Matthew and Susan are both optimizing consumers in the...

Matthew and Susan are both optimizing consumers in the markets for shirts and hats, where they pay $\$ 100$ for a shirt and $\$ 50$ for a hat. Matthew buys 4 shirts and 16 hats, while Susan buys 6 shirts and 12 hats. From this information, we can infer that Matthew's marginal rate of substitution is _____ hats per shirt, while Susan's is _____. a. $2,1$ b. $2,2$ c. $4,1$ d. $4,2$

Matthew and Susan are both optimizing consumers in
the markets for shirts and hats, where they pay $\$ 100$
for a shirt and $\$ 50$ for a hat. Matthew buys 4 shirts
and 16 hats, while Susan buys 6 shirts and 12 hats.
From this information, we can infer that Matthew's marginal rate of substitution is _____ hats per shirt, while Susan's is _____.
a. $2,1$
b. $2,2$
c. $4,1$
d. $4,2$

Key Concepts

Marginal Rate of Substitution (MRS)

The MRS represents the rate at which a consumer is willing to substitute one good for another while maintaining the same level of satisfaction. It is defined as the ratio of the marginal utilities of the two goods, capturing the trade-off in incremental terms.

Consumer Optimization

Consumer optimization is the process by which individuals select the combination of goods that maximizes their utility subject to their budget constraint. In equilibrium, an optimizing consumer chooses a bundle where the marginal rate of substitution equals the ratio of the goods' prices.

Budget Constraint and Price Ratio

A budget constraint delineates the set of all consumption bundles affordable given the consumer's income and the prices of goods. The ratio of the goods' prices determines the slope of the budget line, and at the optimal bundle, it must equal the consumer's marginal rate of substitution, ensuring that the value lost by spending on one good is exactly compensated by the gain from the other.

Transcript

00:01 Okay, so in this question, we have two optimizing consumers, matthew and susan.

00:07 So because they're optimizing consumers, we know that they're going to consume at the point where their utility function, or so sorry, the slope of the utility function is going to be equal to the slope of their budget constraint.

00:23 Okay.

00:24 So essentially, they're consuming on the highest possible utility function given their budget constraint.

00:30 So for matthew, that's going to be given by point a, and for susan, that's going to be given by point b.

00:39 So we know that consume two goods, shirts and hats, and we know the price of a shirt is $100 and the price of a hat is $50.

00:47 Matthew buys four shirts and 16 hats, and we know that susan buys six shirts and 12 hats.

00:53 So we want to infer their marginal rates of substitution.

00:58 All right, so we have the situation represented for matthew in the top diagram.

01:03 So we can see that at the point of optimization, matthew consumes four shirts and 16 hats, and the bottom diagram we see that susan consumes six shirts and 12 hats as we are told from the question.

01:18 Okay.

01:19 And we know that at this point, oh, at these points rather, a and b, we know that the price ratio, okay, between a shirt and a hat is going to be equal to, the margin utility or the ratio of the margin utilities between shirts and hats.

01:40 Okay, so the price ratio is the slope of the budget constraint and the margin rate of substitution, which is the ratio of the margin utilities, okay, mux divided by mui, where mux is margin utility from consuming an extra shirt and mui is margin utility for consuming an extra from consuming an extra hat...

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