In consumer demand theory, the relationship that shows how the consumer’s optimal choice of the quantity of a good varies as its price changes, holding other factors constant, is called the ________ function.

Introduction / Context:
In microeconomics, demand and supply functions describe how quantities demanded or supplied respond to changes in price and other variables. The consumer’s optimisation problem leads to a demand function that links the optimal quantity of a good to its price, income and the prices of other goods. This question asks you to identify the name of the function that specifically relates optimal quantity demanded to the price of a good.

Given Data / Assumptions:
- The consumer chooses an optimal quantity of a good based on preferences, income and prices.
- The question focuses on how this optimal quantity changes when the price of the good changes.
- Other factors such as income and prices of other goods are held constant for this relationship.
- The options are price function, substitution function, supply function and demand function.

Concept / Approach:
The demand function expresses quantity demanded as a function of its own price and other determinants. When we isolate the relationship between quantity demanded and the good’s own price, holding other factors constant, we obtain the individual demand curve. This is the central concept underlying the law of demand. The supply function, by contrast, relates quantity supplied to price from the producer’s perspective. The term substitution function is not standard for this relationship, and a price function would more likely refer to price as a function of other variables.

Step-by-Step Solution:
Step 1: Recall that the consumer’s optimisation leads to a mapping from prices and income to optimal quantities of each good.Step 2: Recognise that when we focus on one good and vary its price, keeping other things equal, we obtain the demand curve for that good.Step 3: Identify that this mapping from price to optimal quantity is called the demand function for that good.Step 4: Note that the supply function would instead describe the producer’s choice of quantity supplied at different prices.Step 5: Conclude that the correct term for the relationship described in the question is the demand function.

Verification / Alternative check:
You can verify by recalling typical notation: Qd = f(P), where Qd is quantity demanded and P is price, is one way to represent the demand function. Graphically, this is shown as a downward sloping curve in the price quantity plane. Supply functions are usually written as Qs = g(P) and are upward sloping under normal conditions. These formal representations confirm that the consumer’s optimal quantity choice as a function of price is indeed the demand function.

Why Other Options Are Wrong:
Option A, price function, is wrong because it would normally refer to price as the dependent variable. Option B, substitution function, is not a standard term; the substitution effect is part of the explanation of how demand changes, but the overall relationship is still called the demand function. Option C, supply function, is wrong because it refers to the behaviour of producers, not consumers.

Common Pitfalls:
Students may sometimes confuse demand and supply because both are drawn in the same type of diagram. It is useful to remember that demand comes from the consumer side and is downward sloping, whereas supply comes from the producer side and is typically upward sloping. Keeping track of whether the question is about the consumer’s or the producer’s optimal choice will point you to the correct function name.

Final Answer:
The relationship described is called the Demand function.