# convention in borrowing money in a multiperiod model

I have a question concerning the idea of consumption in multi period. The following is given

$$C_1=W_0-xS_1+B$$ $$C_2=xS_2-BR$$

where

$W_0$ is initial wealth

$x$ is the weight on an asset with price / value $S_t$

$B$ is a zerobond at some riskless rate $1<R<2$

the question: how can it be justified (or can it at all be?) that the agent's only chance to transfer money into the second period is via buying the risky asset $S_1$ and not by buying zerobonds? would it not be much more realistic if one had

$$C_1=W_0-xS_1\pm B$$ $$C_2=xS_2\pm BR$$?

Thanks in advance for any suggestions.

## 1 Answer

Your question depends on the discount factor you wish to use for pricing. If u use the risk-free rate (from the bond), it wouldn't be in line with the no-abitrage condition to assume an risk neutral agent can't/wouldn't invest in bonds to carry money into next period. To understand this: just assume a 1 period model with two outcomes for S, where both outcomes are more beneficial than investing in the bond.

I believe, if you really want to force the desired behaviour into your economical framework with resonable parameters for your assets, you have to model agents with an really high amount of risk searching attitude. (CRRA/EZ Utility with risk aversion parameters way below 1, most likley even negative (I don't know if EZ is even defined or can be interpreted for s.th. like that)). This, however, wouldn't be a realistic setting.

The best way to check wether your idea is reasonable or not is to ask yourself: Would I do it? My answer is a clear no.