#### Compensated Demand Curve

Compensated Demand Curve:

The demand curves that we have discussed, and used, so far are called ordinary demand curves (or sometimes Marshallion demand curves). These demand curves are in principle observable if we could hold money income and other prices constant and only vary the price of one good. A purely theoretical construction is a compensated demand curve. Along this demand curve we must imagine that money income is constantly adjusted so that the consumer’s utility is kept at some given initial level (U0). For all prices above the initial price, an extra income is given (positive compensation) and for all prices below the initial price, money is taken away (a negative compensation or a tax). This demand curve is also called a Hicksian demand curve and we therefore use the letter “Hj to denote it, for example the Hicksian demand curve for good 1:

Q1 = H(p1, p2,U0).

Note that we do not include money income as an argument in this demand function, but the initial utility instead. Effectively, along this demand curve the income effect is eliminated due to instantaneous income compensation, and the slope of this function shows only the substitution effect. If we choose another, for example, higher utility level, the compensated demand curve will shift out to the right, since more will be demanded at each price by a consumer that prefers more to less.

Also, since for a normal good, the income and substitution effects reinforce each other the Hicksian demand function is steeper than the Marshallian demand function. This is illustrated in figure, where Q(p, Y0) is the Marshallian demand curve and H(p,U0) is the Hicksiand demand curve corresponding to the initial price, utility level, and money income level (p0,U0 and Y0). If the price increases to p1 the quantity demanded will decrease along the Marshallian demand curve to Q1i+s, but only to Q1s along the Hicksian demand curve (i and s stand for income and substitution effects). If we had begun at the final price (p1), but with the same money income (Y0), there is a new Hicksian demand curve going through the coordinate (Q1,Q1i+s), this corresponds to a lower utility level (U1). It is lower since the price has increased but money income is unchanged. For an inferior good the Marshallian demand function is steeper. While it is theoretically possible that the Marshallian demand function of a (very) inferior good has a positive slope, the Hicksian demand function is always negatively sloped.

Remember that the substitution effect is always negative.

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