A consumer with an income of $240 is spending it all on 12 units of good X and 1

Question

A consumer with an income of $240 is spending it all on 12 units of good X and 18 units of good Y. The price of X is $5 and the price of Y is $10. The marginal utility of the last X is 20 and the marginal utility of the last Y is $30. What should the consumer do? a. Nothing, this is the utility maximizing choice. b. Buy more Y and less X because MUy is higher than MUx. c. Buy more Y and less X because the marginal utility per dollar of Y is higher. d. Buy more X and less Y because the price of X is lower. e. Buy more X and less Y because the marginal utility per dollar of X is higher. Please explain

Explanation / Answer

MRS = - MUx / MUy = - 20 / 30 = -2/3 (slope of the Utility function)

I = Px*x + Py*y (rearrange eqn to solve for y)

I/Py - (Px/Py)*x = y (-Px/Py is the slope of the Budget Line)

-Px / Py = -5 /10 = -1/2

At the optimum bundle, the MRS will equal the Price Ratio, since the slopes of the Utility Function and Budget Line are equal at that point.

Does this condition hold?

-Px/Py = -MUx/MUy ----> -1/2 =/= -2/3 ... No, therefore some different bundle will bring greater utility. But which one?

Let's rearrange this identity:

MUx/Px : MUy/Py -----> 20/5 : 30/10 --------> 4 > 3

This implies there is more utility to be gained by acquiring more X and less Y since for the next dollar spent on X, 4 Utils are gained, but only 3 Utils for the next Y.

Therefore, the answer is D, but not only because it is cheaper.

It is because more utility is gained on the next X than what is lost by giving up one Y.

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