X5.18
For the utility function, Ub,c 1
11
,bconfirm that: c
a) The utility function can be rewritten Ub,c b c` .
Ub,c 1 1
1 bc 1 bc b c
bc
bc
bc
b) The consumption bundle, (1, 1) lies on the indifference curve, Evaluating, bc = 1, b + c = 2, so
bc bc`
bc b c`
0.5 .
0.5 .
c) The indifference curve can be expressed in explicit form, c 2b1b . From part b), 2bc = b + c, so (2b – 1)c = b. The result follows by dividing through by 2b – 1. d) If b = 0.5, then it is impossible to evaluate this expression for c. If b = 0.5, then 2b – 1 = 0, so we cannot evaluate the fraction. e) If b < 0.5, then c < 0; and we disregard the consumption bundle. If b < 0.5, then 2b – 1 < 0. Since we require c > 0, this consumption bundle cannot be obtained. f)
As b , c 0.5; which is to say, when b takes larger and larger values, c will become closer and closer to 0.5. b 1 -1 0. As b , c 2b1 2 1 2 since b 1 b
X5.19
Economists frequently refer to utility as being an ordinal rather than a cardinal concept because it is only possible to interpret the ranking of utility values, not their absolute value. a) Suppose that the amount of money that Arun has doubles. Would it be reasonable to suppose that his utility from consumption doubles? No – were it to do so, then this would simply mean that Arun treated utility as identical to wealth.