Answer to Question #219929 in Macroeconomics for Darryl

Question

1) A consumer has a utility function given by 

ln U = 5 ln x1 + 3 ln x2

if the budget constraint is given by 

10x1 + 14x2 = 124, find

i) the optimal quantities of the two goods that the consumer should purchase in order to maximise utility, subject to the budget constraint. 

ii) the value of the consumer’s marginal utility of money at the optimum

iii) the marginal rate of substitution (MRS) of x1 for x2 and determine its direction at the optimum

Answer 1

ln U= 5lnX₁+3lnX₂

$ln U = ln(X1)^5 + ln(X2)^3$

U = X₁⁵ + X₂³

MU_x1 = $\frac{\delta U}{\delta X1}$ = 5X₁⁴

MU_x2 = $\frac{\delta U}{\delta X2}$ = 3X₂²

Utility Maximization

$\frac{MUx}{MUy} = \frac{Px1}{Px2}$

$\frac{(5X1)^4}{(3X2)^2} =\frac{10}{14}$

70X₁⁴ + 30X₂²

$X2= \sqrt(\frac{7}{3}X1^4)$

$X2 = \frac{\sqrt 21}{3} (X1)^2$

Replacing the value of X₂ into the budget line

10X₁+ 14($\frac{\sqrt{21}}{3}$X₁² = 124

21.38535324X₁² + 10X₁ = 124

$a=21.38535324$

$b= 10$

$c= - 124$

X₁= 2.1855 or -2.65311

Using the positive value of X₁ since there is no negative commodity, we get the value of

X₂ = 7.296

Therefore the optimal quantities of X₁ and X₂ are;

X_{1 =} 2.1855

X₂= 7.296

c) The Marginal Rate of Substitution

Since the utility is equal along an indifference curve, we pick another point that will bring the same total utility.

These points would be;

$X1= 3.522$

$X2= 5.9595$

Marginal rate of Substitution is obtained as follows;

MRS $= \frac{\Delta X2}{\Delta X1}$

MRS = $\frac{7.296-5.9595}{2.1855-3.522}$ $= 1$