Non-separable utility in OLG
Here we analyse the behaviour of the savings function when utility is non-separable. In particular, let utility be represented by
We use the substitution method, but in this case we will introduce different alternatives, depending on the derivative we compute. In particular, we introduce different elements of the budget constraint:
and are normal goods
First, we derive conditions such that
Consumption when young
We are interested in computing
Using implicit derivation we have:
Since, in equilibrium
Consumption when old
In this case, we write the optimality condition as
Hence, we can compute the derivative
Savings, wages and interest rate
We now show that savings are increasing in wages, and that they may increase or decrease with the interest rate.
First, we compute the relationship between savings and wages using the optimality condition
Using the same optimality condition we proceed with the relationship between savings and the interest rate:
Hence, the sign of the derivative depends on
The relationship depends on the intertemporal elasticity of substitution (