Question: An agent lives for two periods and has an endowment of one unit of a homogeneous consumption good in the first period, and γ units in the second period. His utility function is given by
ln c1 + ln c2
where ci is consumption in period L The agent can store any feasible quantity of his first-period endowment for consumption at a later time and can get an interest-free loan of up to β units of the good (i.e., s ≥ - β and R = 1).
(i) Calculate the agent's saving function, ignoring the constraint s ≥ - β.
(ii) For what combinations of parameter values will the constraint be binding? In what regions of the (β, γ) plane will we have an interior solution and a corner solution? Write the agent's savings function, taking into account the constraint.
(iii) Write the maximum-value function for the problem as a function of γ V{γ). Verify that V(γ) is continuous at the point at which there is a regime change (i.e., as we go from an interior solution to one in which the constraint is binding). Is the value function differentiable at this point?