Answer to Question #276140 in Microeconomics for Kaju

Question

Consider a consumer who buys two good x and y with utility function u (x,y)=2 under root x+y. The consumer's income is 20 and price of y= 4. Compute the optimal consumption bundle when the price of x =1 and if the price of x rises to 4 what is the new optimal bundle

Answer 1

$u(x,y)=2\sqrt{x}+y\\Mu_x=\frac{1}{\sqrt{x}}\\Mu_y=1\\MRS=\frac{Mu_x}{Mu_y}=\frac{1}{\sqrt{x}}\\p_x=1\\p_y=4\\I=p_xx+p_yy\\x+4y=20$

The optimality condition involves equating MRS to the $\frac{p_x}{p_y}$

$\frac{1}{\sqrt{x}}=\frac{1}{4}\\\sqrt{x}=4\\x=16$

Substituting this to the budget equation:

$16+4y=20\\y=1$

Therefore, the optimal consumption bundle is:

$(x,y)=(16,1)$

If $p_x$ rises to 4

$4x+4y=20\\\frac{1}{\sqrt{x}}=1\\x=1\\4+4y=20\\4y=16\\y=4$

Therefore, the optimal consumption bundle becomes:

$(x,y)=(1,4)$