# The relationship between the expenditure function and many others!

I dont understand the relationships between Hicksian demand, walrasian demand (marshallian), the expenditure function and the indirect utility function (including the value function V(b)). I have found this subject very difficult and cannot comprehend how they relate to each other due to the formality that is used in the books I have available!

I understand how to derive the indirect utility, however, I need to be comfortable to show how I can use it to derive the expenditure function and the rest and how they differ in dualities!

Following up on the excellent MWG diagram in Amstell's answer, the fundamental observation needed is that holding $p$ fixed, $e$ and $v$ are inverses of each other. $e$ tells us the amount we need to spend to get a certain amount of utility $u$, while $v$ tells us the maximum amount of utility we can get from a certain expenditure $w$. Whenever we want to convert from utility to wealth, we use $e$; and whenever we want to convert from wealth to utility, we use $v$.

All the key identities can be derived from this observation. For instance, suppose we want to derive an identity for $\partial v(p,w)/\partial p_i$. We already know the corresponding identity for the expenditure function, $\partial e(p,u)/\partial p_i=h_i(p,u)$. To turn this into an identity for $v$, we substitute $w=e(p,u)$, obtaining $v(p,e(p,u))=u$, and differentiate with respect to $p_i$. The chain rule implies $$\frac{\partial v(p,e(p,u))}{\partial p_i} + \frac{\partial v(p,e(p,u))}{\partial w}\cdot\frac{\partial e(p,u)}{\partial p_i} =0\\ \Longleftrightarrow \frac{\partial v(p,w)}{\partial p_i} = -\frac{\partial v(p,w)}{\partial w}\cdot x_i(p,w)$$ which, if we divide by $-\partial v/\partial w$ on both sides, becomes Roy's identity.

Or, suppose that we want to derive the Slutsky equation, which gives the relationship between the derivatives of Marshallian and Hicksian demand (decomposing a Marshallian demand change into substitution and income effects). Analogously to above, we can substitute $w=e(p,u)$ into Marshallian demand $x(p,w)$ to obtain $x(p,e(p,u))=h(p,u)$. Then, differentiating with respect to $p_i$ on both sides and applying the chain rule gives $$\frac{\partial x(p,e(p,u))}{\partial p_i} + \frac{\partial x(p,e(p,u))}{\partial w}\cdot \frac{\partial e(p,u)}{\partial p_i} = \frac{\partial h(p,u)}{\partial p_i}\\ \Longleftrightarrow \frac{\partial x(p,w)}{\partial p_i}=\frac{\partial h(p,u)}{\partial p_i} - \frac{\partial x(p,w)}{\partial w}\cdot x_i(p,w)$$ In general, I think the heuristic "switch between $w$ and $u$ as needed using $v$ and $e$" allows you to get pretty much everything here. (A similar heuristic is also useful if you ever deal with Frisch demand systems, where marginal utility $\lambda$ plays the same role that $w$ and $u$ do in Marshallian and Hicksian demand systems.)

Of course, there is one other key fact used above, which is $\partial e(p,u)/\partial p_i = h_i(p,u)$, which for $w=e(p,u)$ becomes $\partial e(p,u)/\partial p_i = x_i(p,w)$. This is best viewed, instead, as a direct consequence of the venerable envelope theorem.

($\partial v/\partial p_i$ can also be derived from the slightly more advanced version of the envelope theorem, where constraints as well as the objective are allowed to depend on a parameter. Since varying $p_i$ in the utility maximization problem changes the budget constraint rather than the objective, the envelope theorem says that its effect will depend on the Lagrange multiplier on that constraint, which is the marginal utility $\partial v/\partial w$ of wealth. This is a good intuition for why the expression for $\partial v/\partial p_i$ is more complicated than the expression for $\partial e/\partial p_i$, picking up an extra factor.)

Not sure how much this will help, but the diagram in Mas-Colell p.75 is something I always have in mind when deriving these functions. I'm not sure what books you're using, but Microeconomics by Mas-Colell et al. is the go to graduate resource. But I prefer Microeconomic Analysis by Varian. Much easier to read and still has the important content needed for graduate level work. From my experience, deriving as many Walrasian demands as possible and just working the process is what got me comfortable with understanding. If you are looking for examples I can apply some formulas to show you how it works, but you seem to understand this. I also have pages and pages of practice problems if you need another resource as well. Hope this helps :)

Update : Here are a few practice problems from some of my problem sets. Careful with the last one. Enjoy

If possible, compute Hicksian, Walrasian, Expenditure, and Indirect for each of the following:

1. $e(p,u) = (p_{1} + p_{2})u$

2. $e(p,u) = p_{1} +p_{2} + up_{1}$

3. $h(p,u) = ( \frac{up_{2}}{p_{1}} , \frac{up_{1}}{p_{2}})$

4. $x(p,w) = ( \frac{w}{p_{1}} , \frac{w}{p_{2}} )$

Edit ; Update to explain #4

1. $x(p,w) = ( \frac{w}{p_{1}} , \frac{w}{p_{2}} )$

At first look, you can see that all the wealth is being used up for each demand $(x_{1}, x_{2})$, which is not possible given the income constraint

$p_{1}x_{1}+p_{2}x_{2} = w$.

One of the properties of the Walrasian Demand is that Walras' Law holds.

Walras' Law : $px = w$

A simple way to show that Walras' Law does not hold is to simple plug in the demands for the income constraint.

$p_{1} ( \frac{w}{p_{1}}) + p_{2} ( \frac{w}{p_{2}}) = w$

$2w \neq w$ ; therefore Walras' Law does not hold and this is not a Walrasian demand.