Help with Income Elasticity Exercise in Becker's Economic Theory

Question

On page 18 of his book Economic Theory, Gary Becker provides the reader with the following exercise (no answers given):

Exercise Statement:

Write the formula $\sum_j K_jN_j = 1$ where $N_j$ is the market income elasticity of demand for the $j$th good and $K_j$ is the fraction of total market income spent on $j$, in terms of the $\eta_{ij}$ and $k_{ij}$, where these are the income elasticities and shares of the $i$th person for the $j$th good. First derive the $N_j$ in terms of the $\eta_{ij}$. Is $N_j$ a simple average of the $\eta_{ij}$, or do higher income persons have a greater weight than lower income persons?

My Question:

What does $\sum_j K_jN_j = 1$ represent?

My first intuition was I understand why the sum of all $K_j$ shares would add up to a whole. The sum of all shares equals the total income which could be 1. So $K_j$ would then just be a percentage of the whole. But what then would $K_jN_j$ represent? This is where I got stuck.

EDIT:

Okay here's my first stab at the problem:

We know $\eta_{ij}$ represents the income elasticity of individual $i$ with respect to the $j$th good. So by definition, the income elasticity is defined as \begin{equation} \eta = \frac{\partial Q}{\partial I} \frac{I}{Q} \end{equation}

We also know the fraction of total market income individual $i$ spends on good $j$ is $k_{ij}$. I reason that this should be just the amount spent on the good divided by the total income of individual. So it should be \begin{equation} k_{ij} = \frac{p_jq_j}{I} \end{equation}

I still don't see how this should come out to $1$ though. Unless we are thinking that elasticities cancel out, such that for every $\eta_i$, $\exists$ $\eta_j = \frac{1}{\eta_i}$ such that $\eta_i\eta_j=1$.

Answer 1

Just to show that indeed, $\sum_j K_jN_j = 1$, we have the following:

$K_j$ is the fraction of total expenditure directed to good $j$. Total expenditure is also market income (denote it by $I_m = p_1Q_1+...+p_nQ_n$). so

$$K_j = \frac {p_jQ_j}{I_m}$$

$N_j$ is the market income elasticity of demand for good $j$. So $$N_j = \frac {\partial Q_j}{\partial I_m}\cdot \frac {I_m}{Q_j}$$

Therefore

$$K_jN_j = \frac {p_jQ_j}{I_m}\cdot \frac {\partial Q_j}{\partial I_m}\cdot \frac {I_m}{Q_j}$$

Simplify,

$$K_jN_j = p_j\frac {\partial Q_j}{\partial I_m}$$

Silently, we assume that prices are not affected. So we can insert the price into the partial derivative

$$K_jN_j = \frac {\partial (p_jQ_j)}{\partial I_m}$$

Then

$$\sum_j K_jN_j = \sum_j\frac {\partial (p_jQ_j)}{\partial I_m} = \frac {\partial }{\partial I_m}(p_1Q_1+...+p_nQ_n) = \frac {\partial I_m}{\partial I_m} = 1$$

By weighting the income demand elasticity for each good by the expenditure-weight it has on the whole economy, we essentially arrive at a tautology, saying "if total market expenditure increases by 100 Euros, then ... total market expenditure will increase by 100 euros". What Becker wants is to decompose this tautology into something a bit more insightful, as regards differences in individual incomes.

Answer 2

The precise question the OP asked was what does $K_j N_j$ represent. As Alecos' response says, the statement $\sum_j K_j N_j = 1$, i.e. the weighted average of the income elasticities equals one, means that if total income goes up by one percent then total expenditures/consumption also increases by one percent. The total market expenditure for some goods (with income elasticity of demand > 1) will go up more than one percent, for some goods (with income elasticity of demand between 0 and 1) will go up but less than one percent, and for some goods (with income elasticity <0) will go down. But taking all goods together, expenditures will go up one percent when total income goes up one percent. (Becker makes this point in the context of an individual rather than the complete market on p. 16 and footnote 2 of the text.)

Now I provide the solution to the exercise statement:

Start with the notion that when individual $i$'s income $y_i$ increases by one percent, his demand for good $j$ increases by $\eta_{ij}$ percent. What effect does this have on total market demand for good $j$? It depends on what fraction $w_{ij}$ of the total demand for good $j$ individual $i$ accounts for. Total market elasticity for good $j$, $N_j$ is the weighted sum of the increases in demand of each individual: $N_j = \sum_i w_{ij} \eta_{ij}$

Algebra can get this in terms of $k_{ij}$ and $\eta_{ij}$ (and $y_i$) as the exercise statement requests.

First find an expression for $w_{ij}$, the fraction of total demand (or expenditure) of person $i$ for good $j$: Letting $q_{ij}$ denote $i$'s demand for $j$, this fraction is $$w_{ij}=\frac{q_{ij}}{\sum_i q_{ij}} = \frac{k_{ij} y_i}{\sum_I k_{Ij} y_I}$$ where the second equality comes from the definition $k_{ij} \equiv \frac{p_j q_{ij}}{y_i}$.

Then $N_j = \sum_i f_{ij} \eta_{ij} = \sum_i \frac{k_{ij} y_i}{\sum_I{k_{Ij} y_I}} \eta_{ij}$. The weight/fraction $w_{ij}$ is increasing in $y_i$: for a given $k_{ij}$ a higher income person's elasticity is weight more. (Actually, what matters for how heavily and individual's elasticity is weight is not just income but actually $k_{ij} y_i$, their expenditures on good $j$, relative to the total market expenditures on good $j$. This makes intuitive sense: the elasticity of a heavy consumer of a good will have a bigger effect on that good's market elasticity than that of a light consumer, regardless of income.)

The fraction of total market expenditures spent on good $j$ is $K_j = \frac{\sum_i k_{ij} y_i}{\sum_i y_i}$.

So with some simple algebra we have $K_j N_j = \frac{\sum_i k_{ij} y_i \eta_{ij}}{\sum_i{y_i}}$.

With more algebra and using the identity $\sum_j k_{ij}\eta_{ij} = 1$ (see p. 16) you can confirm that $\sum K_j N_j = \frac{\sum_i (y_i \sum_j k_{ij}\eta_{ij})} {\sum_i{y_i}} = 1$.