David N. gets 3 per week as an allowance to spend any way...

David $\mathrm{N}$. gets $\$ 3$ per week as an allowance to spend any way he pleases. Because he likes only peanut butter and jelly sandwiches, he spends the entire amount on peanut butter $\langle\text { at } \$ 0.05 \text { per ounce })$ and jelly (at $\$ 0.10$ per ounce). Bread is provided free of charge by a concerned neighbor, David is a particular catcr and makes his sandwiches with exactly 1 ounce of jelly and 2 ounces of peanut butter. He is set in his ways and will never change these proportions. a. How much peanut butter and jelly will David buy with his $\$ 3$ allowance in a week? b. Suppose the price of jelly were to rise to $\$ 0.15$ an ounce. How much of each commodity would be bought? c. By how much should David's allowance be increased to compensate for the rise in the price of jelly in part (b)? d. Graph your results in parts (a) to (c). e. In what sense does this problem involve only a single commodity, peanut butter and jelly sandwiches? Graph the demand curve for this single commodity. f. Discuss the results of this problem in terms of the income and substitution effects involved in the demand for jelly.

David $\mathrm{N}$. gets $\$ 3$ per week as an allowance to spend any way he pleases. Because he likes only peanut butter and jelly sandwiches, he spends the entire amount on peanut butter $\langle\text { at } \$ 0.05 \text { per ounce })$ and jelly (at $\$ 0.10$ per ounce). Bread is provided free of charge by a concerned neighbor, David is a particular catcr and makes his sandwiches with exactly 1 ounce of jelly and 2 ounces of peanut butter. He is set in his ways and will never change these proportions.
a. How much peanut butter and jelly will David buy with his $\$ 3$ allowance in a week?
b. Suppose the price of jelly were to rise to $\$ 0.15$ an ounce. How much of each commodity would be bought?
c. By how much should David's allowance be increased to compensate for the rise in the price of jelly in part (b)?
d. Graph your results in parts (a) to (c).
e. In what sense does this problem involve only a single commodity, peanut butter and jelly sandwiches? Graph the demand curve for this single commodity.
f. Discuss the results of this problem in terms of the income and substitution effects involved in the demand for jelly.

Key Concepts

Budget Constraint

The budget constraint represents the limit on the consumption choices available to a consumer given his limited income and the prices of goods. In this problem, it shows how David's total expenditure on peanut butter and jelly is bounded by his allowance, with changes in the price of jelly altering the feasible set of consumption bundles he can purchase.

Perfect Complements

Perfect complements refer to goods that are consumed in fixed proportions where the utility derived comes from the specific combination of the goods, not from their individual quantities. In the context of the problem, peanut butter and jelly are used together in a set ratio for making sandwiches, implying that David will only purchase them in that fixed combination.

Demand Functions Under Fixed Proportions

In scenarios where goods are perfect complements, the demand functions are determined by the fixed consumption ratio in which the goods are combined. This simplifies the analysis since the consumer's choice is characterized by a single decision – the number of sandwiches to produce – rather than choosing the quantities of each good independently.

Price Changes and the Adjusted Budget Constraint

When the price of one commodity changes, the budget constraint is altered, affecting the consumer's ability to purchase the fixed bundle in the same quantity. The problem explores how an increase in the price of jelly shifts the budget line and requires an adjustment (in terms of increased allowance) to maintain the same consumption level of sandwiches.

Income and Substitution Effects

The income and substitution effects analyze how a consumer's purchasing behavior changes in response to a change in the price of a good. With perfect complements, only the income effect is relevant because the fixed consumption ratio does not allow for substitution between the goods, even when relative prices change. This problem examines how the rise in the price of jelly affects David both in terms of his effective purchasing power and the necessity to adjust his budget.

Transcript

00:01 So david n.

00:02 Gets $3 per week as an allowance to spend any way it pleases.

00:07 So because he likes only peanuts, butter, and jelly sandwiches, it spends the entire amount on peanut butter.

00:20 So peanuts butter, he spends everything on peanut butter and jelly.

00:27 So this peanut butter is 0 .05 per ounce.

00:33 And jelly is 0 .0 .5 per ounce.

00:38 10 cents per ounce and bread is provided free are charged by a consignable so bread is free now there it is a particular eater and it makes the sandwiches with exactly one ounce of jelly so one ounce of jelly and two ounces of peanut butter now it sets in his ways and will never change his proportion so you have to have one ounce of jelly and two ounces of peanut butter then the first question says how much peanut butter and jelly would david buy with this three dollar and allowance in a week so first of all you need to know that these goods are complementary goods because you need it needs the peanut butter and the jelly and the bread to make a complete peanut butter jelly sandwich so and complementary goods are the goods that are consumed together by a consumer like the peanut butter jelly and the bread so consuming this the single commodity will not at the utility of others so therefore when the price of one of them increases the demand for the complement would decrease because then nobody wants to buy it so the income of the consumer is the total budget that the individual has to spend on the available commodities now moving on since we're giving that um he likes to spend consuming peanut butter and jelly sand we're using one ounce of jelly and two ounces of butter that means that the utility and the demand function is giving as perfect compliments now it is assumed that x stands for peanut butter and y stands for jelly so let's okay i'm sorry let's go by the way it was so this is two peanut butter and this is jelly so let's say let x be peanut butter and y be jelly so it just it just looks easier so so the utility function with the for the complementary goods will be you is equal to m ian open brackets x comma two y simply because it consumes one of x and two of y so the money income which is m is of three dollars and the price of peanut butter to be five cents and jelly of ten cents so the budget constraints would be as follows so, p of x plus p y of y is equal to m.

03:41 So 0 .05x plus 0 .10y is equal to 3.

03:50 So this is equation 1.

03:51 Now, since the two goods are using fixed proportions, so therefore x is equal to 2y.

03:57 So we need to calculate the demand function.

03:59 So we substitute x equals to 2y in equation 1.

04:02 And we're going to get 0 .05 x plus 0 .10 y is equal to 3 so therefore 0 .05 multiply by 2y where we see x plus 0 .10 y is equal to 3 and from there 0 .20 y is equal to 3 and we can say that y is equal to 15 so now we can calculate the value of x as x is equal to 2y so x is equal to 2 y so x is equal to 2 2 multiplied by 15 and the x is equal to 30 right so therefore it would buy peanut butter um and so therefore buy peanut butter and 15 ounces of jelly so we can say that it would buy and remember that x is x stands for peanut butter so it would buy 30 ounces of peanut butter so it would buy and why stand for jelly so and 15 ounces of jelly so that's the answer so moving on to the second question suppose that the price of jelly were to increase to 0 .15 an ounce so price of jelly went up to 0 .15 an ounce now how much of each commodity would be bought so when the price increases to this and we need to find the quantity the consumer will buy.

05:55 So at the price of jelly increases to 0 .15, it is more costlier than before than the consumer rights because before the price of jelly was at 10 cents and now it's 15 cents, so it is more expensive.

06:10 So therefore, the new budget constraints will be cost to 0 .05x plus 0 .15...

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