During any year, I can consume any amount that doesnot exceed my current wealth. If I consume c dollars duringa year, I earn ca units of happiness. By the beginning of thenext year, the previous year’s ending wealth grows by afactor k.a Formulate a recursion that can be used to maximizetotal utility earned during the next T years. Assume Ioriginally have w0 dollars.b Let ft(w) be the maximum utility earned during years t, t 1, . . . , T, given that I have w dollars at the be-ginning of year t; and ct(w) be the amount that should be consumed during year t to attain ft(w). By workingbackward, show that for appropriately chosen constantsat and bt,ft(w) btwa and ct(w) atwInterpret these results.
During any year, I can consume any amount that doesnot exceed my current wealth. If I consume c dollars duringa year, I earn ca units of happiness. By the beginning of thenext year, the previous year’s ending wealth grows by afactor k.a Formulate a recursion that can be used to maximizetotal utility earned during the next T years. Assume Ioriginally have w0 dollars.b Let ft(w) be the maximum utility earned during years t, t 1, . . . , T, given that I have w dollars at the be-ginning of year t; and ct(w) be the amount that should be consumed during year t to attain ft(w). By workingbackward, show that for appropriately chosen constantsat and bt,ft(w) btwa and ct(w) atwInterpret these results.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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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 ft(w) be the maximum utility earned during years
t, t 1, . . . , T, given that I have w dollars at the be-
ginning of year t; and ct(w) be the amount that should
be consumed during year t to attain ft(w). By working
backward, show that for appropriately chosen constants
at and bt,
ft(w) btwa and ct(w) atw
Interpret these results.
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