During any year, I can consume any amount that does not exceed my current wealth. If I consume c dollars during a year, I earn ca units of happiness. By the beginning of the next year, the previous...

During any year, I can consume any amount that does not exceed my current wealth. If I consume c dollars during a year, I earn ca units of happiness. By the beginning of the next year, the previous year’s ending wealth grows by a factor k.

a Formulate a recursion that can be used to maximize total utility earned during the next T years. Assume I originally have w0 dollars.

b Let f(w) be the maximum utility earned during years t, t + 1, . . . , T, given that I have w dollars at the beginning of year t; and c_t(w) be the amount that should be consumed during year t to attain f_t(w). By working backward, show that for appropriately chosen constants at and bt,

f_t(w) = b_tw^a and c_t(w) = a_tw Interpret these results.