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: $$ P_x \, x + P_y \, y = I $$ where \(P_x\) and \(P_y\) are prices, \(x\) and \(y\) are quantities, and \(I\) is income.
The optimal choice satisfies: $$ \frac{MU_x}{P_x} = \frac{MU_y}{P_y} $$ 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: $$ U(x,y) = x^{0.5}\,y^{0.5} $$
- Compute marginal utilities: $$ MU_x = 0.5\,x^{-0.5}\,y^{0.5} \quad \text{and} \quad MU_y = 0.5\,x^{0.5}\,y^{-0.5} $$
- Set the ratio equal to the price ratio: $$ \frac{MU_x}{P_x} = \frac{MU_y}{P_y} \;\;\Rightarrow\;\; \frac{0.5\,x^{-0.5}\,y^{0.5}}{P_x} = \frac{0.5\,x^{0.5}\,y^{-0.5}}{P_y} $$
- Simplify to find the relationship between \(x\) and \(y\): $$ \frac{y}{x} = \frac{P_x}{P_y} $$
- Use the budget constraint to solve for \(x\):
$$ P_x\,x + P_y\,\left(\frac{P_x}{P_y}\,x\right) = I \;\;\Rightarrow\;\; 2P_x\,x = I $$
Thus: $$ x = \frac{I}{2P_x} $$ - Notice how \(x\) (quantity demanded of good \(x\)) depends on its own price \(P_x\) and income \(I\). The demand curve for \(x\) is $$ Q_x = \frac{I}{2P_x} $$ which slopes downward – as price falls, quantity demanded rises.
Key Take‑aways in a Quick Table
| Step | Result |
|---|---|
| Utility Function | $U(x,y)=x^{0.5}y^{0.5}$ |
| MU Ratio Condition | $\frac{MU_x}{P_x}=\frac{MU_y}{P_y}$ |
| Demand Function | $Q_x=\dfrac{I}{2P_x}$ |
Exam Tips 📚
Remember:
- Show the MU ratio step clearly – it’s the core of the derivation.
- Always link back to the budget constraint; it gives the final quantity.
- Use symbols consistently – students will spot any mix‑ups.
- When asked for a demand curve, state the functional form and explain its slope.
Revision
Log in to practice.