<p>I am working with a Robinson Crusoe economy, i.e. an economy in which there is no production and only consumption is possible: there are only two goods, $x_1$ and $x_2$. </p>

<p>There is just one individual that lives two periods. He has endowments $\omega_1$ in period 1 and $\omega_2$ in period 2. He can borrow one unit of food in period 1 in exchange for $1 + r$ units of food in the next period. And he can save part of $\omega_1$, without getting any return, to consume it in period 2.</p>

<p>I tried to work out something but I have some problems since I obtained a budget constraint that has different slopes for different allocations. Do you have any hint?</p>

CC BY-SA 3.0
2016-03-04T03:11:06.887Z
could you give some more details? namely, what are the budget constraints which you derived?
HRSE
CC BY-SA 3.0
2016-03-04T13:37:52.080Z
The budget constraint is not difficult to write. The technical issue here is that the consumer has also a binary choice to make (borrow or not a prespecified amount), and so the objective function is not differentiable with respect to that choice. Also, clarifications are needed: is, in principle, the consumer allowed to save not only part of $\omega_1$ but the whole of it, and even part of the loan (it may not be optimal to do so, but it is important as regards the feasible choices, and so the constraints, that the consumer faces). Finally, I guess it is assumed that $\omega_2>1+r$.
Alecos Papadopoulos
CC BY-SA 3.0
2016-03-07T14:14:58.283Z
@AlecosPapadopoulos Thank you, with your conditions imposed is clearer than before.
My problem in this case is the distinction I have to made in order to take into account the fact that borrowing has a cost equalt to $r$, whereas saving has no return for the individual. 
Is that there are two different returns?
PostDocing
CC BY-SA 3.0
2016-03-07T14:16:06.430Z
@Alessandro It's good to hear.
Alecos Papadopoulos
<microeconomics><self-study>
Robinson Crusoe economy with saving and borrowing
PostDocing
# Robinson Crusoe economy with saving and borrowing

(Asked by: username_0 on 2016-03-03)

I am working with a Robinson Crusoe economy, i.e. an economy in which there is no production and only consumption is possible: there are only two goods, $x\_1$ and $x\_2$.

There is just one individual that lives two periods. He has endowments $\omega\_1$ in period 1 and $\omega\_2$ in period 2. He can borrow one unit of food in period 1 in exchange for $1 + r$ units of food in the next period. And he can save part of $\omega\_1$, without getting any return, to consume it in period 2.

I tried to work out something but I have some problems since I obtained a budget constraint that has different slopes for different allocations. Do you have any hint?

**Tags:** microeconomics, self-study

> **Comments:**
> **username_1**: could you give some more details? namely, what are the budget constraints which you derived?  
> **username_2**: The budget constraint is not difficult to write. The technical issue here is that the consumer has also a binary choice to make (borrow or not a prespecified amount), and so the objective function is not differentiable with respect to that choice. Also, clarifications are needed: is, in principle, the consumer allowed to save not only part of $\omega_1$ but the whole of it, and even part of the loan (it may not be optimal to do so, but it is important as regards the feasible choices, and so the constraints, that the consumer faces). Finally, I guess it is assumed that $\omega_2>1+r$.  
> **username_0**: @AlecosPapadopoulos Thank you, with your conditions imposed is clearer than before. My problem in this case is the distinction I have to made in order to take into account the fact that borrowing has a cost equalt to $r$, whereas saving has no return for the individual.  Is that there are two different returns?  
> **username_2**: @Alessandro It's good to hear.  

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