MRSy→x at the point (1, 1) of the utility function U(x,y) = 2xy4 is

Question

MRSy→x  at the point (1, 1) of the utility function U(x,y) = 2xy4 is

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We start by identifying the utility function and the point of interest. The utility is U(x,y) = 2 x y^4, and we are evaluating at (x, y) = (1, 1).

First, compute the marginal utilities:
- MU_x = ∂U/∂x = 2 y^4. At (1,1), MU_x = 2 * (1)^4 = 2.
- MU_y = ∂U/∂y = 8 x y^3. At (1,1), MU_y = 8 * 1 * (1)^3 = 8.

The typical definition for the slope of the indifference curve (dy/dx) is dy/dx = - MU_x / MU_y. This slope represents the rate at which y must change to keep utility constant when x changes. Therefore, at (1,1), dy/dx = -2/8 = -1/4.

Some texts refer to the marginal rate of substitution of y for x, MRS_{y→x}, as the negative of this slope, i.e., MRS_{y→x} = - dy/dx = MU_x / MU_y. Using this convention, MU_x/MU_y = 2/8 = 1/4, which is positive. In many exam contexts, however, MRS_{y→x} is presented as the negative slope of the indifference curve, giving MRS_{y→x} = - dy/dx = -1/4.

Now evaluate each answer:
- -1/16: This would imply a very different ratio that does not match the computed MU_x and MU_y (MU_x/MU_y = 1/4 and -MU_x/MU_y = -1/4). The value -1/16 cannot come from the standard MU_x and MU_y at (1,1).
- -1/4: This aligns with the calculation - MU_x / MU_y = -2/8 = -1/4, or equivalently the negative of the slope dy/dx at the point. This is the correct magnitude and sign under the common interpretation of MRS as the negative slope of the indifference curve.
- -1/2: This would require MU_x/MU_y to be 1/2 in magnitude or dy/dx to be -1/2, which does not follow from MU_x = 2 and MU_y = 8 at (1,1).
- -1/8: This would correspond to a ratio of 1/8 in magnitude, which is inconsistent with MU_x = 2 and MU_y = 8 (whose ratio is 1/4).

In summary, the computed information at (1,1) yields a rate of substitution with magnitude 1/4 and a negative sign when expressed as the slope of the indifference curve, matching the option -1/4.

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

MRSy→x at the point (1, 1) of the utility function U(x,y) = 2xy4 is

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