Bridget has a limited income and consumes only wine and cheese; her current consumption choice is four bottles of wine and 10 pounds of cheese. The price of wine is $10 per bottle, and the price of cheese is $4 per pound. The last bottle of wine added 50 units to Bridget's utility, while the last pound of cheese added 40 units. Is Bridget making the utility-maximizing choice? Why or why not? If not, what should she do instead? Why?

Cost = 10 w + 4 c

d (cost) = 10 dw + 4 dc
d (utility )= 50 dw + 40 dc

her present state is
w = 4
c = 10

Cost = 40 + 40 = 80
to stay at the present cost
10 dw + 4 dc = 0
or
dc = - (2.5)dw
so
d Utility at present cost = 50 dw -40(2.5) dw
= 50 dw - 100 dw = -50 dw

so if I increase wine by 1, I decrease cheese by 2.5
AND I decrease my utility by 50!!!
eat more cheese, less wine !!!

To determine if Bridget is making the utility-maximizing choice, we need to compare the marginal utility per dollar spent on wine and cheese.

The concept of marginal utility measures the additional satisfaction or utility gained from consuming an additional unit of a good. In this case, the marginal utility of the last bottle of wine consumed is 50 units, and the marginal utility of the last pound of cheese consumed is 40 units.

To calculate the marginal utility per dollar, we divide the marginal utility by the price. So for wine: 50 units / $10 = 5 units per dollar, and for cheese: 40 units / $4 = 10 units per dollar.

The utility-maximizing choice occurs when the marginal utility per dollar for each good is equal. In this case, Bridget's marginal utility per dollar for cheese is higher (10 units per dollar) compared to wine (5 units per dollar).

Therefore, Bridget is not making the utility-maximizing choice. She should reallocate her consumption to maximize her overall satisfaction. This means she should consume more cheese and less wine.

To find the utility-maximizing choice, Bridget should continue consuming until the marginal utility per dollar of both goods is equal. In other words, she should keep consuming until the marginal utility from the last unit of cheese is equal to the marginal utility from the last unit of wine. At that point, the additional satisfaction for spending an extra dollar on each good is the same.

By doing this, Bridget will achieve the highest possible level of satisfaction given her limited income.