# Deriving a demand curve from a Cobb-Douglas utility

Probably a daft question but I derived an equation for a demand curve from a general Cobb-Douglas utility function $$U(x,y)=\beta x^{\alpha}y^{1-\alpha}$$ given a budget constraint $$M=xP_x+yP_y$$ and found that the quantity of $$x$$ demanded would be $$x=\frac{\alpha M}{P_x}$$.

which surprised me in that this function is independent of the price of good $$y$$. I see some references on these pages which suggests that I didn't make any spectacular errors but, my elementary understanding is that the demand curve is a functions of many thing, not least the price of other related goods. Now surely goods $$x$$ and $$y$$ are substitutes to some extent?

Now, my question is this, is my understanding of this essentially correct and, what sort of utility functions would generate demand curves $$x=f(P_x, P_y, M, etc)$$?

Thank you

If you take the general class of CES utility functions, of which Cobb-Douglas is a special case, you do indeed get a demand function that depends on other prices. Specifically, the CES utility function (over $$n$$ goods, $$x_1,\dots,x_n$$) takes the form $$$$u(x_1,\dots,x_n)=\bigl[\alpha_1x_1^\rho+\cdots+\alpha_nx_n^\rho\bigr]^{1/\rho},$$$$ where $$\rho\in(-\infty,1]\setminus\{0\}$$, $$\alpha_i\in[0,1]$$ and $$\sum_i\alpha_i=1$$. We interpret $$\alpha_i$$ as the consumption share of good $$i$$ and $$\sigma\equiv\frac{1}{1-\rho}$$ as the constant elasticity of substitution. Note also that when $$\sigma=1$$ (or $$\rho\to0$$), we get the Cobb-Douglas utility form.

Solving utility maximization subject to the usual budget constraint, we get the demand for good $$i$$ as $$$$x_i(p_1,\dots,p_n,M)=\frac{M(\alpha_i/p_i)^\sigma}{\sum_{j=1}^n\alpha_j^\sigma p_j^{1-\sigma}},\quad i=1,\dots,n.$$$$ Again, observe that when $$\sigma=1$$ we get the demand associated with Cobb-Douglas utility.

The elasticity of substitution governs how relative expenditures on different goods change as relative prices change. Take a two-good example. An increase in the relative price $$p_1/p_2$$, i.e. good 1 becoming relatively more expensive, causes two effects simultaneously:

1. per unit expenditure on good 1 rises, as good 1 now costs more in relative terms; and
2. quantity demanded for good 1 decreases due to law of demand.

These are opposing effects on the expenditure on good 1 relative to that on good 2. It turns out that the elasticity of substitution determines which effect dominates. If $$\sigma>1$$, the second effect dominates, and if $$\sigma<1$$, the first effect dominates. When $$\sigma=1$$, which is the case of Cobb-Douglas, the two effects cancel each other exactly, so that relative expenditure is independent of the relative prices, and depends only on preference parameters (the $$\alpha_i$$'s).

Of course, CES utility is not the only class of utility functions that generate demands dependent on prices of other goods. Another common form of utility function, the quasi-linear utility function, $$$$u(x_1,\dots,x_n)=x_1+v(x_2,\dots,x_n),$$$$ where $$v(\cdot)$$ is strictly increasing and strictly concave, also generates demand functions that depend on other prices. A common example is $$u(x_1,x_2)=x_1+2\sqrt{x_2}$$. I trust that you can verify that both $$x_1$$ and $$x_2$$ depend on the two prices $$p_1$$ and $$p_2$$.

• Thank you very much, I'll see if I can do the two case. – Jonathan Andrews Mar 31 '20 at 7:28