Consider a utility function given by U= x11/2x23/2. What is the MRS2→1?

Question

BlackTom AI · Accepted Answer

We start by restating the problem clearly: given U = x1^{1/2} x2^{3/2}, determine the MRS_{2→1}.

First, compute the marginal utilities: MU1 = ∂U/∂x1 = (1/2) x1^{-1/2} x2^{3/2} and MU2 = ∂U/∂x2 = (3/2) x1^{1/2} x2^{1/2}.

A standard expression for the marginal rate of substitution of good 2 for good 1 is MRS_{2→1} = - d x2 / d x1 on an indifference curve, which equals MU1 / MU2 when using the usual sign convention. Plugging in the marginals:

MU1 / MU2 = [(1/2) x2^{3/2} / x1^{1/2}] / [(3/2) x1^{1/2} x2^{1/2}] = (1/3) * (x2 / x1) = x2 / (3 x1).

Thus, with the conventional positive ratio, MRS_{2→1} = x2 / (3 x1). Some curricula present MRS with a negative sign to reflect the downward slope, i.e., MRS_{2→1} = - MU1 / MU2. In that convention, the value becomes - x2 / (3 x1).

Now evaluate the given options:
- Option 1: MRS_{2→1} = - 3 x2 /(2 x1). This is incorrect because it would imply MU1/MU2 = 3 x2 /(2 x1), which does not match the correct ratio x2 /(3 x1) under any standard rearrangement of the exponents. The coefficients 3/2 and 1/2 from MU2 and MU1 cancel to 3 in the denominator, not a 3/2 factor, so this option miscomputes the ratio.
- Option 2: MRS_{2→1} = - x2 /(3 x1). This matches the negative-sign convention of MRS_{2→1} = - MU1 / MU2, given MU1/MU2 = x2 /(3 x1). Therefore this option aligns with the standard derivative-based approach when the minus sign is included.
- Option 3: MRS_{2→1} = - 3 x2^{3/2} /(2 x1). This incorrectly copies parts of MU1 or MU2 into the MRS without proper division. The exponent on x2 should reduce to x2^{1} in the ratio, not x2^{3/2}, and the 2 in the denominator and the 3 in the numerator don’t align with the derivation.
- Option 4: MRS_{2→1} = - x2 /(2 x1). This would correspond to a ratio with 1/2 instead of 1/3 as the denominator factor, which is inconsistent with the MU1/MU2 computation (the 3 in the MU2 term yields a 3 in the denominator after division).
- Option 5: MRS_{2→1} = - x2 / x1. This would be far too large by a factor of 3, since MU1/MU2 = x2 /(3 x1). Omitting the 3 in the denominator mischaracterizes the relative marginal utilities.

In summary, the correct structure of the MRS using MU1 and MU2 yields a ratio with a 3 in the denominator, and applying the chosen sign convention leads to the selected option. The remaining options fail because they alter the coefficient or the power of x2 improperly, or ignore the proper division between MU1 and MU2.

ECON5001 Microeconomic Theory Quiz 1 (available 18 March, due 28 March)

Consider a utility function given by U= x11/2x23/2. What is the MRS2→1?

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