Question

What is the form of the inverse demand function for good 1 in the case of perfect complements?

Short answer

The inverse demand for good 1 in perfect complements: \( p_1 = M - px_1 \).

Step-by-step solution

Understanding Perfect Complements

In perfect complements, two goods are consumed together in fixed proportions. For example, if goods 1 and 2 are perfect complements, the consumer might want them in a 1:1 relationship.

Demand Function for Perfect Complements

For perfect complements, the demand is based on the ratio or proportion of the two goods consumed together. The Demand for good 1 is equal to the quantity of good 2 when they are perfect complements. So, the demand for good 1 is given by: \[ x_1 = x_2 \] where \(x_1\) and \(x_2\) are the quantities of goods 1 and 2 respectively.

Inverse Demand Function Concept

The inverse demand function relates the price of a good to the quantity demanded. It essentially answers "what is the price consumers are willing to pay for a certain quantity?" Given the demand function, we need to express price as a function of quantity.

Formulating the Inverse Demand Function

In the case of perfect complements, the price of good 1 (\( p_1 \)) is determined by setting the consumer's budget to equal consumption values. Therefore, if the consumer spends exactly their budget \for these goods, each quantity increment directly affects the price.Since \( x_1 = x_2 \), the inverse demand function can be:\[ p_1 = f(x_1) \]The price \(p_1\) isn't only affected by smooth quantities but depends on how these quantities matchup with respect to overall minimum of \( x_1 \) and \( x_2 \), Assuming prices equal expenditures uniformly: \\[ p_1 = m - p_2x_2 \]For a simple form (assuming similar unit costs and proportions): \\[ p_1 = M - px_1 \] where M is a constant representing fixed costs/budget allocating x_1.\

Key concepts

Perfect Complements

Perfect complements refer to pairs of goods that are used or consumed together in fixed proportions. They are interdependent such that the utility derived from one cannot be achieved without the other. Imagine coffee and sugar as perfect complements where you need one spoon of sugar for every cup of coffee to enjoy your perfect drink. This means for every unit of good 1, you need exactly one unit of good 2.
In economic terms, the demand for one good directly depends on the quantity of its complement. If you have only one cup of coffee and zero sugar, your demand is limited by the missing sugar regardless of having excess coffee. Therefore, when considering goods as perfect complements, the consumption and thus demand strategy is all about reaching a balanced proportion.

Consumer Budget

A consumer budget is the total amount of money that an individual is willing to spend on goods and services. It is a fundamental factor influencing purchasing decisions. In scenarios involving perfect complements, the consumer budget plays a crucial role in determining the quantities of each good to purchase.
For example, if your budget is $10, and both the complementary goods cost $1 each, you can purchase five units of each commodity. Here, your budget directly influences the maximum quantities you can afford while maintaining their required complementary ratio. The budget constraint shapes the consumer's choice, necessitating a balance between spending money on each perfect complement to achieve optimal utility.

Demand Function

The demand function represents the relationship between the quantity demanded of a good and factors that influence it, primarily its price. For perfect complements, the demand function is characterized by the need to maintain a fixed ratio between the quantities of complementary goods.
In mathematical terms, if goods 1 and 2 are perfect complements, their demand function might represent as:

• \[ x_1 = x_2 \]

This indicates a one-to-one consumption ratio. If either good is unavailable in the quantity needed, it restricts the amount of the other good you can use, even if your budget allows for more. The demand function for perfect complements ensures the quantities are consumed together in exact proportions, dictating the precise amounts demanded based on available quantities.

Quantities Relationship

In the context of perfect complements, the quantities relationship is crucial because it defines how goods are paired and consumed. This relationship ensures that for every unit of good 1, there is exactly one corresponding unit of good 2. The purchase of these goods is inseparably linked; buying or consuming an uneven amount won't maximize utility as intended.
This matching of amounts can be mathematically expressed and has practical implications for both consumers and market analysis. The inverse demand function gives insight into how the price consumers are willing to pay shifts when these quantities are tightly restricted by the need for perfect matching. For instance, if the price of good 1 shoots up but you still need the same amount of good 2, you'll adjust purchases based around what your total expenditure will allow, keeping quantities balanced according to your budget.