# Marshallian and Hicksian demands

Marshallian and Hicksian demands stem from two ways of looking at the same problem- how to obtain the utility we crave with the budget we have. Consumption duality expresses this problem as two sides of the same coin: keeping our budget fixed and maximising utility (primal demand, which leads us to Marshallian demand curves) or setting a target level of utility and minimising the cost associated with it (dual demand, which gives us Hicksian demand curves). We must also look at the Lagrangian functions where we obtain the first derivatives.

This leads us to the main difference between the two types of demand: Marshallian demand curves simply show the relationship between the price of a good and the quantity demanded of it. Hicksian demand curves show the relationship between the price of a good and the quantity demanded of it assuming that the prices of other goods and our level of utility remain constant.  This makes sense when we look at consumption duality: for dual (Hicksian) demand, we maintain a fixed level of utility, and so our level of wealth, or income, must remain constant. We simply cannot be as satisfied if we do not maintain equal purchasing power.

Marshallian and Hicksian demand curves meet where the quantity demanded is equal for both sides of the consumer choice problem (maximising utility or minimising cost). For prices above this equilibrium point, consumer wealth is higher with Hicksian demand curves than Marshallian demand curves, because to maintain utility constant, Hicksian demand curves assume real wealth remains unchanged. Marshallian demand assumes only nominal wealth remains equal. The opposite is true for prices below this point: Marshallian demand assumes that as nominal wealth remains the same but price levels drop (negative inflation), the consumer is better off. Hicksian demand assumes real wealth is constant, so the individual is worse off. This is why Marshallian demand curves are more ‘stable’: they reflect both rent effect and substitution effect. Hicksian demand curves only show substitution effects (utility is constant, therefore rent must remain constant), which means that demand varies with price only because other options become more attractive.

Formally,

 Marshallian demand (dX1) is a function of the price of X1, the price of X2 (assuming two goods) and the level of income or wealth (m): X*=dX1(PX1, PX2, m)
 Hicksian demand (hX1) is a function of the price of X1, the price of X2 (assuming two goods) and the level of utility we opt for (U): X*=hX1(PX1,PX2,U)

For an individual problem, these are obtained from the first order conditions (maximising the first derivatives) of the Lagrangian for either a primal or dual demand problem.

Marshallian demand makes more sense when we look at goods or services that make up a large part of our expenses. Here, the income effect is very large. However, for smaller purchases, we are willing to spend more or less any amount as long as we derive the utility we expect to.

Video – Marshallian and Hicksian demand curves: