We can explain saving for retirement for a worker by introducing a production possibility frontier that describes possible combinations of output in two periods.
Given access to financial markets, the worker will find the production point that reaches the best possible budget constraint.
Given access to financial markets, our worker produces at P and consumes at C. Without access to financial markets, the worker's would produce and consume at A, which is on a lower indifference curve.
This diagram illustrates a benefit of financial markets that helps explain why they exist.
Why do people save money?
One approach to this issue is to consider consumption in two time periods to be two different goods and apply the standard two goods - two prices analysis.
It is not good economics to answer questions by putting prices, interest rates, or money into untility functions.
We explain the role of the interest rate instead by observing that it plays an important role in the intertemporal budget constraint.
As you can see, if you raise the interest rate, people save more.