# Intertemporal consumer choice

## Discounted utility

### Discounting

Time inconsistency (hyperbolic discounting). again, this should mirror ai. page overview if necessary. call it that?

## Intertemporal budget constraint

### Budget constraint

\(\sum_{t=T}C^t(1+r)^{-t}=\sum_{t=T}Y_t(1+r_t)^{-t}+W_T\)

\(W_t\) is wealth endowment at time \(T\).

### The Euler equation

If we have exponential discounting we have:

\(U_T=E[ \sum_{t=T}^\infty (1+\delta )^t U(C_t) ]\)

The first-order conditions give us:

\(u'(x_t)=(1+\delta)(1+r_t)u'(x_{t+1})\)

### The Euler equation with liquidity constraints

\(u'(x_t)=(1+\delta)(1+r_t)u'(x_{t+1})+\lambda_{t+1}\)

### The Euler equation with continuous time

### Marginal propensity to consume

## Renewal

### Switching costs

## Household finance

### Household wealth and liquidity

Hold cash, equity, bonds, mortgages.

### Mortgages

### Fixed and variable rates

### Expense and income timing

## Other

### Elasticity of intertemporal substitution

Different utility function.

### Durable goods

Can wait to purchase. depends on expected prices in future.

### Buying or owning

production of other commodities. house produces rentable space eg if you own house, you have rentable space each period.

depreciation? 1 for something like rent, maybe 0.01 for long term asset