Hicksian demand, the expenditure function and duality

The problem of Lesson 5 is the primal: choose x to maximise u(x) subject to p·x ≤ m. Its dual swaps the objective and the constraint: choose x to minimise p·x subject to u(x) ≥ u.

• The cheapest bundle that reaches utility u is the Hicksian (or compensated) demand, h(p, u). It answers: what would she buy if she were paid exactly enough to reach u?
• What it costs is the expenditure function, e(p, u) = p·h(p, u).
• The utility she reaches at income m is the indirect utility function, v(p, m).

The two problems are tied together by one tangency. If the bundle x* solves the primal with income y and gives utility u*, then the bundle that minimises expenditure to reach u* is that same bundle, and the expenditure is y (JNU 2019). Anything cheaper reaches less utility, and anything that reaches more utility costs more.

Likewise the two demands are the same bundle at the starting point: h(p, v(p, m)) = x(p, m) and x(p, e(p, u)) = h(p, u). They differ only when something moves, which is the whole of Lesson 12.

In the left picture the curve and the line touch, whichever problem is given. Switch between “given income” and “given utility” and the bundle stays put when the numbers match. The right picture draws the expenditure function against the price of x for a fixed utility. The dotted tangent at the current price has a slope equal to the quantity of x she buys, a fact called Shephard’s lemma (next but one chapter).

e(p, u) is the minimum spending that reaches utility u (not the maximum: UGC NET and CUET both test this). Its properties:

• Homogeneous of degree one in prices. Double every price and the cheapest way to reach u costs exactly double. (Not degree zero; that is the property of demand.)
• Non-decreasing in prices, and strictly increasing in u.
• Concave in prices. It is a minimum of straight lines (for each fixed bundle, cost is linear in prices), and a minimum of linear functions is concave. When a price rises she can switch away from the good, so spending rises by less than if she could not.
• Continuous. It is not strictly convex, which is the wrong direction.

So CUET 2025 has (C) and (D) only; UGC NET 2023 has homogeneous of degree one, increasing in u and non-decreasing in P, concave in P, and continuous, but not strictly convex.

If she must spend more on x when its price rises, by how much? By about the number of units she buys. That is the result:

Three consequences, each one an exam item. Hicksian demand is homogeneous of degree zero in prices (the derivative of a degree-one function). The compensated demand curve slopes down (the second derivative of a concave function is negative). And the cross effects are symmetric: ∂h_x/∂p_y = ∂h_y/∂p_x, because both are the same second derivative of e.

For Cobb–Douglas u = x^ay^1−a, the whole set can be written out:

The indirect utility function v(p, m) is the maximum utility at prices p and income m. It is defined over prices and income (JNU 2013), not over quantities, and it assumes utility maximisation, not profit maximisation. It is homogeneous of degree zero in prices and income together: double all prices and income and the budget is the same, so the utility is the same. It is not homogeneous of degree one.

It gives demand back by Roy’s identity: a dearer x lowers utility by (the marginal utility of x per rupee) × (the quantity of x), and an extra rupee raises it by λ, so the ratio is the quantity.

Together, the four objects are four ways into the same consumer: start from tastes, or from x*, or from v, or from e, and the others follow. The whole map is: x* from the primal; v = u(x*); e by inverting v; h from the derivative of e.

Put the two demand curves for the same good on one diagram. They cross at the starting price, where income m and utility u match. Away from it they differ by the income effect. For a normal good, the Hicksian curve is steeper than the Marshallian one (UOH 2012): the Marshallian curve includes an income effect that reinforces the substitution effect. For an inferior good the Hicksian curve is the flatter one, and for a Giffen good the Marshallian curve slopes up while the Hicksian one slopes down. So no statement that one is “always” steeper can be right (JNU 2014).

Choose tastes and prices, and the Workbench computes the whole list at once: demand, indirect utility, expenditure, Hicksian demand, all three elasticities, both decompositions of a price change and the money measures. The table at the bottom checks the identities of this lesson against the numbers it has just found: Shephard, Roy, the Slutsky equation, homogeneity and adding-up. It also has a page of its own.