Patience has a utility function U(c 1 , c 2 )  c 1/2 1  0.83c 1/2 2 , where c 1 is her consumption in period 1 and c 2 is her consumption in period 2.Her income in period 1 is twice as large as her income in period 2.At what interest rate will she choose to consume the same amount in period 1 as in period 2?A) 0.40B) 0.10C) 0.20D) 0E) 0.30

Question

Patience has a utility function U(c 1 , c 2 )    c 1/2 1     0.83c 1/2 2 , where c 1  is her consumption in period 1 and c 2  is her consumption in period 2.Her income in period 1 is twice as large as her income in period 2.At what interest rate will she choose to consume the same amount in period 1 as in period 2?A) 0.40B) 0.10C) 0.20D) 0E) 0.30

Milou Vennema · Accepted Answer

Final Answer : C, Explanation :  In order for Patience to consume the same amount in period 1 and period 2, we need to set c 1  = c 2 . We can express her income in period 2 in terms of her income in period 1 as follows: c 1/2 2  = 1/2c 1/2 1 . Substituting these equal consumption levels and income levels into her utility function, we get:U(c 1 , c 2 ) =   c 1/2 1     0.83c 1/2 2 =   c 1/2 1     0.83(1/2)c 1/2 1 =   c 1/2 1  (  )(1/2)(1 + 0.83)To maximize her utility and consume the same amount in each period, she should choose the interest rate that will allow her to spend the same amount in each period, given her income levels in each period. Therefore, we set:c 1/2 1  = (1 + r)c 1/2 2 where r is the interest rate earned between periods 1 and 2. Substituting this expression into her utility function, we get:U(c 1 , c 2 ) =   c 1/2 1     0.83c 1/2 2 =   c 1/2 1  (  )(1/2)(1 + 0.83(1 + r))Taking the derivative of this expression with respect to r and setting it equal to zero, we get:c 1/2 1  0.83 (  )(1/2) = c 1/2 1  (  )(1/2)(0.83)Simplifying this equation, we get:(1 + r) = (0.83/ ) + 1 = 1.83Therefore, the interest rate that will allow Patience to consume the same amount in period 1 as in period 2 is r = 0.83, or 83%. However, since this interest rate is greater than 100%, it is not a feasible solution. The actual feasible solution is:(1 + r) = 1.83r = 0.83 - 1 = -0.17Therefore, the interest rate that Patience should choose to consume the same amount in period 1 as in period 2 is -0.17, or -17%. Note that this is a negative interest rate, which corresponds to holding cash instead of investing it. This makes sense, since Patience values present consumption more highly than future consumption, and is willing to give up the potential returns from investments in order to achieve equal consumption in each period.

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