The Effects of Interest Rate on the Optimal Consumption Path in a Bewley Model with the Co-existence of Currency and Credit

In the Bewley models, the endowment is faced to idiosyncratic risks. But contingent claims markets is restricted or completely excluded by assumption and so households couldn’t insure themselves against these risks. Consequently, households will have strong motive to precautionary saving for self-insurance. Households’ only option is to “self-insure” by managing a stock of a single asset to buffer their consumption against adverse shocks. The bewley models differ mainly with respect to the particular asset that is the instrument for self-insurance: fiat currency, credit (such as IOU's, bank deposits, government bonds and so on) or capital. In these models if the interest rate would be equal to the rate of the time preference then asset and consumption diverge to infinity and so monetary equilibrium doesn't exist. Therefore these models conclude that the use of Friedman rule can be misleading in an incomplete market setup. Therefore these models reduce the interest rate so that asset and consumption converge and consequently the monetary equilibrium exists.
In this paper we extend the bewley models and construct a heterogeneous model with idiosyncratic risks and borrowing constraint where agents hold money and bearing interest assets as government bonds for precautionary motives and self-insurance. We show that the consequences of bewley models in this condition are still true: There should be the interest rate lower than time preference ( ) to insure the existence of monetary equilibrium. With sufficient uncertainty in the income and interest rate sequences, consumption will grow without bound even if the rate of interest is equal to or greater than the discount rate.