The prices of goods x and y are each $1. Gizem has $20 to spend. She has strictly monotonic and strictly convex preferences. At the bundle (x,y)=(10,10), her marginal rate of substitution is -2. Is this an optimal choice for Gizem? If not, which good should she consume more of, and which good should she consume less of?

Ответ:

Since Gizem has strictly monotonic and strictly convex preferences, her optimal consumption bundle should be where the marginal rate of substitution (MRS) equals the price ratio.

In this case, the MRS is given as -2 at the bundle (x,y)=(10,10). If the prices of goods x and y are both $1, then the price ratio is 1:1. Since the MRS is -2, Gizem is willing to give up 2 units of good y to get 1 more unit of good x.

To reach the optimal point where MRS equals the price ratio, Gizem should consume more of the good for which she is willing to give up fewer units (in this case, good y), and less of the good for which she is willing to give up more units (in this case, good x).

Therefore, Gizem should consume more of good y and less of good x to reach the optimal consumption bundle.

Ответ:

Given the marginal rate of substitution (MRS) at the bundle (x, y) = (10, 10) is -2, it means Gizem is willing to give up 2 units of good y for an additional unit of good x while maintaining the same level of satisfaction.

For utility maximization in the context of strictly convex preferences, the consumer should allocate their budget in a way that equates the MRS with the price ratio of the goods.

Since both goods x and y are priced at $1, Gizem's optimal choice would be where the MRS equals the price ratio, which is 1 in this case (because both goods have the same price).

Since -2 is not equal to 1, the current bundle (x, y) = (10, 10) is not an optimal choice for Gizem. To improve her utility, she should consume more of the good for which she is willing to give up fewer units (in this case, good y) and less of the good for which she is willing to give up more units (in this case, good x).