Question

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.

Answer 1

$$ln U=lnx_1^5+lnx_2^3$$

$$U=x_1^5 x_2^3$$

$$Lag=x_1^5x_2^3-\lambda(10x_1+14x_2-124)$$

$$\frac {\varDelta Lag}{\varDelta x_1}=5x_1^4x_2^3-10 \lambda$$

$$\frac {\varDelta Lag}{\varDelta x_2}=3x_1^5x_2^2-14 \lambda$$

$$x_1=\frac{7}{3}x_2$$

$$x_2=3.3$$

$$x_1=7.8$$