Slutsky Equation (Theorem)

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A change in the price and income alters the consumer’s expenditure pattern. So to know the consumer’s expenditure pattern due to changes in price and income, all the variables should be varied simultaneously.

Let,

Utility function be U\; =\; f\; \left( q_{1},\; q_{2} \right)

Subject to y\; =\; p_{1}q_{1}\; +\; p_{2}q_{2}

Or y\; -\; p_{1}q_{1}\; +\; p_{2}q_{2}\; =\; 0

Here we can form a new function with the help of langrange multiplier i.e.,

V\; =\; f\left( q_{1},\; q_{2} \right)\; +\; \lambda \; \left( \; y\; -\; p_{1}q_{1}\; -\; p_{2}q_{2} \right)

First-order derivatives can be written as:

\frac{\partial V}{\partial q_{1}}\; =\; f_{1}\; -\; \lambda p_{1}\; =\; 0

\frac{\partial V}{\partial q_{2}}\; =\; f_{2}\; -\; \lambda p_{2}\; =\; 0

\frac{\partial V}{\partial \lambda }\; =\; y\; -\; p_{1}q_{1}\; -\; p_{2}q_{2}\; =\; 0

Now we can know the effect by total differentials of the above equations:

So,

f_{11}dq_{1}\; +\; f_{12}dq_{2}\; -\; p_{1}d\lambda\; -\; \lambda dp_{1}\; =\; 0

f_{21}dq_{1}\; +\; f_{22}dq_{2}\; -\; p_{2}d\lambda \; -\; \lambda dp_{2}\; =\; 0

dy\; -\; p_{1}dq_{1}\; -\; q_{1}dp_{1}\; -\; p_{2}dq_{2}\; -\; q_{2}dp_{2}\; =\; 0

Now it can also be rewritten as :

f_{11}dq_{1}\; +\; f_{12}dq_{2}\; -\; p_{1}d\lambda \; =\; \lambda dp_{1}

f_{21}dq_{1}\; +\; f_{22}dq_{2}\; -\; p_{2}d\lambda \; =\; \lambda dp_{2}

\; -\; p_{2}dq_{2}\; -\; q_{2}dp_{2}\; =\; -\; dy\; +\; p_{1}dq_{1}\; +\; q_{1}dp_{1}

Here the unknown are dq_{1}\; ,\; dq_{2}\; and\; d\lambda

\begin{pmatrix} f_{11} & f_{12} & -p_{1} \\ f_{21} & f_{22} & -p_{2} \\ -p_{1} & -p_{2} & 0 \\ \end{pmatrix} \begin{pmatrix} dq_{1} \\ dq_{2} \\ d \lambda \\ \end{pmatrix} = \begin{pmatrix} \lambda dp_{1} \\ \lambda dp_{2} \\ -\; dy\; +\; p_{1}dq_{1}\; +\; q_{1}dp_{1} \\ \end{pmatrix}

Now, the first matrix contains the same element as the bordered Hassian determinant. The solution can be done by the Cramers Rule. Their rule is as follows: if AB = C, then B = A-1C. But before applying the Cramers Rule, let us denote the determinant as D and the cofactor with their respective subscripts.

\begin{pmatrix} dq_{1} \\ dq_{2} \\ d \lambda \\ \end{pmatrix} = \begin{pmatrix} f_{11} & f_{12} & -p_{1} \\ f_{21} & f_{22} & -p_{2} \\ -p_{1} & -p_{2} & 0 \\ \end{pmatrix} ^{-1} \begin{pmatrix} \lambda dp_{1} \\ \lambda dp_{2} \\ -\; dy\; +\; p_{1}dq_{1}\; +\; q_{1}dp_{1} \\\end{pmatrix}

Also,

\begin{pmatrix} dq_{1} \\ dq_{2} \\ d \lambda \\ \end{pmatrix} = \frac{1}{D} \begin{pmatrix} D_{11} & D_{21} & D_{31} \\ D_{12} & D_{22} & D_{32} \\ D_{13} & D_{23} & D_{33} \\ \end{pmatrix} \begin{pmatrix} \lambda dp_{1} \\ \lambda dp_{2} \\ -\; dy\; +\; p_{1}dq_{1}\; +\; q_{1}dp_{1} \\ \end{pmatrix}

Solving the above equations for dq_{1}\; ,\; dq_{2}\; and\; d\lambda:

dq_{1}\; =\; \frac{D_{11}\lambda dp_{1}\; +\; D_{21}\lambda dp_{2}\; +\; D_{31}\left( -\; dy\; +\; q_{1}dp_{1}\; +\; q_{2}dp_{2} \right)}{D}

dq_{2}\; =\; \frac{D_{21}\lambda dp_{1}\; +\; D_{22}\lambda dp_{2}\; +\; D_{32}\left( -\; dy\; +\; q_{1}dp_{1}\; +\; q_{2}dp_{2} \right)}{D}

d\lambda\; =\; \frac{D_{13}\lambda dp_{1}\; +\; D_{23}\lambda dp_{2}\; +\; D_{33}\left( -\; dy\; +\; q_{1}dp_{1}\; +\; q_{2}dp_{2} \right)}{D}

Now, consider dq_{1}\; and divide both sides by dp_{1}\;

\frac{dq_{1}}{dp_{1}}\; =\; \frac{\lambda \; D_{11}}{D}\; +\; \frac{D_{31}q_{1}}{D} -------(A)

Income Effect 

The left-hand side of equation (A) shows the partial change in quantity q1 due to change in price. Similarly, we can calculate the effect of income.

Income effect corresponds to \frac{\partial q_{1}}{\partial y} which can now be obtained by dividing the equation by dy. Keeping price as constant.

So, \frac{dq_{1}}{dy} changes to \left( \frac{\partial q_{1}}{\partial y} \right)_{p-constant}

So, \left( \frac{\partial q_{1}}{\partial y} \right)_{p-constant} = (-)\frac{D_{31}q_{1}}{D}

Substitution Effect

Change in the commodity price may take consumer to some other indifference curve. Consider that the price change is compensated by the change in income, so that the consumer remains on the same indifference curve.

Increase in price is accompanied by increase in income so du\; =\; 0

As we know that f_{1}dq_{1}\; +\; f_{2}dq_{2}\; =\; 0

And \frac{f_{1}}{f_{2}}\; =\; \frac{p_{1}}{p_{2}}\;

So, p_{1}dq_{1}\; +\; p_{2}dq_{2}\; =\; 0 is true and also -\; dy\; +\; p_{1}dq_{1}\; +\; p_{2}dq_{2}\; =\; 0

So, (\frac {dq_1}{dp_1})_{U-Constant} = \frac{\lambda D_{11}}{D}

Now equation A can be rewritten as

\frac{\partial q_{1}}{\partial p_{1}}\; = \; \left( \frac{\partial q_{1}}{\partial p_{1}} \right)_{U-constant}\; – q_{1}\left( \frac{\partial q_{1}}{\partial y} \right)_{p-constant}

The above equation gives us the rate of change in the purchase of commodity due to change in price in terms of substitution and income effects.

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