Difficulty: Intermediate to Advanced | Prerequisites: Hicksian and Marshallian demand (Part 1 notes), Lagrangian optimisation
Once you can solve the UMP and EMP, the next question is: what happens to the optimum when parameters (prices, income, target utility) change? The envelope theorem answers this cleanly, without re-solving the whole problem. Meanwhile, duality formalises the mirror-image relationship between the two problems, and convexity of preferences governs whether the solutions you find are unique. These results tie together the entire consumer theory toolkit and are heavily tested.
The envelope theorem lets you differentiate value functions (indirect utility, expenditure function) with respect to parameters by treating the optimal choices as fixed. Shephard's lemma recovers Hicksian demands from the expenditure function; Roy's identity recovers Marshallian demands from indirect utility. Duality says the UMP and EMP are two sides of the same coin. Strict convexity of preferences guarantees unique demand.
Envelope theorem
A result that says: the derivative of the optimised value function with respect to a parameter equals the partial derivative of the Lagrangian with respect to that parameter, evaluated at the optimum. You do not need to account for how the optimal choices shift.
Think of it as a shortcut. Instead of re-solving the problem when a parameter changes, just differentiate the Lagrangian directly.
Shephard's lemma
The envelope theorem applied to the EMP. It says ∂e(p,U)/∂pᵢ = hᵢ(p,U). The derivative of the expenditure function with respect to a price gives the Hicksian demand for that good.
In simple terms, if the price of good i goes up by one tiny unit, minimum expenditure rises by exactly the amount of good i you are buying.
Roy's identity
The envelope theorem applied to the UMP. It says xᵢ*(p,m) = −(∂V/∂pᵢ)/(∂V/∂m). You recover Marshallian demand from the indirect utility function by taking the ratio of partial derivatives and flipping the sign.
Duality (in consumer theory)
The principle that the UMP and EMP are inverse problems. Solving one gives you enough information to solve the other. The key identities: e(p, V(p,m)) = m and V(p, e(p,U)) = U.
In simple terms, if you know the cheapest way to reach any utility level, you also know the best utility you can reach at any income, and vice versa.
Strictly convex preferences
Preferences where the "at least as good as" set for any bundle is strictly convex. Graphically, indifference curves are strictly bowed toward the origin with no flat segments. This guarantees a unique optimal bundle.
Convex preferences (non-strict)
Preferences where the "at least as good as" set is convex but may contain flat segments on indifference curves. Multiple bundles can be optimal, but the set of optimal bundles is itself convex.
General statement
The EMP value function is the expenditure function e(p, U). The Lagrangian is L = p·x + λ(U − u(x)).
By the envelope theorem, differentiating e with respect to pᵢ:
∂e/∂pᵢ = ∂L/∂pᵢ evaluated at the optimum = xᵢ = hᵢ(p, U)
This is Shephard's lemma: the derivative of minimum expenditure with respect to the iᵗʰ price is the Hicksian demand for good i.
Differentiating with respect to U
∂e/∂U = ∂L/∂U evaluated at the optimum = λ
The Lagrange multiplier in the EMP equals the marginal cost of utility, the rate at which minimum expenditure rises as the utility target increases.
General statement
The UMP value function is the indirect utility function V(p, m). The Lagrangian is L = u(x) + λ(m − p·x).
Differentiating V with respect to pᵢ:
∂V/∂pᵢ = ∂L/∂pᵢ at the optimum = −λxᵢ*
Differentiating V with respect to m:
∂V/∂m = ∂L/∂m at the optimum = λ
Dividing:
xᵢ* = −(∂V/∂pᵢ) / (∂V/∂m)
This is Roy's identity. It recovers the Marshallian demand from the indirect utility function without re-solving the optimisation problem.
Why Roy's identity has a negative sign
When a price rises, utility falls (∂V/∂pᵢ < 0), and the magnitude of that fall is proportional to how much of good i the consumer buys. Dividing by the marginal utility of income (∂V/∂m > 0) converts the utility loss into a quantity.
Expenditure function envelopes
The expenditure function for Cobb-Douglas is e(p,U) = U · (p₁/α)ᵅ · (p₂/(1−α))¹⁻ᵅ.
Shephard's lemma:
∂e/∂p₁ = αU · (p₁/α)ᵅ⁻¹ · (1/α) · (p₂/(1−α))¹⁻ᵅ = h₁(p,U)
∂e/∂p₂ = (1−α)U · (p₁/α)ᵅ · (p₂/(1−α))⁻ᵅ · (1/(1−α)) = h₂(p,U)
Differentiating with respect to U:
∂e/∂U = (p₁/α)ᵅ · (p₂/(1−α))¹⁻ᵅ = λ (the marginal cost of utility)
Indirect utility function envelopes
The indirect utility function is V(p,m) = (α/p₁)ᵅ · ((1−α)/p₂)¹⁻ᵅ · m.
Roy's identity check:
∂V/∂p₁ = −α · (α/p₁)ᵅ · ((1−α)/p₂)¹⁻ᵅ · m/p₁
∂V/∂m = (α/p₁)ᵅ · ((1−α)/p₂)¹⁻ᵅ
−(∂V/∂p₁)/(∂V/∂m) = αm/p₁ = x₁*, which matches the Marshallian demand
Are both goods normal?
Yes. x₁* = αm/p₁ and x₂* = (1−α)m/p₂ both increase linearly in m (since α ∈ (0,1) and prices are positive). This holds for any Cobb-Douglas with positive exponents summing to 1.
The two problems side by side
The UMP maximises utility on a budget set. The EMP minimises cost on an indifference curve. Graphically, both problems find the point where an indifference curve is tangent to a budget line, but they approach that point from different directions:
UMP: fix the budget line, find the highest indifference curve that touches it
EMP: fix the indifference curve, find the lowest budget line that touches it
The duality identities
e(p, V(p, m)) = m. Start with the UMP: find the best utility V you can reach with income m. Then ask: what is the minimum cost of reaching V? The answer is m itself, because you are already there.
V(p, e(p, U)) = U. Start with the EMP: find the minimum cost e of reaching U. Then ask: what is the best utility with income e? The answer is U, because you can just afford it.
From one demand to the other
h(p, U) = x*(p, e(p, U)). Plug the expenditure function into the Marshallian demand to get the Hicksian demand.
x*(p, m) = h(p, V(p, m)). Plug the indirect utility into the Hicksian demand to get the Marshallian demand.
Graphical intuition
Picture a standard indifference curve, budget line diagram. The tangency point is the same whether you approached it by maximising utility or minimising expenditure. The only difference is which object was fixed (the budget line or the indifference curve) and which was varied (the indifference curve or the budget line).
Strictly convex preferences and the UMP
If preferences are strictly convex (indifference curves have no flat segments), the optimal Marshallian demand at any given prices and income is unique. The intuition: a strictly convex indifference curve can touch a straight budget line at only one point. If it touched at two, the line segment between them would lie above the indifference curve (by strict convexity), meaning you could do better, contradicting optimality of both.
Convex (not strictly) preferences and the UMP
If preferences are merely convex, indifference curves can have flat segments. If a flat segment of the highest reachable indifference curve lies along the budget line, every point on that segment is optimal. So demand need not be unique. However, the set of optimal bundles is convex: any weighted average of two optimal bundles is also optimal (it lies on the same flat segment and satisfies the budget).
Strictly convex preferences and the EMP
The same logic applies. If preferences are strictly convex and strictly monotone, the Hicksian demand is unique. A strictly convex indifference curve touches the lowest budget line at exactly one point.
Nonconvex preferences and the EMP
With nonconvex preferences (indifference curves that bend outward), an indifference curve can touch a budget line at multiple points that are not connected by a convex set. The optimal solution may not be unique, and the set of optimal bundles need not be convex. This is a key distinction: convexity of the optimal set is lost when preferences are nonconvex.
Why this matters practically
Uniqueness of demand is what lets us write demand as a function x*(p, m) rather than a correspondence (a set-valued mapping). If demand is not unique, comparative statics become more complicated, and you lose the clean results that the rest of consumer theory relies on.
Shephard's lemma is the workhorse behind cost-of-living indices. When economists estimate how much a household's cost of living changes after a price shock, they differentiate the expenditure function with respect to prices, which gives the compensated demand. This avoids the bias that arises from using Marshallian demand, which conflates the price effect with the income effect.
Students often confuse Shephard's lemma with Roy's identity. Shephard's lemma applies to the expenditure function (EMP); Roy's identity applies to the indirect utility function (UMP). The two have different signs and different structures.
A common error is thinking duality means the UMP and EMP always give the same numerical answer. They give the same optimal bundle, but the value functions are different objects: V(p,m) is a utility level, e(p,U) is a dollar amount.
Students frequently assume that convex preferences guarantee a unique solution. They do not. Only strict convexity does. With merely convex preferences, you can have a flat segment of optimal bundles.
Some students think nonconvex preferences are exotic or rare. In fact, any preference that exhibits increasing marginal rates of substitution is nonconvex. This can arise with goods that complement each other very strongly.
⚠️ Be able to state and apply Shephard's lemma and Roy's identity from memory. These are standard exam items.
⚠️ Know the duality identities: e(p, V(p,m)) = m and V(p, e(p,U)) = U.
⚠️ Be prepared to explain why strict convexity gives uniqueness and mere convexity does not, with a graphical argument.
⚠️ Questions about nonconvex preferences often ask whether the optimal set is convex. The answer is no, and you should be able to draw a picture showing two tangency points that cannot be connected by a convex combination on the indifference curve.
True or False: Shephard's lemma says ∂V/∂pᵢ = hᵢ(p,U). (False. Shephard's lemma is about the expenditure function: ∂e/∂pᵢ = hᵢ.)
Fill in the blank: Roy's identity recovers ________ demand from the indirect utility function. (Marshallian)
True or False: If preferences are merely convex, the set of optimal demands is always a single point. (False. It can be a convex set of multiple points.)
Fill in the blank: e(p, V(p, m)) = ____. (m)
True or False: With nonconvex preferences, the set of optimal demands at a given price and income must be convex. (False.)
Q: State Shephard's lemma and explain its economic intuition.
A: Shephard's lemma says ∂e(p,U)/∂pᵢ = hᵢ(p,U). If the price of good i rises by a small amount, the minimum cost of reaching utility U rises by approximately hᵢ units of currency, because you are buying hᵢ units of good i and each unit just became slightly more expensive.
Q: Derive Roy's identity from the envelope theorem.
A: From the UMP Lagrangian L = u(x) + λ(m − p·x), the envelope theorem gives ∂V/∂pᵢ = −λxᵢ* and ∂V/∂m = λ. Dividing: xᵢ* = −(∂V/∂pᵢ)/(∂V/∂m).
Q: Why does strict convexity guarantee uniqueness of demand but convexity alone does not?
A: With strict convexity, the indifference curve is strictly bowed, so it can be tangent to a straight budget line at only one point. With mere convexity, the indifference curve can have a flat segment that lies along the budget line, making every point on that segment optimal.
Q: In the duality picture, what is "fixed" and what is "varied" in each problem?
A: In the UMP, the budget line is fixed and you search over indifference curves (varying utility). In the EMP, the indifference curve is fixed and you search over budget lines (varying expenditure). The tangency point is the same.
The envelope theorem results feed directly into the Slutsky equation (covered in Part 3 notes), which decomposes the total price effect into substitution and income effects. Shephard's lemma gives the substitution effect cleanly, which is why the Hicksian framework is essential for welfare economics. Convexity assumptions carry forward into producer theory (convex production sets) and general equilibrium.
Envelope theorem, Shephard's lemma, Roy's identity, duality, indirect utility function, expenditure function, value function, Lagrange multiplier, marginal utility of income, marginal cost of utility, convex preferences, strictly convex preferences, nonconvex preferences, uniqueness of demand, demand correspondence, Cobb-Douglas, ECN 312, intermediate microeconomics, consumer theory, welfare economics, compensated demand