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# Calculus

Given the utility function of good x and y as, \U=\x^{3/4}\y^{1/4}

Prices of good X and Y \Px=6\and\Py=3

Budget \B=\$120

The point of tangency or the combination of good X and Y (maximum satisfaction) \U=\frac{dUx}{dUy}=frac{Px}{Py}.....(1)

\frac{dUx}{dx}=\x^{3/4}\y^{1/4}

differentiating with respect to x,

\=\3/4\x^{3/4-1}\y^{1/4}

\=\3/4\x^{-1/4}\y^{1/4}......(ii)

\frac{dUy}{dy}=\x^{3/4}\y^{1/4}

differentiating with respect to y

\=\x^{3/4}\1/4\y^{1/4-1}

\=\x^{3/4}\1/4\y^{-3/4}.......(iii)

substituting the values in equation (i)

\frac{3/4\y^{3/4}\y^{1/4}}{x^{3/4}\1/4\x^{-1/4}}=frac{6}{3}

Diving the upper portion of the equation by 3, we get;

\frac{1/4\y^{3/4}\y^{1/4}}{1/4\x^{3/4}\x^{1/4}}=frac{2}{3}

Cross multiplying;

\3y=2x

or, \x=\frac{3y}{2}.......(iv)

The equation of budget line can be written as Px.X+Py.Y=120........(v)

Substitution the equation (iv) prices of x and y good in equation (v), we get;

\6\frac{3y}{2}+3y=120

or, \9y+3y=120

or, 12y=120

or, y=10

Similarly again substituting the value of y in equation (iv) weget;

\x=\frac{3×10}{2}

or, \x=\frac{30}{2}

or, \x=15

Hence the combination of 15 units of good x and 10 units of good y will be the point of tangency to the indifference curve.