# The formula for expansion path

## Is there a way how to precisely compute the expansion path?

I know a consumer's utility function $$U(\boldsymbol{x})$$, I know the budget constraint $$\sum P_i x_i \leq M$$, I am able to compute the marshallian demands $$\boldsymbol{D}(\boldsymbol{P}, M)$$, therefore I am able to gain optimal point for any given prices and income.

If I plot this, using sliders, I can observe how the optimal point moves with changes in one single price. This movement, however, seems to be on a curve, the expansion path. Is there some way how to compute it analythically? Would the differentiation of demands be reasonable approach or is this too naive:

$$\qquad \qquad x_1^{EP} = \frac{\partial D_1(\boldsymbol{P}, M)}{\partial P_1} \qquad \qquad x_2^{EP} = \frac{\partial D_2(\boldsymbol{P}, M)}{\partial P_1}$$

What would be the next step? This would be two variables but I would like to plug it into single curve...

Or is there another approach?

EDIT: The expansion path I search for is also called price consumption curve in literature. I could find it by fitting few points (see illustration below), however, how can I find this analythically?

• So you want $x_i$ as a function of $P_i$ is that correct? At first glance I suggest maximizing the Lagrangian. Mar 23 at 14:06

The answer depends on what you mean by "computing the expansion path".

Variant (i): For fixed $$p_2$$ and $$M$$, you want to compute the optimal points $$\boldsymbol{x}=(x_1,x_2)$$ as a function of $$p_1$$.

That's easy: $$\boldsymbol{x}=\boldsymbol{D}(p_1,p_2,M)$$.

Variant (ii): For fixed $$p_2$$ and $$M$$, you want to compute the $$x_2$$-component of the optimal points $$\boldsymbol{x}=(x_1,x_2)$$ as a function of $$x_1$$.

This needs some work: You are given the functions $$x_1=D_1(p_1,p_2,M)$$ and $$x_2=D_2(p_1,p_2,M)$$. From the equation for $$x_1$$ express (if possible) $$p_1$$ as a function $$p_1=P_1(x_1,p_2,M)$$ and substitute for $$p_1$$ in $$D_2$$ to get $$x_2=D_2(P_1(x_1,p_2,M),p_2,M)$$. For fixed $$p_2$$ and $$M$$, this gives $$x_2$$ as a function of $$x_1$$, the graph of which is the red curve in $$x_1$$-$$x_2$$-space.

• Thank you very much! Yes, I was searching for variant (ii). Mar 23 at 16:17

You compute it by setting $$MRS$$ equals relative prices:

$$\frac{\frac{\partial U}{\partial x}}{\frac{\partial U}{\partial y}} = \frac{p_x}{p_y}$$,

solving for one variable in terms of the other.

• Thank you very much! However, this is not correct answer. I will edit the question such that it is more clear what I want. The answer you proposed gives me just a line between optimum and the zero/zero point. It is not the expansion path for prices. The expansion path for prices should not change with the change in price. Mar 23 at 8:40

To illustrate for cobb douglas preferences $$U(x_1,x_2)=x_1^\alpha x_2^{1-\alpha}$$ the marshallian demands for $$x_1$$ and $$x_2$$ are: $$x_1(p_1,m)=\frac{\alpha m}{p_1}, x_2(p_1,m)=\frac{(1-\alpha)m}{p_2}$$
all you need to do is hold $$m$$ constant to see the shape of these "single price changing" expansion paths.