Household Intertemporal Optimization

In Perfect Credit Markets §

ass. Financial Markets are perfectly competitive, i.e.:

• they have no market power, and no profits
• interest rates are unaffected by household optimization Q. Intertemporal household optimization problem. We start with the two-period situation with the following budget constraints:

$$\begin{array}{rl}{C}_t+S_t& =Y_t^D\\ C_{t+1}& =Y_{t+1}^D+S_t(1+r)\end{array}$$

where:

• $Y_t^D:=Y_t-T_t$: disposable income
• $r$: interest rate And solving for $S_t$ we get: lem. Two-period intertemporal household lifetime budget constraint:

$$\begin{array}{rlr}{C}_t+\frac{C_{t+1}}{1+r}& =\stackrel{\text{ NPV Lifetime Wealth }}{\overbrace{Y_t^D+\frac{Y_{t+1}^D}{1+r}}}& \text{at τ=t}\\ C_{t+1}& =Y_{t+1}^D+(1+r)(Y_t^D-C_t)& \text{equivalently at τ=t+1}\end{array}$$

thus we get the optimization problem:

$$\underset{C_t,C_{t+1},{\ell }_t,{\ell }_{t+1}}{\max }u(\cdot )\text{ s.t. }{C}_{t+1}=Y_{t+1}^D+(1+r)(Y_t^D-C_t)$$

we can make this into an unconstrained optimization problem by substituting $C_{t+1}$ in the utility function with the budget constraint. Rearrange the NPV lifetime wealth, which we term $W_t:=Y_t^D+\frac{Y_{t+1}^D}{1+r}$:

$$\begin{array}{r}{W}_t=C_t+\frac{C_{t+1}}{1+r}\\ C_{t+1}=W_t(1+r)-(1+r)C_t\end{array}$$

Which is the budget constraint plotted below:

where:

• $E$: consumption endowment. We always start here
• $\bar{u}$: maximum utility without borrowing/lending (=w/o financial markets)
• $\bar{u}$: maximum utility with borrowing/lending
• Slope of the budget constraint is $-(1+r)$ as derived from the lifetime budget constraint; simply because this is the rate at which you can trade current and future consumption Intuition. Financial markets allow a household to shift consumption forwards/backwards in time. this is called consumption smoothing. Remark. The household still has convex utility, meaning it has a taste for variety. This means often that households will try to balance current and future consumption Remark. ${\text{MRS}}_{C_t,C_{t+1}}=\text{Slope of Indifference Curve}$ thus $=1+r$ at optimal

Taxation Effects §

Timing of $T_t,T_{t+1}$ does not distort interest rates, thus does not affect optimization

In Imperfect Credit Markets §

Motivation. In reality credit markets do not follow the assumptions stated above. In reality, we consider risk. thm. Profit function of Banks

$$\pi =(1+r^{\text{borrow}})L-(1+r^{\text{lend}})D$$

where:

• $r^{\text{borrow}}$ is the interest rate for (businesses/HHs) borrowing investment for consumption
• $L$ is the total amount loaned out
• $r^{\text{lent}}$ is the interest rate for consumers “lending” to banks
• $D$ is the total amount deposited Remark. $r^b-r^{\ell }$ is also known as the credit spread. ass. From now on, we assume $L=D$ thus:

$$\pi =(1+r^b)L-(1+r^{\ell })L$$

Modelling risk §

A loan amount $L$ is a Bernouilli trial with:

• probability $a\in [0,1]$ of successful repayment → repay $L(1+r^b)$
• probability $1-a$ of default → repay $0$
• Expected bank profit for loan $L$ is $\mathbb{E}(\pi )=[a(1+r^b)-(1+r^{\ell })]L$ In perfect competition, we have $\pi =0$ thus:

$$1+r^{\ell }=a(1+r^b)$$

Thus when $a<1$ we have $r^b>r^{\ell }$.

Imperfect Credit Household Optimization §

The intertemporal budget constraint is broken into two parts:

$$C_t=\{\begin{array}{ll}{Y}_t^D+\frac{Y_{t+1}^D}{1+r^b}-\frac{C_{t+1}}{1+r^b}& \text{when borrowing}\\ Y_t^D+\frac{Y_{t+1}^D}{1+r^{\ell }}-\frac{C_{t+1}}{1+r^{\ell }}& \text{when saving}\end{array}$$

Thus:

$$\begin{array}{r}{C}_{t+1}=\{\begin{array}{ll}{W}_t(1+r)-(1+r^b)C_t& \text{if net borrow}\\ W_t(1+r)-(1+r^{\ell })C_t& \text{if net save}\end{array}\end{array}$$

Drawing only relevant parts from the endowment point $E$:

where:

• $r^{\ell }<r^b$ due to imperfect markets
• $A^{\ast }$ is the potential optimal point if Financial Markets were perfect

Effects of… §

Taxation §

Intuition. Government tax cuts are always good, and always better right now, because the government essentially borrows for you at the lower rate $r^{\ell }$:

Recession/Market Risks §

Intuition. Changes in $a$ i.e. default risk will simply change the borrowing rate:

Counterparty Risk and Collateral §

Motivation. Banks don’t know if a borrower is credible or not, due to counterparty risk $a<1$ (=Asymmetric Information), banks require a collateral (usually a house) for consumers to borrow. The budget constraint changes thusly:

Public Pension Systems §