Consumer's optimum and derivation of the demand curve

Detail

1. The consumer's problem

• Rational consumer: a person who chooses the bundle she likes best among all the bundles she can afford.
• Bundle: a combination of two goods, written (x₁, x₂). Example: 3 bananas and 2 mangoes.
• Budget set: all bundles that cost her no more than her income. p₁x₁ + p₂x₂ ≤ M.
• Budget line: the bundles that cost exactly her income. p₁x₁ + p₂x₂ = M. Its slope is −p₁/p₂.
• Indifference curve (IC): a curve joining bundles that give her equal satisfaction. She does not prefer any point on it over any other point on it [2].
• A standard IC slopes down from left to right and is convex to the origin [2].

• A higher IC (further from the origin) means more satisfaction.

• Her goal: reach the highest IC she can while staying inside her budget set.

2. The optimum lies on the budget line, not below it

• Monotonic preferences: she always likes more of at least one good, with no less of the other.
• Points below the line are wasteful.
• Below the line, some income is unspent.
• So a point on the line exists with more of one good and no less of the other.

• With monotonic preferences, that point is better. So a point below the line can never be the optimum.

• Points above the line cannot be bought. They cost more than M.

• So the optimum must lie on the budget line.

3. The optimum is at the tangency point (Fig 2.12)

• The optimum bundle (x₁*, x₂*) is where the budget line just touches (is tangent to) an IC.
• Any other point on the budget line lies on a lower IC.
• Any higher IC lies wholly outside the budget set, so she cannot reach it.
• Britannica says the same thing: the best bundle is found where the budget line is tangent to the consumer's ICs [2].
• This point is called consumer equilibrium (the point where she has no reason to change her choice).

4. The equilibrium condition

• Marginal rate of substitution (MRS): how many units of good 2 she is willing to give up to get one more unit of good 1 and stay equally happy. It is the (absolute) slope of the IC.
• Price ratio p₁/p₂: how many units of good 2 the market makes her give up for one more unit of good 1. It is the (absolute) slope of the budget line.
• Ordinal condition: MRS = p₁/p₂
• Slope of IC = slope of budget line.

• What she is willing to swap = what the market lets her swap.

• Link to utility: the slope of the IC equals the ratio of the marginal utilities of the two goods, MRS = MU₁/MU₂ [2].

• Cardinal version (utility measured in numbers): MUx/Px = MUy/Py
• Marginal utility (MU): the extra satisfaction from one more unit of a good.

• The last rupee spent on each good must give the same extra satisfaction.

• Worked example (cardinal):

• Px = ₹10, Py = ₹5. At her current bundle, MUx = 40 utils, MUy = 10 utils.
• MUx/Px = 40/10 = 4 utils per rupee. MUy/Py = 10/5 = 2 utils per rupee.
• A rupee spent on X gives more. So she moves spending from Y to X.
• As she buys more X, MUx falls (law of diminishing marginal utility). As she buys less Y, MUy rises.
• She stops when the two ratios are equal, e.g. MUx = 30, MUy = 15 → 3 = 3.

5. Why a point with MRS ≠ price ratio is not optimal

• Case MRS > price ratio (NCERT example):
• MRS = 2: she would give up 2 mangoes for 1 more banana.
• Both goods cost the same, so p₁/p₂ = 1: the market asks for only 1 mango.
• She buys the extra banana by giving up just 1 mango.
• She now has one mango more than the "2-mango" trade would have left her. That trade already kept her equally happy, so this bundle is better.

• She keeps doing this until MRS falls to 1.

• Case MRS < price ratio: the same reasoning works the other way.

• Example: MRS = 1, p₁/p₂ = 2. She values 1 banana at only 1 mango. But giving up 1 banana gets her 2 mangoes in the market.

• So she sells bananas for mangoes and becomes better off, until MRS rises to 2.

• Rule: any point where MRS ≠ p₁/p₂ allows a gainful swap. So it cannot be the optimum.

• Corner solution (footnote 9):
• Sometimes the best point is on an axis. She spends all her income on one good.
• At the corner, the tangency rule need not hold. Example: she likes good 2 so little that even the first unit is not worth its price.

6. Demand and the demand function

• Demand: the quantity a consumer is willing to buy and able to afford. It depends on:
• the price of the good,
• prices of other goods,
• her income,

• her tastes and preferences.

• Willingness alone is not demand. Wanting a car without the money to buy it is not demand.

• Demand function: X = f(P).
• It links her optimal quantity of a good to that good's own price.

• Everything else (other prices, income, tastes) is held constant (ceteris paribus, "other things being equal").

• NCERT box on functions: axis convention

• In maths, the independent variable (x) goes on the horizontal axis.
• Economics reverses this. Price (independent) goes on the vertical axis and quantity (dependent) on the horizontal axis [3].

• This habit comes from Alfred Marshall's diagrams. It is a common exam trap.

• Demand curve: the graph of the demand function.

• It shows the quantity demanded at each price, with other prices, income and preferences held constant.
• It generally slopes down from left to right. Price and quantity demanded move in opposite directions [3].

7. Deriving the demand curve from ICs (Fig 2.14)

• Step 1: starting price.
• Price of good 1 = P′₁. Price of good 2 = P′₂. Income = M.

• Equilibrium at C. She buys X′₁. Plot the point (X′₁, P′₁) on a price–quantity graph.

• Step 2: first price fall.

• Price falls to P̄₁. P′₂ and M stay fixed.
• The budget line swings outward around its good-2 intercept and becomes flatter. Slope = −p₁/p₂, and p₁ has fallen.

• The new tangency is at D, on a higher IC. She buys X̄₁ > X′₁. Plot (X̄₁, P̄₁).

• Step 3: second price fall.

• Price falls again to P̂₁. She buys X̂₁. Plot (X̂₁, P̂₁).

• Step 4: join the points.

• The result is a negatively sloped demand curve.

• Each point on the demand curve is a consumer-optimum point. So the demand curve is a list of best choices at different prices.

• The line joining the tangency points C, D and so on in the IC diagram is called the price-consumption curve. The demand curve is that same information redrawn in price–quantity space.

• Worked example:
• M = ₹120, p₂ = ₹10. Suppose her tastes make her always spend half her income on good 1, so x₁ = 60/p₁.

p₁	Optimal x₁	Point on demand curve
₹20	3	(3, 20)
₹12	5	(5, 12)
₹10	6	(6, 10)

• Price falls → quantity rises. The demand curve slopes down.

8. Individual demand and the demand schedule (Class 9, The Price Puzzle)

• Individual demand: the quantity one consumer wants to buy at different prices, other factors held constant.
• Demand schedule: a table showing quantities demanded at different prices.
• Srivalli's mangoes:

Price per kg	Srivalli's demand
₹150	1 kg (point A)
₹100	2 kg (point B)
₹50	3 kg (point C)

• Joining A, B and C gives the downward-sloping demand curve DD′.
• Every ₹50 fall in price raises her demand by 1 kg. So this schedule fits a straight-line demand curve (see §9).
• Needs vs wants (Class 7, Understanding Markets): needs are essentials such as food and clothing. Wants are extra things we would like. Either becomes demand only when it is backed by the ability to pay.
• Indian consumption data (context for "ability to pay"):
• Average MPCE (monthly per capita consumption expenditure, i.e. how much each person in a household spends on consumption in a month):
  • Rural: ₹1,430 (2011-12) → ₹3,773 (2022-23) → ₹4,122 (2023-24) [4].
  • Urban: ₹2,630 (2011-12) → ₹6,459 (2022-23) → ₹6,996 (2023-24) [4].
• Food share: about 47% of rural MPCE (2023-24). Non-food items make up about 53% (rural) and 60% (urban) of MPCE (2023-24) [5].
  • Food's share falls as incomes rise. This is Engel's law (the poorer the household, the larger the share of its budget spent on food).
• Gini coefficient of consumption (a measure of inequality: 0 = perfect equality, 1 = perfect inequality):
  • Rural: 0.266 (2022-23) → 0.237 (2023-24) [4].
  • Urban: 0.314 (2022-23) → 0.284 (2023-24) [4].
• Survey: Household Consumption Expenditure Survey (HCES), conducted by MoSPI/NSO [4][5].

9. Linear demand curve

• Formula:
• d(p) = a − bp for 0 ≤ p ≤ a/b
• d(p) = 0 for p > a/b

• Here a > 0 and b > 0 are constants.

• Key points:

• At p = 0, demand = a (the most she would take if the good were free).
• At p = a/b, demand = 0 (the choke price, the price at which she stops buying).

• Demand falls by b units for each ₹1 rise in price.

• NCERT error (Class 12): be careful.

• NCERT calls a the "vertical intercept" and −b the "slope". This is wrong for a graph with price on the vertical axis. The correct reading is:

  • a = quantity (horizontal) intercept.
  • a/b = price (vertical) intercept.
  • Slope of the drawn curve = −1/b (Δp/Δq).
  • −b = Δq/Δp. It is the slope only if quantity were on the vertical axis.

• Worked example 1: d(p) = 10 − 2p.

• Quantity intercept = 10 (at p = 0).
• Price intercept = 10/2 = ₹5 (choke price).
• Slope of drawn curve = −1/2. Each ₹1 rise cuts demand by 2 units.

• At p = ₹3: d = 10 − 6 = 4 units. At p = ₹6: d = 0 (price is above a/b = 5).

• Worked example 2 (Srivalli's schedule):

• The data fit q = 4 − (1/50)p, so a = 4 and b = 1/50.
• Check: p = 150 → q = 4 − 3 = 1 ✔; p = 100 → 2 ✔; p = 50 → 3 ✔.
• Price intercept = a/b = 4 ÷ (1/50) = ₹200. At ₹200 she buys no mangoes.
• Slope of DD′ as drawn = −1/b = −50 (price must fall by ₹50 for 1 kg more).

Prelims Hooks

• Consumer equilibrium (ordinal approach): MRS = p₁/p₂. Slope of IC = slope of budget line, at the tangency point.
• Cardinal equilibrium: MUx/Px = MUy/Py. MRS = MU₁/MU₂ [2].
• With monotonic preferences, the optimum is never below the budget line. Points above it cannot be afforded.
• Corner solution: the optimum is on an axis and all income goes to one good. The tangency condition need not hold there.
• Demand = willingness plus ability to pay. Desire alone is not demand.
• Demand curve axes: price on the vertical axis, quantity on the horizontal axis. This is the reverse of the maths convention [3].
• A fall in p₁ (p₂ and M fixed) makes the budget line flatter. It swings outward around the good-2 intercept.
• Linear demand d = a − bp: quantity intercept = a, price intercept = a/b, slope of drawn curve = −1/b. Trap: NCERT's "vertical intercept a" is wrong.
• Individual demand schedule = a table. Demand curve = its graph. Both hold other factors constant.
• HCES 2023-24 (MoSPI): average MPCE is ₹4,122 rural and ₹6,996 urban. The consumption Gini fell to 0.237 rural and 0.284 urban [4].

Mains Points

• Demand theory underlies welfare policy.
• A price cut through a subsidy (for example free or cheap food grains) swings the budget line outward. The household reaches a higher IC.
• A cash transfer shifts the budget line out in parallel.

• Comparing the two tangency points shows why economists argue over in-kind vs cash transfers (DBT) for the same fiscal cost (GS-III: subsidies, PDS).

• The MRS = price ratio condition explains why distorted prices waste welfare.

• Taxes, subsidies or controls push the market price ratio away from true costs.

• Consumers then set their MRS equal to a wrong ratio. This gives a case for rationalising GST rates and cutting untargeted subsidies.

• Rising MPCE and a falling food share point to rising real incomes and changing demand.

• Rural non-food share is about 53% and urban about 60% (2023-24) [5].

• This matters for CPI weights, farm policy (diversification towards milk, vegetables and processed food), and demand-led manufacturing (GS-III: growth, inclusive development).

• Limits of the model.

• It assumes rational, fully informed consumers.
• Behavioural economics (habits, the pull of the present, framing) explains why "nudges" (e.g. Swachh Bharat, JAM-based savings) sometimes work better than price signals alone.