Demand curves, shifters and consumer surplus

The optimum of Lesson 5 depends on three things: the price of x, the price of y and income. Write it x*(p_x, p_y, m). A demand curve for x holds two of those three fixed (income and the price of y) and lets the third, the price of x, vary. Plot the quantity she chooses against the price, and the picture is a slice of the larger function.

Two small examples are worth having by heart. For u = xy the demand for x is m/(2p_x): half of income, divided by the price. Plotted, it is a rectangular hyperbola. For the quasi-linear u = √x + y it is (p_y/2p_x)², from the tangency of Lesson 6, and income does not appear at all.

Two words need keeping apart. Quantity demanded is one point on the curve. Demand is the whole curve (or the whole schedule of prices and quantities). A change in the price of the good changes the quantity demanded and moves her along the curve. Anything else that matters changes demand and shifts the curve.

Does the curve always slope down? Nothing in the axioms of Lesson 1 forces it to. For almost every good it does, and that is the law of demand. The exception, a Giffen good, is Lesson 12.

Choose tastes and slide the price of x. On the left is the choice problem of Lesson 5, with the dotted path showing where she would choose at other prices. On the right the same choice becomes a point on the demand curve. Then change income or the price of y and see what happens to the curve: it moves off the dashed baseline.

Try quasi-linear tastes and move income. The curve does not budge, because for these tastes the quantity of x depends only on the two prices. Then try the Cobb–Douglas tastes: raising income shifts the curve to the right at every price.

Since the curve is x*(p_x, p_y, m) at fixed p_y and m, the shifters are exactly the other arguments of the function, together with anything that changes the function itself:

Each row has been an exam question. A fall in income shifts the demand curve for a normal good to the left (CUET 2018). If chicken and fish are substitutes, a rise in the price of chicken shifts the demand curve for fish to the right (CUET 2018, and UOH 2013, which asks which event shifts a demand curve right and has a rise in the price of a substitute as the answer). The demand for JIF peanut butter falls if the price of bread, a complement, rises (CUET 2018). Which of these cannot lead to larger purchases at each price? Rising income for an inferior good, because it lowers the demand for such a good (CUET 2025).

The curve so far treats each consumer as choosing alone. Sometimes her demand depends on how many others buy. In a bandwagon effect she wants the good more because others have it (a popular phone, a fashionable school). In a snob effect she wants it less for the same reason. A Veblen good is bought because it is expensive.

A bandwagon makes demand more elastic. When the price falls, more people buy, and that makes the good more attractive to everyone else, so the quantity rises by more than it would have done with no such effect (CUET 2022). The snob effect works the other way: demand is less responsive to price. The measure of how responsive is the elasticity of Lesson 10.

Questions give a demand function and ask for its limits. Two values are worth finding at once, by setting one variable to zero.

• Set the price to zero to find the most she could ever buy. For Q = 24 − 3P that is Q = 24 (JNU 2020).
• Set the quantity to zero to find the price at which she stops buying, the choke price. For Q = 36 − 3P it is 36 ÷ 3 = 12 (JNU 2020).

Economists draw price on the vertical axis, so the “slope” of a curve written as Q = a − bP is −1/b, not −b. To plot it, join the point (Q = a, P = 0) to the point (Q = 0, P = a/b).

A buyer’s reservation price for a unit is the most she would pay for it (UOH 2021). If the price is below that, she gains by buying. The gain is the gap between the two, and it is her consumer surplus on that unit.

Trade needs the numbers to overlap. A seller will not take less than her own reservation price, and a buyer will not pay more than hers. If A will sell a book for no less than Rs 400 and B will pay no more than Rs 360, no price suits both and no trade takes place (UOH 2015).

With many units the reservation price falls as she buys more: the first unit is worth a lot, the tenth less. Marshall explained why through diminishing marginal utility, and that is the basis of consumer surplus in his analysis (UGC NET 2024). Add up the gaps on every unit she buys:

A small case: her reservation prices for four units are Rs 40, 32, 25 and 10, and the price is Rs 20. She buys the first three (the fourth is worth less than the price) and gains 20 + 12 + 5 = Rs 37.

Put price on the vertical axis. The height of the demand curve above a unit is her reservation price for it. So the area under the curve up to the quantity she buys is what the whole lot was worth to her. She pays the rectangle price × quantity. The difference is the surplus: the triangle-like region between the curve and the price line.

For a straight line the region is a triangle and the integral is just half the base times the height. Choose a curve below and slide the price. The blue region is the surplus; the gold rectangle is what she pays. Notice that the surplus shrinks as the price rises, and that for the last curve (q = 30/p²) the area stretches without end along the axis but still has a finite value.