Question

1)     Assume the logarithmic transformation f a utility function, for the consumption of two commodities is given by

ln U = ln4 + 0.5ln X + 0.25lnY

(a) if the price of X is GHS2.50 and that of Y is GHS4.00, calculate the optimal combination for an income of GHS50.00.

b)  Determine and interpret the value of the Lagrange multiplier.

 

Answer 1

$log U =log 4+0.5 log X+0.25 log Y$

$50=2.50X+4Y$

$L=log 4+ 0.5log X +0.25 log Y+ \lambda [50-2.50X-4Y]$

$\frac {\delta L}{\delta X}=\frac {0.5}{X}-\lambda (2.50)=0$

$\frac {\delta L}{\delta Y}=\frac {0.25}{Y}-\lambda(4)=0$

$\frac {0.5}{2.50X}=\frac {0.25}{4Y}$

$\frac {0.2}{X}=\frac {0.0625}{Y}$

$\frac {X}{Y}= \frac {0.2}{0.0625}$

$X= 3.2Y$

$50=2.50X+4Y$

$50=2.50(3.2Y)+4Y$

$50=12Y$

$Y=4.16$

$X=13.33$