Mathematical organization of demand curves

date

Jun 23, 2024

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mathematical-organization-of-demand-curves

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Published

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Note

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🦋 brain

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I conducted a systematic review of the demand curve, a fundamental concept in microeconomics, using a mathematical approach. After carefully examining existing economic literature and critically analyzing their theoretical frameworks, I formulated a new approach. This was done because it was suggested that there is room for further clarification and refinement in the conceptualization and representation of demand curves.

Formulation

We consider a specific single market. We also assume that the utility function is of the Cobb-Douglas type.

Let's define the following variables:

: Type of good

: Goods space

: Consumer

: Quantity of good $i$ consumed by consumer $j$

: Set of all $x_{ji}$ for consumer $j$

: Price of good

: Set of all possible prices for good ()

: Income of consumer

: Budget constraint for consumer

: Preference of consumer for good ()

: Utility function of consumer

Given these definitions, the demand curve can be expressed as follows:

: Demand curve of consumer for good

(where is the function representing the demand curve in this form)

Furthermore, the market demand curve is expressed as:

: Market demand curve for good

This formulation provides a more rigorous and comprehensive mathematical representation of demand curves, taking into account individual consumer preferences, budget constraints, and market aggregation.

Specific Example

Let's consider a scenario with one specific market for Calpis (a Japanese uncarbonated soft drink), and another good, which we'll assume is only Cola.

We'll assign values as follows:

: Calpis

: Cola

: Price of Calpis (variable),

: Price of Cola (fixed)

: Alice

: Bob

: Alice's income

: Alice's consumption of Calpis

: Alice's consumption of Cola

: Alice's consumption of all goods

: Alice's budget constraint

: Alice's preference for Calpis

: Alice's preference for Cola

: Alice's utility function (she likes Calpis slightly more than Cola)

The demand curve is as follows:

: Alice's demand curve for Calpis

Assuming Bob's demand curve is derived similarly and looks like this:

: Bob's demand curve for Calpis

Then, the market demand curve would be:

: Market demand curve for Calpis

Specifically, the graph of Alice's budget constraint and indifference curve would look as follows:

The source code is:

Additionally, the graph of Alice's demand curve for Calpis is as follows:

The source code is:

Reference

Mighty Microeconomics, Michihiro Kandori