Question

Samantha has an income of $50 and is currently consuming a bundle with 10 pizza slices and 5 glasses of beer. Samantha has typical shaped (convex) indifference curves, and at this bundle she is willing to give up 3 beers for one more slice of pizza since her marginal utility of beer is 4 and her marginal utility of pizza is 12 at this bundle. The price of a glass of beer is $2 and the price of a slice of pizza is $4.

a) Graph Samantha’s budget line on a graph with pizza on the horizontal axis and beer on the vertical axis. Identify both intercept quantities on the graph.

b) Write the equation of this particular budget line.

c) Is Samantha maximizing her utility? Explain.

d) Would she change her current consumption bundle to one with more pizza and less beer, one with less pizza and more beer, or keep her current consumption bundle?

Answer 1

Given:

Consumption bundle = (10 slices of pizza, 5 glasses of beer)

Income: (M) = $50

As samantha is willing to give 3 beers for 1 pizza, it means that Samantha's marginal rate of substitution is:

Given; Marginal utility of pizza (MU_P) = 12

Marginal utility of beer (MU_B) = 4

Price of a glass of beer (P_B) = $2

Price of a pizza slice (P_P) = $4

1.) The budget constraint shows the combination of two goods that a consumer can afford to consume given his income or budget.

Budget constraint;

Income = (Price of pizza) pizza silzes+ (Price of beer) Gasses of beer

Putting the values of prices and income of Samantha's

finding the intercepts of the budget ocnstraint:

the intercepts of the budget constraint shows the maximum amount of pizza and beer that she should buy with her income, if she spends all her income on one.

Therefore, the quantity of pizza slices when she buys zero beer.

Putting B = 0, P

Similary  the number of glasses of beer when she buys zero pizza slices

Putting P = 0, B

As we are representing pizza on the horizontal axis, therefore the horizontal intersept is (12.5) and the vertical intercept is (25). That is we have two points,

(12.5 slices of pizza, 0 beer of glasses) and (0 slices of pizza, 25 beer of glasses)

Therefore the budget constraint is drawn as:

Description of the diagram:

The pizza is on the x-axis and beer on the y-axis. where the x intercept is 12.5 and the horizontal intercept is 25.

That is,

(12.5,0): (0,25)

2.) Therefore, The equation of the budget constraint of Samantha is:

3.) To check whether Samantha is maximizing her utility, we have to set the consumption maximization problem. The consumption of an individual is maximized when the ratio of the marginal utilities of the two goods is equal to the ratio of the prices.

That is,

Checking if the consumption is maximized at the current consumption bundle of:

(10 slices of pizza, 5 glasses of beer)

As samantha is willing to give 3 beers for 1 pizza, it means that Samantha's marginal rate of substitution is:

Given; Marginal utility of pizza (MU_P) = 12

marginal utility of beer (MU_B) = 4

Therefore,

Price of a glass of beer (P_B) = $2

Price of a pizza slice (P_P) = $4

Therefore, as we can see:

Therefore this consumption bundle is not the optimal bundle for Samantha. As the ratio of marginal utilities from both goods is more than the ratio of prices. it means that she is deriving more per dollar satisfaction from the slices of pizza than glasses of bear . So it is not where the consumption is maximized.

As the ratio of marginal utility is more than the ratio of prices, it means that the current consumption bundle is not optimum. And since the marginal rate of substitution (MU_P/MU_B) is more than the ratio prices, It means that the slope of the indifference curve is more than the slope of the budget constraint. Therefore it means that the consumption bundle doesnt lie where the indifference curve cuts the budget constraint. As the consumption maximization condition is not fulfilled. So it is not the optimum bundle.

d.) She would change her consumption to one with more pizza and less beer.

As, The ratio of marginal utilities or the Marginal rate of substitution is more than the ratio of prices, therefore it means that consumers are deriving more satisfaction from pizza (x-axis) than the beer (y-axis).

As

This shows that,

Therefore Samantha's marginal utility per dollar spent on pizza is more than the marginal utility per dollar spent on glasses of beer. It means that now one more pizza is adding more to her satisfaction than one more beer. The rational consumer Samatha is, she would try to maximize her consumption and consume more of what gives her more additional satisfaction. And she can only increase the consumption by giving up another. Therefore she will increase the consumption of pizza and decrease the consumption of beer until the marginal utility per dollar spent on both equalizes.

Therefore she will consume more pizza slices rather than stopping at this consumption. She increases her consumption of pizza and decreases her consumption of beer, rather than sticking to the current consumption bundle