8/17/2019 Macroeconomics exercise

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BEEB5023 A152

Homework #3 (April 27, 2016) [Maximum Marks = 15; Due on May 8, 2016 (Sun)]

Consider the general utility function u = u(c), where u is utility and c is consumption. Then, u (c)is the marginal utility so that the growth rate of consumption, denoted by G c, and the growth

rate of marginal utility, denoted by G u , can be defined as

.)('

)('and ' cu

dt cduG

c

dt dcG

uc

Now define elasticity of intertemporal substitution (or intertemporal elasticity of substitution) asa measure of responsiveness of the growth rate of consumption to the growth rate of marginalutility, or equivalently, as the negative of the percentage (or relative) change in the growth rate ofconsumption due to the percentage (or relative) change in the growth rate of marginal utility.Denoting this elasticity concept by F, it may be expressed as

.)(')('

)(')('

)(')('

' dt cdu

cu

c

c

dt cdu

cu

c

dt dc

cu

dt cduc

dt dc

G

G F

u

c

a) Given the last equality, show that ..)(")('

ccu

cu F

b) Now consider the specific utility function ,1

11

cu

where 1 is a parameter. Derive F.

Note : This is two-person group homework.

F is essentially the reciprocal of E, which is the elasticity of marginal utility with respect to consumption.