AIDS Demand Systems and Budget-Share Equations

Why Model Budget Shares?

When consumers divide a fixed expenditure across goods, the resulting budget shares describe how that allocation responds to prices and total expenditure. A useful demand system should represent those responses while respecting basic economic requirements: the shares must add to one, demand must be homogeneous in prices and nominal expenditure, and compensated price responses must obey Slutsky symmetry. The Almost Ideal Demand System (AIDS) builds these properties into a flexible share equation through restrictions on its parameters.

The AIDS is a system for $K$ goods. For good $i$, let $p_i>0$ be its price, $q_i$ its quantity, and $x>0$ total expenditure on the goods in the system. Its budget share is $w_i=p_iq_i/x$, so the shares sum to one when the $K$ goods exhaust that expenditure. The AIDS relates each share to log prices and real expenditure, where “real” expenditure is measured relative to a price index.

Definition: The AIDS budget-share equation is $$ w_i=\alpha_i+\sum_{j=1}^{K}\gamma_{ij}\ln p_j+\beta_i\ln(x/P), \qquad i=1,\ldots,K, $$ where $P$ is a translog price index, $\alpha_i,\gamma_{ij},\beta_i$ are parameters, and $x/P$ is expenditure relative to that index. The price and expenditure units are held fixed when taking logarithms.

Deriving the Share Equation

The equation follows from an expenditure function with a particular form. An expenditure function gives the minimum expenditure needed to attain utility level $u$ at prices $p$. In the AIDS, write its logarithm as

\ln c(u,p)=A(p)+uB(p), \qquad A(p)=\alpha_0+\sum_{i=1}^{K}\alpha_i\ln p_i+ \frac{1}{2}\sum_{i=1}^{K}\sum_{j=1}^{K}\gamma_{ij}\ln p_i\ln p_j, \qquad B(p)=\beta_0\prod_{i=1}^{K}p_i^{\beta_i}.

Here $c(u,p)$ is minimum expenditure, $\alpha_0$ is an intercept in the price index, and $\beta_0$ scales the utility-dependent term. Define the translog price index by $\ln P=A(p)$. Shephard’s lemma states that, for a differentiable expenditure function, its derivative with respect to a good’s price is the Hicksian (utility-compensated) quantity of that good: $h_i=\partial c(u,p)/\partial p_i$. Therefore $p_i h_i/c=\partial\ln c/\partial\ln p_i$, the expenditure share at the chosen utility level.

Differentiate the specified expenditure function with respect to $\ln p_i$. If the price-index coefficients satisfy $\gamma_{ij}=\gamma_{ji}$, the derivative of its quadratic term is $\alpha_i+\sum_j\gamma_{ij}\ln p_j$. Also, $\partial B(p)/\partial\ln p_i=\beta_iB(p)$. Hence

\frac{\partial\ln c(u,p)}{\partial\ln p_i} =\alpha_i+\sum_{j=1}^{K}\gamma_{ij}\ln p_j+u\beta_iB(p).

At expenditure $x=c(u,p)$, the expenditure function gives $\ln(x/P)=uB(p)$. Substitution yields the AIDS share equation. This derivation also shows why the share equation is not an arbitrary collection of regressions: its coefficients are linked to a common expenditure function.

Worked Example: A Share at Equal Prices

Consider three goods with $\alpha=(0.40,0.35,0.25)$, $\beta=(0.10,-0.05,-0.05)$, and $\alpha_0=0.20$. Let all three prices equal one unit, and consider expenditure $x=2P$. The log prices are zero, so $\ln P=\alpha_0=0.20$, and $P=e^{0.20}\approx1.2214$. Thus $x\approx2.4428$ expenditure units and $\ln(x/P)=\ln2\approx0.6931$.

For good 1, the share equation gives $$ w_1=0.40+0.10(0.6931)=0.4693. $$ The corresponding Hicksian quantity at these prices and the utility level associated with $x$ is $h_1=xw_1/p_1$, so $$ h_1=\frac{2.4428(0.4693)}{1}\approx1.146 $$ units of good 1. The same share equation gives $w_2=0.35-0.05(0.6931)=0.3153$ and $w_3=0.25-0.05(0.6931)=0.2153$. Their sum is $0.4693+0.3153+0.2153=0.9999$, which is one up to rounding.

Restrictions Implied by Economic Consistency

The coefficients cannot be chosen independently if the share equations are to describe a coherent demand system. Three sets of restrictions are central. The first two follow from adding-up and homogeneity; the third expresses Slutsky symmetry.

Theorem: The standard AIDS restrictions are $$ \sum_{i=1}^{K}\alpha_i=1,\qquad \sum_{i=1}^{K}\beta_i=0,\qquad \sum_{i=1}^{K}\gamma_{ij}=0\ \text{for each }j, \qquad \sum_{j=1}^{K}\gamma_{ij}=0\ \text{for each }i, \qquad \gamma_{ij}=\gamma_{ji}. $$ The first three ensure adding-up of the shares. The zero row sums ensure homogeneity of the share equations, and symmetry of the $\gamma$ matrix imposes Slutsky symmetry.

Adding-up

If the modeled goods exhaust expenditure $x$, their shares must satisfy $\sum_i w_i=1$. Summing the AIDS equations gives $$ \sum_i w_i=\sum_i\alpha_i+\sum_j\left(\sum_i\gamma_{ij}\right)\ln p_j+ \left(\sum_i\beta_i\right)\ln(x/P). $$ For this equality to hold at all prices and expenditure levels, the constant must be one and the coefficients on each varying term must be zero. Thus $\sum_i\alpha_i=1$, $\sum_i\gamma_{ij}=0$ for every column $j$, and $\sum_i\beta_i=0$. This is an identity across the system, not a condition that must be checked only at one observed price vector.

Homogeneity

Demand is homogeneous of degree zero in prices and nominal expenditure: multiplying every price and $x$ by the same positive factor must leave quantities, and therefore shares, unchanged. In the AIDS equations this requires $\sum_j\gamma_{ij}=0$ for every good $i$, as well as $\sum_i\beta_i=0$. To see the price part, increase every log price by $t$. The price terms in share $i$ change by $t\sum_j\gamma_{ij}$, which is zero under the row-sum restriction.

The price index itself has the matching property when $\sum_i\alpha_i=1$ and the $\gamma$ row and column sums are zero: replacing every $p_i$ by $e^t p_i$ increases $\ln P$ by $t$. Scaling $x$ by the same factor therefore leaves $x/P$ unchanged. The zero sum of the $\beta_i$ also makes $B(p)$ invariant to a common price scaling in the expenditure function.

Slutsky symmetry

Slutsky symmetry requires the compensated effect of price $p_j$ on Hicksian quantity $h_i$ to equal the compensated effect of $p_i$ on $h_j$: $\partial h_i/\partial p_j=\partial h_j/\partial p_i$. In the expenditure-function representation, shares are derivatives of $\ln c(u,p)$ with respect to log prices. Their cross derivatives at fixed utility must therefore be symmetric. Specifically, differentiating the share equation along the expenditure function gives $$ \frac{\partial w_i}{\partial\ln p_j}\bigg|_u =\gamma_{ij}+\beta_i\beta_j uB(p). $$ The second term is symmetric in $i,j$. Thus symmetry of these cross derivatives requires $\gamma_{ij}=\gamma_{ji}$. This is the AIDS parameter restriction corresponding to Slutsky symmetry.

Worked Example: Checking a Three-Good Parameter System

Take $\alpha=(0.40,0.35,0.25)$, $\beta=(0.10,-0.05,-0.05)$, and the symmetric matrix $$ \Gamma= \begin{pmatrix} 0.08 & -0.03 & -0.05\\ -0.03 & 0.06 & -0.03\\ -0.05 & -0.03 & 0.08 \end{pmatrix}. $$ The $\alpha_i$ sum to $1.00$, and the $\beta_i$ sum to $0.10-0.05-0.05=0$. Each row of $\Gamma$ sums to zero; for example, the first is $0.08-0.03-0.05=0$. Each column also sums to zero, and the matrix is symmetric. The system therefore satisfies these adding-up, homogeneity, and symmetry restrictions.

Now change only $\gamma_{13}$ from $-0.05$ to $-0.04$. The first row sums to $0.08-0.03-0.04=0.01$, and the third column sums to $0.01$; also $\gamma_{13}\ne\gamma_{31}$. With the other restrictions unchanged, the sum of predicted shares becomes $1+0.01\ln p_3$, rather than one for arbitrary $p_3$. For example, if $p_3=2$, it is $1+0.01\ln2\approx1.0069$. The altered system violates adding-up and symmetry, and the nonzero first-row sum also violates homogeneity.

Worked Example: Computing Shares in a Restricted System

Use the valid three-good system above, set $\alpha_0=0.20$, and take prices $p=(2,1,4)$. The log-price vector is approximately $(0.6931,0,1.3863)$. The price index is calculated from the translog formula: $$ \ln P=0.20+0.40(0.6931)+0.25(1.3863)+\frac12(0.0961) \approx0.8719. $$ The quadratic contribution is $z'\Gamma z\approx0.0961$, where $z$ is the log-price vector. Hence $P\approx e^{0.8719}=2.392$. Choose $x=10P$, so $\ln(x/P)=\ln10\approx2.3026$ and $x\approx23.92$ expenditure units.

For good 1, the price contribution is $0.08(0.6931)-0.05(1.3863)=-0.0139$, giving $$ w_1=0.40-0.0139+0.10(2.3026)\approx0.6164. $$ For good 2, the price contribution is $-0.03(0.6931)-0.03(1.3863)=-0.0624$, so $w_2=0.35-0.0624-0.05(2.3026)\approx0.1725$. For good 3, it is $-0.05(0.6931)+0.08(1.3863)=0.0762$, so $w_3=0.25+0.0762-0.05(2.3026)\approx0.2111$. The computed shares sum to $0.6164+0.1725+0.2111=1.0000$, up to rounding. In this hypothetical allocation, approximately 61.64%, 17.25%, and 21.11% of expenditure goes to goods 1, 2, and 3, respectively.

Restrictions Are Not the Same as Regularity

Adding-up, homogeneity, and symmetry are structural restrictions on the AIDS parameters. They do not, by themselves, guarantee that the specified system is economically regular at every possible price and expenditure level. In particular, predicted shares should be consistent with nonnegative expenditure allocations in the relevant range, and the expenditure function should be concave in prices, corresponding to a negative semidefinite compensated substitution matrix. An arbitrary parameter estimate can satisfy the algebraic restrictions yet fail one of these regularity checks. Those checks must be evaluated for the prices and expenditure levels relevant to the application.

The AIDS is useful because its share equations are flexible in log prices and real expenditure while retaining transparent cross-equation restrictions. The restrictions matter for interpretation and prediction: without adding-up, predicted allocations need not exhaust the budget; without homogeneity, a change in currency units could change predicted shares; without symmetry, compensated price responses cannot be rationalized by the stated expenditure function.

Common Mistake

Wrong reasoning: “The predicted shares add to one for the observed prices, so the system satisfies adding-up.” A single numerical coincidence does not establish the required identity across prices and expenditure.
Correction: Check the coefficient restrictions directly: $\sum_i\alpha_i=1$, $\sum_i\beta_i=0$, and $\sum_i\gamma_{ij}=0$ for every $j$. Check the row sums and symmetry separately for homogeneity and Slutsky symmetry. Then assess regularity in the relevant range rather than assuming these restrictions guarantee it.

Key takeaway: The AIDS share equation comes from differentiating a common expenditure function. Its adding-up, homogeneity, and Slutsky-symmetry properties are encoded by distinct restrictions on the intercepts, expenditure coefficients, and price coefficients.

Check Your Understanding

Use the equations and restrictions above to answer these questions.

Why does $\sum_i\beta_i=0$ matter for adding-up? What role does it also play when prices and expenditure are scaled together?
For a three-good AIDS system, what restrictions on $\Gamma$ correspond to homogeneity, and what restriction corresponds to Slutsky symmetry?
If the shares sum to one at one observed price vector, does that establish adding-up for the system? Explain.
A proposed system has $\sum_i\alpha_i=1$ and symmetric $\Gamma$, but one column of $\Gamma$ does not sum to zero. Which property fails?
Why can an AIDS system satisfy the standard parameter restrictions but still fail an economic regularity check?