization problem using sequential budget con$-$

straints. Indicate which term captures the saving or borrowing of this household in the sequential

budget constraints. Using this term, describe when this household saves and when borrows.

$($c$)$ Derive the intertemporal budget constraint and explain its economic meaning $($using the con$-$

cept of the present discounted value$).$

$($d$)$ Rewrite this household's intertemporal optimization problem using the intertemporal budget

constraint. Explain why the gross interest rate $(1+r)$ can be interpreted as a relative price be$-$

tween current and future consumption by comparing this optimization problem with the static

optimization problem over two goods specified in Q$1($b$).$

$($e$)$ Set up a Lagrangian equation and derive the optimal conditions.

$($f$)$ Derive the Euler equation and provide an economic reason why this equation has to hold at

the optimum.

$($g$)$ Solve the intertemporal optimization problem. Does this household save? Or borrow? By how

much?

$($h$)$ Explain what 'consumption smoothing' means. Does this household smooth its consumption?

$($i$)$ On the $c_0-c_1$ plot, illustrate the point of income endowment, the optimum, the intertemporal

budget constraint, and the indifference curve passing the optimum. Indicate the amount of bor$-$

rowing $/$ saving on this plot.

$($j$)$ Assume that the interest rate changes from $r=\frac{1}{9}$ to $r=\frac{2}{9}.$ Find the new optimum. Does this

household still smooth consumption?

$($k$)$ Describe the three channels $($the substitution effect, the positive wealth effect, and the negative

wealth effect$)$ through which the interest rate change described in question $($j$)$ affects the optimal

consumption bundle.

$(1)$ On the $c_A-c_B$ plot, illustrate how the optimum, the intertemporal budget constraint, and the

indifference curve passing the optimum have changed as a result of the interest rate change de$-$

scribed in question $($j$).$ On the same plot, graphically decompose the shift of the optimum into

the shifts caused by the wealth effect $($i$.$e$.$ the combined effect of the positive and negative wealth

effects$)$ and the substitution effect.

$($m$)$ Central banks decrease interest rates when they want to boost up the aggregate demand of

the economy. $($Such an action by a central bank is often called an 'expansionary monetary policy'.$)$

In a fully$-$fledged, large$-$scale macroeconomic model that many central banks practically use, the

substitution effect dominates among the three channels $($the substitution effect, the positive wealth

effect, and the negative wealth effect$)$ because the positive and negative wealth effects nearly

cancel out each other. Using the substitution effect we study in the two$-$peri