Question

A consumer’s utility function is given by 𝑈 =∝ 𝐼𝑛𝑥1 + 𝛽𝐼𝑛𝑥2

. Find the values of 𝑥1

and 

𝑥1 which maximize utility subject to the budgetary constraint 𝑃1𝑥1 + 𝑃2𝑥2 = 𝑀

Answer 1

$$\frac {\delta U}{\delta x_1}=\frac {\alpha}{x_1}$$

$$\frac {\delta U}{\delta x_2}=\frac {\beta}{x_2}$$

$$\frac {\alpha}{x_1p_1}=\frac {\beta}{x_2p_2}$$

$$x_2p_2=\frac{\beta}{\alpha}x_1p_1$$

$$x_1=\frac {2M}{p_1(\alpha+\beta)}$$