The formula for expansion path

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

As previously stated by VARulle you simply compute your Marshallian demand for your goods, fix your income and just vary your prices.

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.