Question

The utility function of the consumer is 5x
0.5y
0.5

The income of the consumer is 5000
The price of good x is 1,000 and the price of good y is 500.
Determine the optimal consumption bundle.

Answer 1

Solution:

Utility Function:$U_(x,y) =5x^{0.5} y^{0.5}$

$MRS_{x,y} =\frac{MUx}{MUy} \;where\;MUx=\frac{dU}{dy} =2.5x^{-0.5} y^{0.5}$

$where\;MUy=\frac{dU}{dy} =0.5x^{0.5} y^{-0.5}$

$MRS_{x,y} =\frac{MUx}{MUy}$

$=\frac{2.5x^{-0.5} y^{0.5} }{0.5x^{0.5} y^{-0.5} } =\frac{2.5x(\frac{y}{x} )^{0.5} }{0.5x(\frac{x}{y}) ^{0.5}}=5(\frac{y}{x})$

$MRS=5(\frac{y}{x})$

$We \;know\; that\; MRS=\frac{Px}{Py}$

$5(\frac{y}{x})=\frac{1000}{500}$

$5(\frac{y}{x})=2$

$y=\frac{2}{5} x$

Plug this into the budget line:

Budget Line:

$I=PxX+PyY$

$5000=1000X+500Y$

$5000=1000X+500(\frac{2}{5} x)$

$5000=1000X+200X$

$5000=1200X$

$X=\frac{5000}{1200}$

$X=4.17$

Plug X into the budget line to get the value of Y:

$5000=1000X+500Y$

$5000=1000(4.17) +500Y$

$5000=4170 +500Y$

$5000-4170=500Y$

$830=500Y$

$Y=\frac{830}{500}$

$Y=1.66$

The optimum consumption bundle is therefore (x,y) = (4.17, 1.66)