Suppose that your utility function for X and Y is U = X 1/3 Y 2/3 = . The price

Question

Suppose that your utility function for X and Y is U = X 1/3 Y 2/3 = . The price of X and Y will be gcncrically Px and PY, and your income is I, so your budget constraint is PxX + PyY = I. Calculate the formula for the marginal rate of substitution. Hint 1: MUX = . Hint 2: Although the MU's are a little nasty, the MRS will be a simple formula with no fractional exponents. Use the "equal slopes" condition to derive a condition about optimal spending shares: when you arc maximizing your utility given the budget constraint, how docs your optimal spending on X compare to your optimal spending on Y? The math is easy: cross multiply. The trick is interpreting the expression... Plug this result back in to the budget constraint and solve for X* and Y* as a function of I, Px, and Py. Presto! You've just calculated the demand curve for X and Y. These demand curves are pretty simple, but they aren't linear.

Explanation / Answer

a>MU(x)=partial derivative of utility w.r.t x =1/3 *(Y/X)^2/3 MU(y)=partial derivative of utility w.r.t y =2/3 *(X/Y)^1/3 marginal rate of substitution=MU(x)/MU(y)=1/2 * (Y/X) b>condition for optimal spending is MU(x)=MU(y) =>1/3 *(Y/X)^2/3=2/3 *(X/Y)^1/3 =>Y=2X c>first replace Y=2X we get X[P(x)+2P(y)]=I then replace X=Y/2 we get Y[P(x)/2+P(y)]=I these are equations of demand curve.

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