A consumer$’$s preferences for consumption across two time periods c$1$ and c$2$ can be represented by the utility function U $($C$1,$ C$2)=$ C$1^0\mathrm{.}7$ C$2^0\mathrm{.}3.$ The consumer$’$s endowment of income in each period is m$1=\pounds 2,500,$ m$2=\pounds 2,100.$ Assume that the price per Module of consumption is constant at p$1=$ p$2=1$ and that there is no inflation between time periods. Income can initially be borrowed or lent between the two periods at an interest rate of $40\%($r $=0\mathrm{.}4).$ The interest rate then increases so that r $=0\mathrm{.}5.$ 

Calculate the optimal allocation of consumption between the two periods both before and after the change in the interest rate and comment on the consumer$’$s change in welfare.  

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SOLUTION Part 1 Optimal Allocation of Consumption before the Change in Interest Rate To calculate the optimal allocation of consumption before the cha... View the full answer