derivation of an individual demand curve

Utility and Demand: Deriving the Individual Demand Curve

What is Utility? 🤔

Utility is a way economists measure how much satisfaction or happiness a person gets from consuming goods. Think of it like the “taste score” you give to a pizza slice – the more slices you eat, the higher the total taste score, but each extra slice might add a little less than the previous one.

Marginal Utility (MU) 📈

Marginal Utility is the extra satisfaction from consuming one more unit of a good.

\$ MU = \frac{\Delta U}{\Delta Q} \$

• First slice of pizza: huge MU.
• Second slice: still good, but MU drops.
• Third slice: MU keeps dropping.

This is the Law of Diminishing Marginal Utility – each additional unit adds less satisfaction than the previous one.

Utility Maximization: The Budget Constraint 🎯

A consumer wants to get the most utility while staying within their budget.

The budget constraint is:

\$ Px \, x + Py \, y = I \$

where \(Px\) and \(Py\) are prices, \(x\) and \(y\) are quantities, and \(I\) is income.

The optimal choice satisfies:

\$ \frac{MUx}{Px} = \frac{MUy}{Py} \$

This means the last dollar spent on each good gives the same extra utility.

Deriving the Individual Demand Curve 📉

Let’s use a simple Cobb‑Douglas utility function:

\$ U(x,y) = x^{0.5}\,y^{0.5} \$

1. Compute marginal utilities:

   \$ MUx = 0.5\,x^{-0.5}\,y^{0.5} \quad \text{and} \quad MUy = 0.5\,x^{0.5}\,y^{-0.5} \$

2. Set the ratio equal to the price ratio:

   \$ \frac{MUx}{Px} = \frac{MUy}{Py} \;\;\Rightarrow\;\; \frac{0.5\,x^{-0.5}\,y^{0.5}}{Px} = \frac{0.5\,x^{0.5}\,y^{-0.5}}{Py} \$

3. Simplify to find the relationship between \(x\) and \(y\):

   \$ \frac{y}{x} = \frac{Px}{Py} \$

4. Use the budget constraint to solve for \(x\):

   \$ Px\,x + Py\,\left(\frac{Px}{Py}\,x\right) = I \;\;\Rightarrow\;\; 2P_x\,x = I \$

   
   Thus:

   \$ x = \frac{I}{2P_x} \$

5. Notice how \(x\) (quantity demanded of good \(x\)) depends on its own price \(P_x\) and income \(I\). The demand curve for \(x\) is

   \$ Qx = \frac{I}{2Px} \$

   which slopes downward – as price falls, quantity demanded rises.

Key Take‑aways in a Quick Table

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