On passing from the discrete to the continuous form of Slutsky’s identity in microeconomics

slutskyI 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. Elementary discussions usually employ a discrete version of Slutsky’s equation which assumes a relatively large price change. This is typically illustrated in a `pivot-shift’ type diagram which is familiar to all undergraduate microeconomics students. In more advanced (typically postgraduate) treatments, however, the discussion suddenly jumps to using a full blown partial differential equation form of Slutsky’s identity assuming an infinitesimal price change. The partial differential equation form is usually expressed in terms of a Hicksian demand function. I have not been able to find any appealing discussions of how the set-up that relates to the discrete form of Slutsky’s equation naturally evolves into the partial differential equation form as we pass from a large price change to the limit of an infinitesimally small price change. In this note I want to explore how the discrete form naturally evolves into the partial differential equation form when we make the price change infinitesimally small.

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

u(x, y) = x^{1/2}y^{1/2}

The mathematical problem is to find the combination of x and y which maximises this utility function subject to the budget constraint

p_1x + p_2y = m

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

L = \frac{1}{2}\ln x + \frac{1}{2}\ln y + \lambda (m - p_1x -p_2y)

Taking first-order partial derivatives with respect to x, y and \lambda and setting them equal to zero we get

\frac{\partial L}{\partial x}  = \frac{1}{2x} - \lambda p_1 = 0

\frac{\partial L}{\partial y} = \frac{1}{2y} - \lambda p_2 = 0

\frac{\partial L}{\partial \lambda} = m - p_1x - p_2y = 0

This is a system of three equations in three unknowns. Dividing the first equation by the second and rearranging one obtains

\frac{y}{x} = \frac{p_1}{p_2}

Solving the third equation for x we get

x = \frac{m}{p_1} - \frac{p_2}{p_1}y

and substituting this into the equation above we get

\frac{y}{\frac{m}{p_1} - \frac{p_2}{p_1}y} = \frac{p_1}{p_2}

\iff

y = \frac{m}{2p_2}

This is the uncompensated demand function (often also called the Marshallian demand function) for good y. By symmetry, the uncompensated demand function for good x is

x = \frac{m}{2p_1}

(Note that rearranging the demand function for y we get p_2y = \frac{m}{2} which says that the consumer will spend exactly half of the income on y, and similarly for x. Whenever the Cobb-Douglas utility function is in a form in which the exponents on the goods are fractions which sum to 1, these fractions tell us the proportions of income which will be spent on the corresponding goods. Our utility function was of the form u(x, y) = x^{1/2}y^{1/2} 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 x and suppose that m = \pounds 1000 while p_1 = \pounds 10. The amount of x demanded at this income and price is then

x(p_1, m) = \frac{1000}{(2)(10)}= 50

This corresponds to the amount of x in the bundle A in the diagram below.

slutskypricerise

Now suppose that the price rises to p_1^{*} = \pounds 20

The amount of x demanded at the original income and this new price falls to

x(p_1^{*}, m) = \frac{1000}{(2)(20)} = 25

This corresponds to the amount of x 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 x at the new price. This amount of additional income is obtained as

p_1^{*}x - p_1x = (20)(50) - (10)(50) = \pounds 500

Therefore, `compensating’ the consumer by increasing the income level from m = \pounds 1000 to m^{*} = \pounds 1500 enables them to buy their original bundle A with x = 50. 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 x the consumer will actually demand at the new price and new income level will be

x(p_1^{*}, m^{*}) = \frac{1500}{(2)(20)} = 37.5

This corresponds to the amount of x in the bundle B in the diagram above, and is usually referred to in the literature as the compensated demand for x (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 x when the price of x changes to p_1^{*} and at the same time the income level changes to m^{*} to keep the consumer’s purchasing power constant:

x(p_1^{*}, m^{*}) - x(p_1, m) = 37.5 - 50 = -12.5

This is the change in the amount of x 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 x from 50 at point A to 25 at point C. The other half of the drop in the demand for x 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 x when we change the income level from m^{*} back to m, holding the price of x fixed at p_1^{*}. Thus, the income effect is computed as

x(p_1^{*}, m) - x(p_1^{*}, m^{*}) = 25 - 37.5 = -12.5

The substitution effect plus the income effect together account for the full drop in the demand for x as a result of moving from bundle A to bundle C in response to the price rise of x.

In this simple numerical example Slutsky’s decomposition equation takes the discrete form

x(p_1^{*}, m) - x(p_1, m) = \big \{ x(p_1^{*}, m^{*}) - x(p_1, m) \big \} + \big\{ x(p_1^{*} , m) - x(p_1^{*}, m^{*}) \big \}

As we pass to the limit of an infinitesimally small price change, however, Slutsky’s decomposition equation takes the form of a partial differential equation which can be derived quite naturally from the discrete form by the following arguments. Suppose we start at point A in the above diagram again, and suppose the price changes by an infinitesimal amount \delta, i.e.,

p_1 \longrightarrow p_1 + \delta

The (infinitesimal) amount of income needed to compensate the consumer so that the bundle at A remains affordable after the price change is

(p_1 + \delta ) x - p_1 x = \delta x

We can then rewrite the discrete form of Slutsky’s equation as

x(p_1 + \delta, \ m) - x(p_1, \ m) =

\big \{ x(p_1 + \delta, \ m + \delta x) - x(p_1, \ m) \big \} + \big \{x(p_1 + \delta, \ m) - x(p_1 + \delta, \ m + \delta x) \big \}

\iff

x(p_1 + \delta, \ m) - x(p_1, \ m) =

\big \{ x(p_1 + \delta, \ m + \delta x) - x(p_1, \ m) \big \} - \big \{x(p_1 + \delta, \ m + \delta x) - x(p_1 + \delta, \ m) \big \}

Dividing through by \delta and taking the limit as \delta \rightarrow 0 we get

\frac{\partial x}{\partial p_1} = \lim_{\delta \to 0} \frac{x(p_1 + \delta, \ m + \delta x) - x(p_1, \ m)}{\delta} - \lim_{\delta \to 0} \frac{x(p_1 + \delta, \ m + \delta x) - x(p_1 + \delta, \ m)}{\delta}

The second term on the right-hand side, the income effect, can be written as

\lim_{\delta \to 0} \frac{x(p_1 + \delta, \ m + \delta x) - x(p_1 + \delta, \ m)}{\delta} = x \lim_{\delta x \to 0} \frac{x(p_1 + \delta, \ m + \delta x) - x(p_1 + \delta, \ m)}{\delta x} = x \frac{\partial x}{\partial m}

Our partial differential equation form of Slutsky’s identity has then so far evolved to

\frac{\partial x}{\partial p_1} = \lim_{\delta \to 0} \frac{x(p_1 + \delta, \ m + \delta x) - x(p_1, \ m)}{\delta} - x \frac{\partial x}{\partial m}

Now consider the first term on the right-hand side, namely

\lim_{\delta \to 0} \frac{x(p_1 + \delta, \ m + \delta x) - x(p_1, \ m)}{\delta}

This is the effect of the (infinitesimal) price change when we eliminate the income effect by changing the income to m + \delta x, i.e., it is the pure price substitution effect. What we are going to do is replace the two terms in the numerator by corresponding Hicksian demand functions, usually denoted by the letter h. Consider first the original demand function x(p_1, \ m) at point A in the diagram above. This was obtained by maximising utility subject to the price p_1 and income level m. However, we can also regard it as having been obtained by solving the dual problem of minimising the expenditure required to achieve the level of utility u(x^A, y^A) associated with the bundle A in the diagram above, given the price p_1. We would find that the minimised expenditure level would be

E(p_1, \ u(x^A, y^A)) = m

and the amount of x demanded would be the same as the original amount:

x(p_1, \ m) = h(p_1, \ E(p_1, \ u(x^A, y^A)))

Now consider the problem of minimising the expenditure required to achieve the level of utility u(x^A, y^A) given the changed price p_1 + \delta. The minimised expenditure level would be

E(p_1 + \delta, \ u(x^A, y^A))

Now, it is clearly the case that this minimised expenditure is `sandwiched’ between m and m + \delta x, i.e.,

m < E(p_1 + \delta, \ u(x^A, y^A)) < m + \delta x

so the difference between the three quantities in the inequality must become vanishingly small as \delta \rightarrow 0. Therefore as long as we are taking limits as \delta \rightarrow 0, we can replace x(p_1 + \delta, \ m + \delta x) with the Hicksian demand function

h(p_1 + \delta, \ E(p_1 + \delta, \ u(x^A, y^A)))

(What I am basically saying here is that as \delta \rightarrow 0, we can replace the point B in the diagram above with a point that is on the same indifference curve as the original bundle at point A. In this case the consumer would be compensated not by returning them to the original purchasing power after the price change, but to the original level of utility they were experiencing. As \delta becomes infinitesimally small, it makes no difference which of these two points we think about for the pure substitution effect. When the price change is large, these two points diverge and we then have a distinction between a Slutsky substitution effect which involves restoring the original purchasing power, and a Hicksian substitution effect which involves restoring the original utility level. The distinction between these two disappears when the price change is infinitesimal). We then have

\lim_{\delta \to 0} \frac{x(p_1 + \delta, \ m + \delta x) - x(p_1, \ m)}{\delta} = \lim_{\delta \to 0} \frac{h(p_1 + \delta, \ E(p_1 + \delta, \ u(x^A, y^A))) - h(p_1, E(p_1, \ u(x^A, y^A)))}{\delta}

=

\frac{\partial h}{\partial p_1}

We now have the final partial differential equation form of Slutsky’s decomposition equation in the way it is usually written using the Hicksian demand function:

\frac{\partial x}{\partial p_1} = \frac{\partial h}{\partial p_1} - x \frac{\partial x}{\partial m}

As before, this says that the total effect of a price change is composed of a pure substitution effect (with income adjusted to exactly compensate the consumer for the wealth effect of the price change) and an income effect.

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 the demand function with the above Cobb-Douglas preferences would be

x = \frac{m}{2p_1}

we have

\frac{\partial x}{\partial p_1} = - \frac{m}{2(p_1)^2}

By solving a simple expenditure minimisation problem it can easily be shown that

h = \frac{(p_2)^{1/2}}{(p_1)^{1/2}} u

(The expenditure minimisation problem would be to minimise E = p_1 x + p_2 y subject to x^{1/2} y^{1/2} = u. Solving the constraint equation for y and substituting into the objective function gives an unconstrained minimisation problem in the variable x only. Solving this yields the above expression for the Hicksian demand function h).

Therefore we also have

\frac{\partial h}{\partial p_1} = -\frac{1}{2}\frac{(p_2)^{1/2}}{(p_1)^{3/2}} u

= -\frac{1}{2}\frac{(p_2)^{1/2}}{(p_1)^{3/2}}\big(\frac{m}{2p_1}\big)^{1/2}\big(\frac{m}{2p_2}\big)^{1/2}

= -\frac{m}{4(p_1)^2}

and

-\frac{\partial x}{\partial m}\cdot x = -\frac{1}{2p_1}\frac{m}{2p_1} = -\frac{m}{4(p_1)^2}

Putting these into the partial differential form of Slutsky’s equation we see that the equation is satisfied.