1. Demand Theory

1.1 Utility maximization

There are \(N\) goods in the world and a market; the consumer faces a price vector and chooses the bundle that maximizes utility subject to the budget:

• Consumption bundle \(X=(X_1,X_2,\dots,X_N)^T\): \(X_i\) is the quantity of good \(i\);
• Price vector \(P=(P_1,P_2,\dots,P_N)^T\): \(P_i\) is the price of good \(i\);
• Utility function \(u:X\to\mathbb{R}\);
• Budget constraint \(P^TX=\sum_{i=1}^N X_iP_i\le M\), where \(M\) is total dispensable income (footnote: \(X_i\) and \(M\) must be measured in the same unit over the same length of time, e.g. both unit/month or USD/week).

Equivalently, use the Lagrangian:

Maximize \(\mathcal{L}\) over every choice variable and minimize over the multiplier \(\lambda\); the first-order conditions (f.o.c.) are

There are three interpretations of the f.o.c. above.

1.1.1 Interpretation 1: relative cost and value

Rearranging (1.2):

The LHS is the value of good \(i\) measured in units of good \(j\); the RHS is the cost of good \(i\) in units of good \(j\). Everyone consumes to a point where they agree with the same relative market price between any two goods. The market pins down the relative exchange price; people value goods differently but allocate different proportions of resources so that all agree on the same relative value of each pair on the margin. Intuitively, someone who loves a good buys tons of it until it no longer seems so attractive.

1.1.2 Interpretation 2: marginal value and marginal cost

Take the total differential of the utility function:

This is important: the RHS of (1.4) is the observable change in total expenditure, which evaluates the unobservable utility change measured in dollars on the LHS (footnote: \(\lambda\) is the utility of one dollar, the common scale converting utility into dollars; \(\tfrac{\partial U}{\partial X_i}/P_i\) can thus be aggregated across people). The same interpretation applies over time: the change in total expenditure over time represents the change in utility over time measured in dollars.

1.1.3 Interpretation 3: efficient allocation

Rearranging (1.2) again:

The marginal utility per dollar must be equal across goods — otherwise reallocation could improve. So \(\lambda\) means "utility per dollar" for the specific person (not tied to any good).

1.2 Different types of demand functions

1.2.1 The Marshallian demand

Solving all f.o.c. in the utility maximization problem gives the optimal demand for each good:

It is a function of the price vector \(P\) and dispensable income \(M\), called the Marshallian demand function. Crucially, Marshallian demand holds income constant while maximizing utility. Around it we further define:

• Price elasticity: the elasticity of good \(i\)'s demand with respect to good \(j\)'s price

i.e. the percentage change in demand for good \(i\) per 1% rise in \(P_j\). When \(i\ne j\), \(\varepsilon_{ij}^M\) is the cross-price elasticity; when \(i=j\), \(\varepsilon_{ii}^M\) is the own-price elasticity. If \(i,j\) are substitutes then \(\varepsilon_{ij}>0\); complements then \(\varepsilon_{ij}<0\).

• Income elasticity: percentage change in demand for good \(i\) per 1% rise in income \(M\)

• Normal vs inferior good: \(\eta_i>0\) normal, \(\eta_i<0\) inferior. A good can be normal at some income and inferior at another (e.g. rice: poor people eat more rice as they get richer, but rich enough to access more nutritious food, rice consumption falls).
• Within normal goods: \(\eta_i\in(0,1)\) is a necessity, \(\eta_i>1\) is a superior good. Intuition: a necessity we consume anyway — even poor we don't go too low, even richer we don't need much more; a superior good gets increasing spending as income rises. This yields the identity \(\sum_{i=1}^N\eta_i s_i=1\) (\(s_i\) = budget share): the average income elasticity across all goods is always one — a 1% income rise is spent somewhere.

1.2.2 The Hicksian demand

Consider the expenditure-minimization problem, the dual of utility maximization:

Form the Lagrangian \(\mathcal{L}=\sum_{i=1}^N X_iP_i+\mu[\bar U-U(X_1,\dots,X_N)]\); the f.o.c. are

This gives \(X_i^H(P_1,P_2,\dots,P_N,\bar U)\), the Hicksian demand function for good \(i\). If \(\bar U\) is set to the maximized utility attainable with income \(M\), then Marshallian demand exactly equals Hicksian demand.

• Price elasticity: Hicksian price elasticity \(\varepsilon_{ij}^H\equiv\tfrac{\partial X_i^H}{\partial P_j}\tfrac{P_j}{X_i}\). Marshallian elasticity is obtained under constant expenditure, Hicksian under constant utility. So the Hicksian own-elasticity measures only the substitution effect (income compensated away to hold utility fixed), always non-positive:

The Marshallian own-elasticity measures the combination of substitution and income effects, more negative (both work the same direction). But for substitutes the Hicksian cross-elasticity is always non-negative (pure substitution), while the Marshallian cross sign is ambiguous (substitution and income work against each other).

• Cost function: the minimum expenditure to attain utility \(\bar U\) under prices \(P_1,\dots,P_N\):

The cost function satisfies:

1. \(C\) is concave in each component of \(P\);
2. \(C\) is non-decreasing in prices;
3. \(C\) is non-decreasing in utility \(\bar U\);
4. \(C\) is homogeneous of degree 1 in prices;
5. \(C\)'s derivative with respect to \(P_i\) is the Hicksian demand for good \(i\) (Shephard's lemma):

Figure 1.1 (concavity of the cost function, paraphrased): the horizontal axis is good \(i\)'s price \(P_i\), the vertical axis is Cost. The point \((P_i^0,C_0)\) is the minimized cost at \(i\)'s original price. If we change \(P_i\) (everything else fixed) and do not allow the consumer to adjust the bundle, cost moves along a straight line through that point (footnote: with quantities fixed, utility \(\bar U\) is still attained and cost moves linearly in \(P_i\)). But if adjustment is allowed, solving the minimization makes cost never increase. This holds at every price point, so the traced curve is the cost function, tangent to that straight line only at \((P_i^0,C_0)\) — hence the cost function is concave.

1.3 Law of demand

(1.7) gives the law of demand when only one price changes. Now the general case. Take two price vectors \((P_1^0,\dots,P_N^0)\) and \((P_1^1,\dots,P_N^1)\) with corresponding cost-minimizing bundles \((X_1^0,\dots,X_N^0)\) and \((X_1^1,\dots,X_N^1)\) yielding the same utility \(U(X^0)=U(X^1)=\bar U\). Since \(X^0\) minimizes cost under \(P^0\) and \(X^1\) under \(P^1\):

Add the two inequalities:

This generalized law of demand doesn't require every good to obey the law of demand — only the aggregate inequality holds. Here all prices change together, while in (1.7) only one price changes, so they don't contradict.

1.4 Useful properties of Marshallian demand and Hicksian demand

1.4.1 Slutsky equation: the relationship between two types of demand

Let the budget \(M\) equal the minimum expenditure for \(\bar U\), i.e. \(M=C(P_1,\dots,P_N,\bar U)\). Then

i.e. the two demand functions generate the same demand. This identity is the Slutsky correspondence. Differentiate both sides with respect to \(P_j\):

Use \(\tfrac{\partial C}{\partial P_j}=X_j^H=X_j\) at the point \(M=C(P,\bar U)\) to rearrange:

This is the Slutsky equation. In particular, \(i=j\) gives \(\varepsilon_i^M=\varepsilon_i^H-\eta_i s_i\). The Hicksian own-elasticity is pure substitution (no income effect in compensated demand), and the law of demand holds for Hicksian: \(\tfrac{\partial X_i^H}{\partial P_i}\le 0\Leftrightarrow\varepsilon_i^H\le 0\). By the Slutsky equation, a normal good \(\eta_i>0\) has

so the law of demand holds for Marshallian demand too, and since income and substitution effects reinforce, the Marshallian curve has a steeper slope. But if the income effect is large enough (\(\eta_i\) negative enough and budget share \(s_i\) not too small), it can dominate substitution, giving \(\varepsilon_i^M>0\) — a Giffen good. Giffen \(\subset\) inferior, and hard to find since high-share inferior goods are rare (most high-share goods are normal). The Slutsky equation also reveals the slope gap: for normal goods the income effect reinforces substitution, so \(|\varepsilon_i^M|\) always exceeds \(|\varepsilon_i^H|\) — Marshallian demand is flatter than Hicksian; but since most goods have small shares, the gap is usually small.

1.4.2 Symmetry

By the envelope theorem \(\tfrac{\partial C}{\partial P_i}=X_i^H\) (i.e. (1.9)). Since \(C\) is continuous, the mixed second partials commute:

Rewrite in elasticity form:

For the Marshallian version, substitute \(\varepsilon_{ij}^H=\varepsilon_{ij}^M+\eta_i s_j\) via the Slutsky equation:

1.4.3 Homogeneity of degree 1 (h.o.d. 1)

The cost function is homogeneous of degree 1 in prices. Marshallian demand has a corresponding homogeneity property: if income and all prices are doubled together, the feasibility set is unchanged and the utility-maximizing choice is unchanged, so the following is an identity:

Since this is an identity, differentiate with respect to \(t\) and evaluate at \(t=1\):

Hicksian version: Hicksian demand is homogeneous in prices only (footnote: the same relative prices give the same expenditure-minimization solution), \(X_i^H(tP_1,\dots,tP_N,\bar U)=X_i^H(P_1,\dots,P_N,\bar U)\), so similarly

Or equivalently, use the Slutsky equation to convert the Marshallian result into the Hicksian one.

1.4.4 Adding-up

Marshallian demand has another identity, called adding-up:

Differentiate both sides with respect to \(P_j\):

For the Hicksian version, substitute via the Slutsky equation:

where the last step uses \(1-\sum_{i=1}^N\eta_i s_i=0\), which comes from differentiating adding-up with respect to \(M\):

1.4.5 Summary of the properties of two types of demand

We first used the Slutsky equation as a bridge between the two demand systems, deriving one's property from the other.

• Slutsky equation: cross-elasticity \(\varepsilon_{ij}^M=\varepsilon_{ij}^H-\eta_i s_j\); own-elasticity \(\varepsilon_i^M=\varepsilon_i^H-\eta_i s_i\).

Then for both systems we discussed three properties:

1. Symmetry: (a) Marshallian \(\varepsilon_{ij}^M s_i+\eta_i s_j s_i=\varepsilon_{ji}^M s_j+\eta_j s_i s_j\); (b) Hicksian \(\varepsilon_{ij}^H s_i=\varepsilon_{ji}^H s_j\).
2. Homogeneous of degree 1: (a) Marshallian \(\sum_{j=1}^N\varepsilon_{ij}^M+\eta_i=0\); (b) Hicksian \(\sum_{j=1}^N\varepsilon_{ij}^H=0\).
3. Adding-up: (a) Marshallian \(\sum_{i=1}^N\varepsilon_{ij}^M s_i+s_j=0\); (b) Hicksian \(\sum_{i=1}^N\varepsilon_{ij}^H s_i=0\); (c) derivative w.r.t. income \(\sum_{i=1}^N\eta_i s_i=1\).

To relate the three, first assume the cost function is continuous and twice differentiable to obtain symmetry — so symmetry cannot be implied by the other two, but adding-up + symmetry \(\Rightarrow\) h.o.d.1, and h.o.d.1 + symmetry \(\Rightarrow\) adding-up.

With these properties we can solve: in an economy with \(N\) goods, given the own-cross elasticity matrix, budget share vector, and income elasticity vector

There appear to be \(N^2+2N\) parameters to estimate, but only \(\left(\tfrac{N}{2}+2\right)(N-1)\) are needed: after estimating any \(N-1\) elements of the budget share vector and \(N-1\) of the income elasticity vector, adding-up gives the last element of each; symmetry gives the lower triangle from the upper; and h.o.d.1 or adding-up gives the last element in each row of the elasticity matrix.

1.5 Price indices

Why care about price change? Totally differentiate the budget-constraint identity:

The first term \(\delta P\) measures the expenditure change due purely to price change (quantities held constant). Since utility can be written \(\bar U=U(X_1,\dots,X_N)\), we regard \(\delta P\) as \(dC(P_1,\dots,P_N,\bar U)\), because by the envelope theorem

Note \(\delta P\) is only a first-order approximation of \(dC\), since the envelope theorem assumes infinitesimal price change does not let people adjust quantities (true theoretically, but real price changes aren't infinitesimal, so this overestimates the cost rise when prices rise and underestimates the cost fall when prices fall, footnote: because the cost function is concave in prices). The second term \(\delta X\) measures the expenditure change due to real consumption change, and its utility change. By the f.o.c. (1.4)

Divide (1.10) by \(\sum_{i=1}^N P_iX_i=M\) to get the percentage form:

We can also rewrite (1.10) per good:

Both decompositions attribute expenditure growth to two parts: real consumption growth and inflation. Expenditure and price are measurable but utility is hard to observe, so these decompositions back out the change in people's utility.

1.5.1 Paasche price index: use new bundle with old and new prices

Consider two bundles \((X_1^0,\dots,X_N^0)\), \((X_1^1,\dots,X_N^1)\) with \(M^0=\sum X_i^0P_i^0\), \(M^1=\sum X_i^1P_i^1\). As derived:

Define the Paasche price index (using the new bundle \((X_1^1,\dots,X_N^1)\) to measure price change):

Similarly the Paasche quantity index (using the new price vector):

1.5.2 Laspeyres price index: use old bundle with old and new prices

Similarly, use the old bundle \((X_1^0,\dots,X_N^0)\) to measure price change, defining the Laspeyres price index:

And the Laspeyres quantity index using the old price vector:

1.5.3 Fisher price index: a geometric average

In fact, both Paasche and Laspeyres mismeasure price change. Figure 1.2 (paraphrased): horizontal axis = quantity of good \(i\), vertical axis = price \(P_i\); when \(i\)'s price falls from \(P_i^0\) to \(P_i^1\) (other prices fixed), the Laspeyres price index corresponds to the area of rectangle \(abcd\), and the Paasche price index to rectangle \(bcfe\). Laspeyres assumes people cannot adjust any quantities, so it underestimates the cost cut; Paasche shifts the old bundle to a higher utility level (point \(e\)), so it overestimates the cost cut. To correct both errors, introduce the Fisher price index, the geometric average of Paasche and Laspeyres:

Similarly,

1.5.4 Multiple periods: Chain-Fisher price index

To measure a price index over multiple periods (not only between two), dynamically adjust the basket of goods, e.g. since 1994:

This is important: once a base year is chosen, it drifts further into the past as time flows. Consumption goods gradually change in nature (slow in the short run, dramatic in the long run), so we adjust the goods used yearly to avoid being trapped by unreasonable comparisons.

1.6 Aggregate Demand

Let \(X_{ih}^M(P_1,\dots,P_N,M_h)\) be household \(h\)'s (total income \(M_h\)) individual demand for good \(i\). Aggregate demand is

Sometimes the market-level consumption of a good is cared about by individuals when deciding (e.g. fashion goods), so \(X_i^{\text{Market}}\) enters individual utility and thus the demand function. Rewrite aggregate demand:

Differentiate with respect to \(P_j\):

If we assume \(0<\sum_{h=1}^H\tfrac{\partial X_{ih}^M}{\partial X_i^{\text{Market}}}<1\), then

i.e. the price effect is enlarged by the positive feedback mechanism.

1.6.1 An example: discrete choice of consumption

Consider two margins:

1. Extensive margin: each unit size is fixed; extension happens via increased number of units;
2. Intensive margin: the number of units is fixed; extension happens via enlarged size per unit.

We can break people's consumption choice into the extensive margin (more people buy) and the intensive margin (each person buys more).

In the discrete-choice model, people cannot change the quantity of purchase; they make a 0–1 choice (0 = not buying, 1 = buying). Recall the reservation price \(p\) is the willingness to pay: in the 0–1 model, if the price is at or below a person's reservation price, they buy; if above, they walk away. So the aggregate demand function can be written as

where \(N\) is the total number of people, and \(F\) is the fraction of people whose reservation price is at or below \(p\). We can assume different functional forms for \(F\) (e.g. probit, logit, etc.) and estimate their parameters to predict the aggregate demand curve.