derivation of an individual demand curve

Utility, Consumer Optimisation and the Derivation of an Individual Demand Curve

1. Key Concepts (Cambridge Economics – Core Vocabulary)

• Scarcity & Choice – limited resources force individuals and societies to make choices.
• Opportunity Cost – the value of the next best alternative foregone.
• Margin & Decision‑Making – choices are made by comparing marginal benefits with marginal costs (AO1, AO2).
• Equilibrium – the point where opposing forces (e.g., MU and price) are balanced.
• Efficiency – resources are allocated so that no one can be made better off without making someone else worse off.
• Time Horizon – short‑run vs. long‑run considerations affect consumer behaviour.
• Government Role & Development – policies can correct market failures and influence welfare (covered later in the syllabus).

2. What Is Utility?

3. The Consumer’s Optimisation Problem (AO2)

A rational consumer chooses a bundle of goods \(x=(x_1,x_2,\dots ,x_n)\) that maximises total utility subject to the budget constraint:

\[ \max_{x}\;U(x)\qquad\text{s.t.}\qquad\sum_{i=1}^{n}p_i x_i = I, \] where \(p_i\) is the price of good \(i\) and \(I\) is disposable income (AO1).

4. Solving the Problem – The Lagrangian Method (AO2)

Introduce the Lagrange multiplier \(\lambda\) (the marginal utility of income) to incorporate the constraint:

\[ \mathcal{L}(x,\lambda)=U(x)+\lambda\Bigl(I-\sum_{i=1}^{n}p_i x_i\Bigr). \]

4.1 First‑Order Conditions (FOCs) – The Equi‑Marginal Principle (AO2)

• For each good \(i\): \[ \frac{\partial U}{\partial x_i}= \lambda p_i\quad\Longleftrightarrow\quad \frac{MU_i}{p_i}= \lambda . \] Interpretation: the last dollar spent on every good yields the same marginal utility (the equi‑marginal principle).
• For the multiplier: \[ I-\sum_{i=1}^{n}p_i x_i =0 . \]

Dividing any two FOCs eliminates \(\lambda\) and gives the familiar marginal rate of substitution (MRS) condition:

\[ \frac{MU_i}{MU_j}= \frac{p_i}{p_j}\qquad\forall i,j . \]

5. Marshallian (Uncompensated) Demand Functions (AO3)

Solving the system of FOCs for each \(x_i\) yields the Marshallian demand functions:

\[ x_i^{*}=f_i(p_1,\dots ,p_n,I),\qquad i=1,\dots ,n . \]

These functions describe how the quantity demanded of each good varies with its own price, the prices of other goods and income, holding everything else constant (ceteris paribus).

6. Worked Example – Cobb–Douglas Utility (AO2 + AO3)

Consider a two‑good world with the Cobb–Douglas utility function

\[ U(x_1,x_2)=x_1^{\alpha}\,x_2^{1-\alpha},\qquad 0<\alpha<1 . \]

Step 1 – Write the Lagrangian

\[ \mathcal{L}=x_1^{\alpha}x_2^{1-\alpha}+\lambda\bigl(I-p_1x_1-p_2x_2\bigr). \]

Step 2 – First‑order conditions

\[ \begin{aligned} \frac{\partial\mathcal{L}}{\partial x_1}&=\alpha x_1^{\alpha-1}x_2^{1-\alpha}-\lambda p_1=0,\\[4pt] \frac{\partial\mathcal{L}}{\partial x_2}&=(1-\alpha)x_1^{\alpha}x_2^{-\alpha}-\lambda p_2=0,\\[4pt] \frac{\partial\mathcal{L}}{\partial\lambda}&=I-p_1x_1-p_2x_2=0 . \end{aligned} \]

Step 3 – Apply the Equi‑Marginal Principle

\[ \frac{\alpha}{1-\alpha}\,\frac{x_2}{x_1}= \frac{p_1}{p_2} \;\Longrightarrow\; x_2=\frac{\alpha}{1-\alpha}\,\frac{p_2}{p_1}\,x_1 . \]

Step 4 – Substitute into the Budget Constraint

\[ p_1x_1+p_2\Bigl(\frac{\alpha}{1-\alpha}\frac{p_2}{p_1}x_1\Bigr)=I \;\Longrightarrow\; x_1^{*}= \frac{\alpha I}{p_1}. \]

Step 5 – Obtain the Second Demand Function

\[ x_2^{*}= \frac{(1-\alpha)I}{p_2}. \]

Result – Marshallian Demand Functions

Good	Marshallian demand
\(x_1\)	\(\displaystyle \frac{\alpha I}{p_1}\)
\(x_2\)	\(\displaystyle \frac{(1-\alpha) I}{p_2}\)

Evaluation (AO3)

• The Cobb–Douglas form assumes strictly convex indifference curves and constant expenditure shares – a useful benchmark but rarely exact in reality.
• It ignores income effects on the price elasticity of demand; empirical data often show non‑linear relationships.
• Nevertheless, the model clearly illustrates the mechanics of the optimisation process and the inverse relationship between price and quantity demanded.

7. From Marshallian Demand to an Individual Demand Curve (AO2)

To obtain the demand curve for a single good (e.g., good 1) we hold income (\(I\)) and the price of the other good (\(p_2\)) constant and plot the relationship

\[ x_1^{*}= \frac{\alpha I}{p_1} \]

against varying values of \(p_1\). The curve is a straight line through the origin with slope \(-\alpha I\), demonstrating the law of demand: as price rises, the quantity demanded falls.

8. Graphical Illustration

9. Quick Reference – Cambridge AS & A‑Level Topics (1‑11)

Topic Code	Title (AS)	Title (A‑Level)
1.1‑1.3	Scarcity, Choice & Opportunity Cost	Same (foundation for all later analysis)
2.1‑2.4	Demand & Supply, Elasticities	Market equilibrium, price mechanisms
3.1‑3.3	Government Intervention – taxes, subsidies, price controls	Welfare analysis of micro‑policy
4.1‑4.6	Aggregate Demand & Supply, Growth, Unemployment, Inflation	Short‑run vs. long‑run macro equilibrium
5.1‑5.4	Fiscal & Monetary Policy, Supply‑side measures	Policy mix and macro‑stability
6.1‑6.5	International Trade, Protectionism, Balance of Payments, Exchange Rates	Trade policy and development
7.1‑7.8	Utility, Consumer Choice, Market Failure, Firm Theory, Pricing	Advanced micro – cost‑revenue analysis, market structures
8.1‑8.3	Government Micro‑policy, Labour Market	Wage determination, unemployment types
9.1‑9.4	Multiplier, Money & Banking	Monetary transmission mechanisms
10.1‑10.3	Government Macro‑policy, Policy Inter‑relationships	Fiscal‑monetary interaction, supply‑side effects
11.1‑11.5	Development, Aid, Globalisation	Economic growth, sustainability, inequality

Each of the above topics can be linked back to the core concepts introduced in Section 1 (e.g., scarcity underpins all demand‑supply analysis; marginal thinking appears in utility, cost‑revenue and policy evaluation).

10. Key Takeaways (AO1 + AO2 + AO3)

• Utility foundations: Total and marginal utility, plus the law of diminishing MU, explain why consumers make trade‑offs.
• Equi‑marginal principle: Optimal consumption occurs where \(\displaystyle\frac{MU_i}{p_i}\) is equalised across all goods (or where MRS = price ratio).
• Marshallian demand follows from solving the utility‑maximisation problem with a Lagrangian; it shows the systematic effect of price, other prices and income on quantity demanded.
• Individual demand curve is obtained by holding income and other prices constant and plotting the Marshallian demand for a single good against its own price – a visual representation of the law of demand.
• Evaluation: Real‑world consumers may not satisfy all model assumptions (perfect rationality, fixed preferences, no income effects). Recognising these limitations is essential for AO3.
• Link to the wider syllabus: The same marginal reasoning and equilibrium ideas reappear in topics on market failure, firm behaviour, macro‑policy and international economics.