I have noticed that there is a sharp ‘jump’ in the literature concerning Slutsky’s decomposition equation for the effects of a price change on demand, from elementary discussions of a discrete version for relatively large price changes (which can be illustrated in a diagram), to the full blown partial differential equation form for infinitesimal price changes expressed in terms of a Hicksian demand function (in more advanced texts). I have not been able to find any appealing discussions of how the set-up that leads to the discrete form relates to the partial differential equation form as we pass to the limit of infinitesimal price changes. In this note I want to explore the link between these two versions.
Slutsky’s decomposition equation expresses the effects of a price change on Marshallian demand in terms of a pure substitution effect, which is always negative, and an income effect, which can be positive or negative depending on whether the good is a normal good or an inferior good respectively. I will employ a simple two-good numerical example to illustrate the discrete form of Slutsky’s equation, using a Cobb-Douglas utility function of the form
The mathematical problem is to find the combination of and which maximises this utility function subject to the budget constraint
The Cobb-Douglas utility function is globally concave and smooth so we are guaranteed to find a unique interior solution by partial differentiation. One normally proceeds by taking the natural logarithm of the utility function (this is a monotonic transformation so does not affect the preferences represented by the original utility function) and setting up the Lagrangian for the problem, namely
Taking first-order partial derivatives with respect to , and and setting them equal to zero we get
This is a system of three equations in three unknowns. Dividing the first equation by the second and rearranging one obtains
Solving the third equation for we get
and substituting this into the equation above we get
This is the uncompensated demand function (often also called the Marshallian demand function) for good . By symmetry, the uncompensated demand function for good is
(Note that rearranging the demand function for we get which says that the consumer will spend exactly half of the income on , and similarly for . Whenever the Cobb-Douglas utility function is in a form in which the exponents on the goods are fractions which sum to , these fractions tell us the proportions of income which will be spent on the corresponding goods. Our utility function was of the form so one-half of the total income is spent on each good, as we confirmed with the above calculation).
To illustrate Slutsky’s decomposition of the effects of a price change into a pure substitution effect and an income effect, consider the above uncompensated demand function for and suppose that while . The amount of demanded at this income and price is then
This corresponds to the amount of in the bundle A in the diagram below.
Now suppose that the price rises to
The amount of demanded at the original income and this new price falls to
This corresponds to the amount of in the bundle C in the diagram.
Slutsky’s decomposition of this total change in demand begins by asking what change in income would be enough to enable the consumer to buy the original amount of at the new price. This amount of additional income is obtained as
Therefore, `compensating’ the consumer by increasing the income level from to enables them to buy their original bundle A with . This increase in the income level corresponds to a shift outwards in the new budget line to a position represented by the blue budget line in the diagram.
In the sense that the original bundle A is affordable again (so purchasing power has remained constant), the consumer is now as well off as before, but the original bundle A is no longer the utility-maximising one at the new price and the higher income level. The consumer will want to adjust the bundle until the utility function is maximised at the new price and new income. The amount of the consumer will actually demand at the new price and new income level will be
This corresponds to the amount of in the bundle B in the diagram above, and is usually referred to in the literature as the compensated demand for (as opposed to the uncompensated demand at point A). The pure substitution effect of the price rise (i.e., abstracting from the income effect) is then the change in demand for when the price of changes to and at the same time the income level changes to to keep the consumer’s purchasing power constant:
This is the change in the amount of represented by the shift from bundle A to bundle B in the diagram above.
In this numerical example, the pure substitution effect of the price rise accounts for exactly half of the total drop in the demand for from 50 at point A to 25 at point C. The other half of the drop in the demand for is accounted for by the income effect (sometimes called the `wealth’ effect) of the price rise, which is represented in the diagram above by a parallel shift inwards of the blue budget line to the position of the final budget line on which bundle C lies. This is the change in demand for when we change the income level from back to , holding the price of fixed at . Thus, the income effect is computed as
The substitution effect plus the income effect together account for the full drop in the demand for as a result of moving from bundle A to bundle C in response to the price rise of .
In this simple numerical example Slutsky’s decomposition equation takes the discrete form
As we pass to the limit of an infinitesimally small price change, Slutsky’s decomposition equation takes the form of a partial differential equation which can be derived for the simple numerical example above (involving the Cobb-Douglas utility function) as follows. Let and be the amounts of and consumed at point A in the diagram above, and let
be the compensated demand for at point B in the diagram above, i.e., the demand for when the consumer is compensated for the rise in the price of by a rise in income enabling the purchase of the original consumption bundle at A. Partially differentiating both sides of this identity with respect to we get
Rearranging this gives the partial differential equation form of Slutsky’s decomposition equation:
As before, this says that the total effect of a price change is composed of a substitution effect (with income adjusted to maintain the initial purchasing power) and an income effect.
Note that the compensated demand for good at point B, namely , can be obtained as the solution to the problem of minimising the expenditure required to achieve the level of utility associated with bundle A in the diagram given the prices and . Explicitly, we seek to minimise
subject to the condition
From the constraint we have
Substituting this into the objective function, the problem reduces to that of minimising
Taking the first derivative and setting it equal to zero we get
This is the required compensated demand function.
Demand functions obtained in this way are called Hicksian demand functions and are usually denoted by the letter . Thus, we have
and we can write Slutsky’s decomposition equation in the more usual form encountered in the literature (using the Hicksian demand function) as
To check that this partial differential equation works in the context of our Cobb-Douglas example above, we can compute the partial derivatives explicitly. Since
We also have
Putting these into the partial differential form of Slutsky’s equation we see that the equation is satisfied.