✏️ Is Dana Optimizing? Example

✏️ At her present levels of consumption of goods X and Y, Dana is spending her entire budget, and her
MRS _X,Y = 5, P_X = $9, and P_Y = $2.
Is Dana consuming the optimal amount of goods X and Y?

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To analyze this, we have to check the two conditions for a consumer optimum:

First Condition for a Consumer Optimum:  a) Your optimal consumption bundle (X*,Y*) should be on the Budget Constraint. In other words, you must use all of your money. Alternate Version:  b) $X^∗×P_X +Y^∗×P_Y = B$

Second Condition for a Consumer Optimum:  a) Slope of IC = Slope of BC  b) $\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$ ("Bang for the buck" formulation) This ensures that you spend your money on the things that bring you the most utility per dollar. Alternate versions:  c) $MRS=\frac{P_x}{P_y}$  d) Ratio of MUs: $\frac{MU_x}{MU_y}=\frac{P_x}{P_y}$

She’s spending her entire budget, so first condition is satisfied.

For the second condition, we have to use the “MRS” version of the second optimization condition. It is not satisifed.

In other words, the following formula must be true:

$$MRS = \frac{P_X}{P_Y}$$

However, that equation doesn’t hold here:

$$MRS = 5 \neq \frac{P_X}{P_Y} = \frac{\$9}{\$2} = 4.5$$

We can also see this based on some of Bruce’s slides.

At point A, we know that Dana can increase her utility by “sliding down the BC to point O. Imagine this as her taking good Y out of her shopping cart so sh e can afford to put good X in it (just like Rob had to take processor speed out of his shopping cart to add a better graphics card for Zoom).

That means that good Y is not as good of a value for her compared to good X.
The value of good Y is $\frac{MU_Y}{P_Y}$, so that means that
$\frac{MU_Y}{P_Y} < \frac{MU_X}{P_X}$