Microeconomics Question from Walter E. Williams:Edit

In a situation with two commodities, X and Y, and starting with an individual's interior (non-corner) endowment, derive his demand curve for commodity X. Is the demand curve derived one with real income constant in the Hicksian or Slutsky sense? First, however, explain the Hicksian versus Slutsky demand curves.



Derivation of the Marshallian demand curve

At the first point (say A) on the upper graph there is a point of utility maximization subject to the first budget constraint. If the price of X is lowered (more X is able to be purchaced with the same amount of income) then the line rotates outward. This does not affect the amount of Y that could be consumed when no X is consumed.

Since the new price is lower than the old price we see a movement downard as the increased quantity of X is available. This is a move along the demand curve from A to B.

This does not take into account the effect that lower price of X has on the consumption of Y (which is all other goods).


Derivation of the Hicksian demand curve

This is where the Hicksian demand curve comes from (an attempt to compensate for the income change). If you shift the new budget line back to the original utility curve (a point like C on the red line) then you have not improved the utility (or subjective income) but you have changed the allocation between X and Y (all other goods). Note that the Elastisity of the Hicksian Demand curve is necessarily less than the elasticity of the Marshallian Demand curve. This will not be true for the Slutsky curve.

With the Slutsky curve, the utility curves are unknown. Therefore the adjustment can only be made to a budget line passing through the original point. This can be a minor change, but is not identical to the Hicksian budget line. When the price change is positive the Slutsky curve is less elastic than the Hicksian demand curve. When the price change is negative the Slutsky curve is more negative than the Hicksian demand curve. The reason being is that the Slutsky has a bit of the income effect still included.

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