Answer to Question #196901 in Microeconomics for Ebi David Uyi

Question

Consider the following utility function belonging to “Consumer A”.

U^A(x,y,z,v) = x^0.1y^0.2z^0.3v^0.4.

Suppose that prices of x, y, z and v are 1$, 2$, 3$ and 4$. The consumer’s income is 10$.

a. Find the values of x, y,z and v that maximize the consumer’s utility subject to his/her budget constraint.

b. Suppose that “Consumer B” has a utility function which is a logarithmic transformation of the consumer A’s utility function, i.e.

 U^B =ln U^A.

Suppose that prices and income are the same. Would the amounts of x, y, z and v consumed by “Consumer B” be the same as the amounts consumed by “Consumer A”?

Answer 1

(a)$UA(x,y,z,v) = x^{0.1}y^{0.2}z^{0.3}v^{0.4}$

$\alpha= x^{0.1}y^{0.2}z^{0.3}v^{0.4}+[^{30}(x+2y+3z+4v)]$

$\alpha =x^{0.1}y^{0.2}z^{0.3}v^{0.4}+\lambda(30-x-2y-3z-4v)$

$\frac{\delta \alpha}{\delta x}=$ ${0.1}y^{0.2}z^{0.3}v^{0.4}(x)^{-0.9}-\lambda=0$

$\frac{\delta \alpha }{\delta y}=x^{0.1}z^{0.3}v^{0.4}(0.2)^{-0.8}-2\lambda=0$

$\frac{\delta \alpha }{\delta z}=x^{0.1}y^{0.2}v^{0.4}(0.3)^{-0.}-3\lambda=0$

$\frac{\delta \alpha }{\delta v}=x^{0.1}y^{0.2}z^{0.3}(0.4)^{-0.6}-4\lambda=0$

$1\div 2$ $\frac{1y}{2x}=\frac{\lambda}{2\lambda}\implies y=x$

$3\div4$ $\frac{3v}{4z}=\frac{3\lambda}{4\lambda}\implies v=z$

$2\div3\space y=z$

$x+2y+3z+4v=30$

$10x=30$

$x=3$

$x=3=y=z=r$

(b)logarithm transformation does not change the frequencies of an individual.

Monotonic transformation is a way of transforming a set of numbers into another set that preserves the order of the original set, it is a function mapping real numbers into real numbers, which satisfies the property, that if x>y, then f(x)>f(y), simply it is a strictly increasing function.

 

Applying a monotonic transformation to a utility function representing a preference relation simply creates another utility function representing the same preference utility of utility, basically what this means is that when monotonic transformation of utility is applied the marginal rate of substitution does not change here is why.