Question

Donald, a frugal graduate student, consumes only coffee (C) and buttered toast \((B T) .\) He buys these items at the university cafeteria and always uses two pats of butter for each piece of toast. Donald spends exactly half of his meager stipend on coffee and the other half on buttered toast. a. In this problem, buttered toast can be treated as a composite commodity. What is its price in terms of the prices of butter \(\left(P_{B}\right)\) and toast \(\left(P_{T}\right) ?\) b. Explain why \(\partial C / \partial P_{B T}=0\) c. Is it also true here that \(\partial C / \partial P_{B}\) and \(\partial C / \partial P_{T}\) are equal to zero?

Short answer

Answer: The partial derivative of coffee consumption C with respect to the prices of butter (PB), toast (PT), and buttered toast (PBT) is equal to zero because Donald spends exactly half of his stipend on coffee and the other half on buttered toast. As the prices of the components change, Donald adjusts the quantity of buttered toast he consumes to maintain the same level of coffee consumption. Therefore, his coffee consumption does not change in response to changes in the prices of butter, toast, or buttered toast.

Step-by-step solution

Calculate the price of the composite commodity

To find the price of buttered toast (PBT) in terms of the prices of butter (PB) and toast (PT), we simply need to add the prices of its components, keeping in mind that two pats of butter are used for each toast. So, P_{B T}=P_{T}+2 P_{B} #b. Explain why the partial derivative of coffee consumption C with respect to the price of buttered toast (PBT) is equal to zero.#

Analyze budget constraint

According to the problem, Donald spends half of his stipend on coffee C and the other half on buttered toast BT. Let his total stipend be S. Then his budget constraint can be written as C * P_{C} + B T * P_{B T} = S

Show that \(\partial C / \partial P_{B T}=0\)

Since Donald spends exactly half of his stipend on coffee, when the price of buttered toast increases, Donald has no choice but to reduce the quantity of buttered toast he consumes in order to maintain the same level of coffee consumption. As a result, his coffee consumption does not change in response to changes in the price of buttered toast. Mathematically, this implies that the partial derivative of coffee consumption with respect to the price of buttered toast is equal to zero: \(\frac{\partial C}{\partial P_{B T}} = 0\) #c. Is it also true here that the partial derivative of coffee consumption C with respect to the prices of butter (PB) and toast (PT) is equal to zero?#

Show that \(\partial C / \partial P_{B}=0\) and \(\partial C / \partial P_{T}=0\)

It is important to note that when the price of one component of the composite commodity changes, it also affects the price of the composite commodity. If the composite commodity's price changes due to changes in the component prices, Donald's coffee consumption could also potentially change. However, given Donald's fixed budget and the specific amounts of butter and toast composition to maintain, as mentioned in part b, his coffee consumption still won't change in response to a change in these prices. Hence the partial derivative of coffee consumption with respect to the prices of butter and toast will also equal to zero: \(\frac{\partial C}{\partial P_{B}} = 0\) and \(\frac{\partial C}{\partial P_{T}} = 0\)

Key concepts

Budget Constraint

Understanding the concept of a budget constraint is pivotal in analyzing consumption behavior. A budget constraint represents the combination of goods and services that a consumer can purchase given their income and the prices of those goods and services. It is essentially an equation that equates the cost of a consumer's chosen bundle of goods to their available income. For instance, if Donald, our frugal student, has a stipend of \(S\), and he has to decide between spending on coffee \(C\) or buttered toast \(BT\), his budget constraint can be expressed as:

\[ C \times P_C + BT \times P_{BT} = S \]
Here, \(P_C\) and \(P_{BT}\) denote the price of coffee and buttered toast, respectively. This constraint means that if the price of either good increases, Donald must adjust his consumption accordingly to stay within his budget. His consumption pattern will therefore directly reflect the constraint imposed by his stipend and the relative prices of coffee and toast.

Partial Derivative

The partial derivative is a mathematical tool used to show how a function changes as only one of its input variables changes. In economic analyses of consumption behavior, partial derivatives can demonstrate how a change in the price of a good affects the quantity consumed, while holding other variables constant. For instance, in our problem, we have the partial derivative \( \frac{\partial C}{\partial P_{BT}} = 0 \), which states that the consumption of coffee \(C\) does not change when the price of buttered toast \(P_{BT}\) changes. This is because Donald's stipend is split equally between coffee and buttered toast, resulting in his coffee consumption being unaffected by the price variation of buttered toast. It reflects the idea that Donald's consumption is rigid in this aspect, adhering strictly to the budget constraint.

Consumption Behavior

Consumption behavior analyzes how consumers allocate their income to different goods and services. This behavior is influenced by a variety of factors, including income, prices, preferences, and the budget constraint. In our scenario with Donald, the consumption behavior is specified: he allocates half of his income to coffee and the other half to buttered toast. His behavior is also influenced by the concept of a composite commodity, which in this case is buttered toast made up of toast and butter.

The composite commodity concept simplifies the consumption analysis by treating a group of goods as a single good. Here, since Donald uses two pats of butter for each piece of toast, the composite commodity 'buttered toast' has a fixed proportion of those two goods. Therefore, changes in the prices of butter or toast will not independently affect Donald's consumption of coffee because it is the price of the composite commodity that matters in his budget allocation. This simplistic model of consumption assumes no substitution between goods and thus helps us understand a controlled aspect of consumer choice behavior.