# Marshallian and Hicksian demand for an inferior good

How will the Marshallian and Hicksian demands for an inferior good differ assuming the price goes up? Also what happens if the demand is not affected by changes in income?

I am assuming it will fall it will fall faster for hicksian demands due to more real purchasing power. In the second case, I believe it will be the same for both.

But I am not sure about my solution.

Let $$p$$ be price vector, let $$m$$ be income and let $$u$$ be utility.

Let $$e(p,u)$$ be the expenditure function which gives the minimal expenditure necessary to get utility level $$u$$ and let $$v(p,m)$$ be the indirect utility function, which gives the maximal utility that can be obtained ad prices $$p$$ given income $$m$$.

The Hicksian demand for good $$i$$, $$h_i(p,u)$$ and the Marshallian demand for good $$i$$, $$x_i(p,m)$$ are related in the following way: $$h_i(p,u) = x_i(p,e(p,u)) \text{ and } h_i(p,v(p,m)) = x_i(p,m)$$ Take the first identity and take the derivative with respect to $$p_i$$ to get: $$\frac{\partial h_i(p,u)}{\partial p_i} = \frac{\partial x_i(p,e(p,u))}{\partial p_i} + \frac{\partial x_i(p, e(p,u))}{\partial m} \frac{\partial e(p,u)}{\partial p_i}$$ Next, the Hicksian demand equals the derivative of the expenditure function $$h_i(p,u) = \frac{\partial e(p,u)}{\partial p_i}$$ so: \begin{align*} \frac{\partial h_i(p,u)}{\partial p_i} &= \frac{\partial x_i(p,m)}{\partial p_i} + \frac{\partial x_i(p,m)}{\partial m} h_i(p,u),\\ &=\frac{\partial x_i(p,m)}{\partial p_i} + x_i(p,m) \frac{\partial x_i(p,m)}{\partial m} \end{align*} This is the Slutsky equation.

If the good is inferior then the second term is negative. As such, the derivative of the Hicksian demand is smaller (more negative) than the one of the Marshallian demand: $$\frac{\partial h_i(p,u)}{\partial p_i} < \frac{\partial x_i(p,m)}{\partial p_i}$$

So indeed, if $$p_i$$ increases and the good is inferior, the Hicksian demand will decrease more (the Hicksian demand is steeper).

Intuitive: if the price increases, the Hicksian demand will go down due to the substitution effect. However, there is also a "decrease in income" (due to the income effect). For inferior goods, this income effect, will increase the demand of the good. As such, the Marshallian demand will decrease less than the Hicksian demand.

If the income effect is zero, then the Marshallian and Hicksian demand effects are equal as the second term vanishes: $$\frac{\partial h_i(p,u)}{\partial p_i} = \frac{\partial x_i(p,m)}{\partial p_i}$$

• The slopes are negative, right? That is why the equation is equivalent to lets say (hicksian slope is -2 and marshallian is -1/5) -2< -1/5 and therefore, the hicksian is decreasing more. Also, I dont understand the intuition. For Hicksian demand, we keep utility constant so an increase in price will lead to substitution and a decrease in demand. For Marshallian demand, the Income is fixed and therefore, real purchasing power decreases. However as the item is an inferior good, then there will be an increase in the demand of the good as well. Is this how you think about it? Oct 3 at 19:06
• Hi @tdm. Great answer. When I studied this kind of thing however, the only thing I felt was lacking would be: 'what conditions allow us to decompose demand into a Hicksian + Marshallian demand.' Are they supposed to represent different markets, are they more suitable at different ends of the income scale etc? Would be good to hear your thoughts. Oct 4 at 16:20