How To Derive A Demand Function from a CES Utility Function

Warning: Boring technical stuff that’s only here because I needed it for the model in the Helicopter Money paper, and there are no other good references online.

All the existing resources on the internet that I’ve found are either vague or inconsistent with their σ/ρ notation (where \(ρ=\frac{σ-1}{σ}\) and σ is the elasticity of substitution, which makes the exponents in the utility function look cleaner, though the exponents in the demand function will be correspondingly messier), or don’t derive it with the coefficients. Almost none go step by step (with the exception of this video) or carry through the coefficients with exponents, and literally none derive a function for more than two goods.

We’re going to do all of these: a fully general derivation of demand functions from an n-good CES utility function, carrying through the actual elasticity of substitution as a parameter. I’ll use sum notation throughout, which you can easily expand to a definite number of goods. It’s worth noting though that the elasticity of substitution has to be the same between all pairs of goods, otherwise there’s no fully general form.

We start by writing our CES function this way, raising our coefficient to the 1/σ and summing over a set of n goods. You might wonder why we’re raising our coefficient to an exponent too. It’ll make our demand function slightly cleaner in the end, and since it’s a parameter, you can just define α_n = β_n^1/σ and substitute that back in at the end.

(1) $$U=\left(\sum_n β_n^{1/σ}G_n^\frac{σ-1}{σ} \right)^\frac{σ}{σ-1} $$

A function of this form means that the elasticity of substitution between any pair of goods is σ. Our budget constraint, then, is

(2) $$I=\sum_nP_nG_n$$

We want to maximize (1) subject to (2). So we set up our lagrangian, and derive with respect to each good plus λ, which gives us n+1 first-order conditions.

(3) $$\mathcal{L} = \left(\sum_n β_n^{1/σ}G_n^\frac{σ-1}{σ} \right)^\frac{σ}{σ-1} + λ\left(I-\sum_nP_nG_n\right)$$

Since both parts of the Lagrangian are sums, and the parameters of the various goods are all siloed into their own terms, this is actually fairly straightforward to derive with respect to any G variable. So we’ll pick any two goods, say a and b, and derive with respect to G_a and G_b, which we can use to find the relative price of a with respect to b. We’ll solve for the demand function for G_a, so any additional goods c, d,… will come out with symmetrical relative price equations.

To get the derivative of the first part of the Lagrangian, remember the chain rule for deriving f(g(x)): \(\frac{∂ f}{∂ x} = \frac{∂ f}{∂ g}\frac{∂ g}{∂ x}\). Our f(g) will be \(g^\frac{σ}{σ-1}\), and our g(x) will be the sum inside the parentheses. Carrying down the \(\frac{σ}{σ-1}\) exponent from \(\frac{∂ f}{∂ g}\) cancels out the \(\frac{σ-1}{σ}\) exponent carried down from the G_n in \(\frac{∂ g}{∂ x}\). This gives us:

(4) $$\frac{∂\mathcal{L}}{∂G_a} = \left(\frac{β_a}{G_a}\right)^\frac{1}{σ} \left(\sum_n β_n^{1/σ}G_n^\frac{σ-1}{σ}\right)^\frac{1}{σ-1} – \lambda P_a = 0$$

(5) $$\frac{∂\mathcal{L}}{∂ G_b} = \left(\frac{β_b}{G_b}\right)^\frac{1}{σ} \left(\sum_n β_n^{1/σ}G_n^\frac{σ-1}{σ}\right)^\frac{1}{σ-1} – \lambda P_b = 0$$

(6) $$\frac{∂\mathcal{L}}{∂ \lambda} = I-\sum_nP_nG_n = 0$$

We have β_a/G_a in (4) because \(\frac{σ-1}{σ} – 1 = \frac{-1}{σ}\), which lets us put G_a in the denominator of the coefficient raised to the \(\frac{1}{σ}\). Similarly, the outer \(\frac{1}{σ-1}\) exponent comes from \(\frac{σ}{σ-1} – 1 = \frac{1}{σ-1}\). Moving the sum to the right-hand side of (6) gives us back the budget constraint, i.e. (2).

From here, we want the relative price of P_a to P_b (and, by extension, P_c, and whatever other hypothetical goods we have FOCs for). Moving the λP terms of (4) and (5) to the right-hand side, we can divide λP_a/λP_b to make the λs cancel out, as well as a good bit of (4) and (5).

(7) $$\frac{P_a}{P_b} = \frac{(β_a/G_a)^\frac{1}{σ}}{(β_b/G_b)^\frac{1}{σ}} = \left(\frac{β_a G_b}{β_b G_a}\right)^\frac{1}{σ}$$

The right hand side is the marginal rate of substitution between G_a and G_b. This equation simply tells us that the price ratio between these two goods has to equal their marginal rate of substitution, which is a basic result in microeconomics. This tells us we’re on the right track.

Solving for G_b, we have

(8) $$G_b = \frac{β_bG_a}{β_a}\left(\frac{P_a}{P_b}\right)^σ$$

And similarly for G_c, etc. What we’re aiming to do here is to replace all the G variables in (2) except G_a with expressions in terms of G_a, so we can then solve for G_a and get an equation defining it only in terms of the β parameters. This will be our demand curve.

So, substituting (8) and its brothers back into the budget constraint (2) gives us

(9) $$I=P_aG_a + \sum_{n\neq a}P_n\frac{β_nG_a}{β_a}\left(\frac{P_a}{P_n}\right)^σ$$

$$= G_a \sum_{n}\frac{β_n}{β_a}P_a^σ P_n^{1-σ}$$

We’ve substituted every G term of the sum, except the original G_a term, with an expression in terms of G_a. But notice that the expression in the sum evaluates to P_aG_a when n=a. We can therefore reincorporate the stray term and pull out G_a from the sum.

All that remains is to solve for G_a.

(10) $$ G_a = \frac{I}{\sum_{n}\frac{β_n}{β_a}P_a^σ P_n^{1-σ}} $$

$$= \frac{I P_a^{-σ}}{\sum_{n}\frac{β_n}{β_a}P_n^{1-σ}}$$

So there’s our demand function for G_a, and mutatis mutandis for any of the other goods. You can verify it works, because in conjunction with the formulas in the appendix to “Helicopters and the Neutrality of Money“, it lets you translate from a weight-in-the-utility-function (i.e. a value of β) to a specific quantity demanded (i.e. a value of G/P), in this case, real money balances.