When addressing the question of optimal savings behavior in the presence of a liquidity constraint and different interest rates for savings and borrowing, we must consider the individual's time preference for consumption, often represented by the parameter δ (delta). In the proposed scenario, the individual can save at an interest rate of 3%, but must pay 9% to borrow. The optimal saving choice would be influenced by the individual's inter-temporal preferences, meaning how they value present consumption against future consumption.
The higher the value of δ, the greater the preference for present consumption compared to future consumption. If δ is high enough that the person would rather consume now than save for later consumption, the individual will not save even if the interest rate is positive. However, there is a threshold level of δ where the person will be indifferent between consuming now and saving for later. If δ is lower than this threshold, it will be optimal for the person to save money. The exact threshold level of δ where the person switches from preferring present consumption to future consumption will depend on the specifics of the individual's utility function and the ratio of the interest rates.
Let's consider some examples to illustrate this behavior. If the person's utility from present consumption is significantly higher than future consumption, they may prioritize immediate consumption over saving due to their high δ, despite the return available from saving. On the other hand, if the utility from future consumption is comparable or favored by the person, and if δ is sufficiently low, the person may choose to save in order to accumulate wealth and consume more in the next period, thereby benefitting from the interest rate on savings.
The liquidity constraint affects this decision as well because it makes borrowing expensive. People facing a high penalty for borrowing (in this case, a 9% interest rate) are more likely to save at 3% interest rate if their δ is low, as it allows them to build a financial cushion and avoid the high cost of borrowing in the future.