Derive the marshallian demand curve, hicksian demand curve, slutsky demand curve. Which one is the steepest?
The correct answer and explanation is :
Marshallian Demand Curve
The Marshallian demand curve shows the relationship between the quantity of a good demanded and its price, holding income and other prices constant. To derive it, we need to use the consumer’s utility maximization problem. A consumer maximizes their utility ( U(x_1, x_2) ) subject to the budget constraint:
[
p_1x_1 + p_2x_2 = I
]
Where:
- ( p_1, p_2 ) are the prices of goods 1 and 2.
- ( x_1, x_2 ) are the quantities of goods 1 and 2 consumed.
- ( I ) is the consumer’s income.
The Marshallian demand function for good 1, ( x_1(p_1, p_2, I) ), is the solution to this utility maximization problem. This demand curve shows how much of good 1 the consumer will buy as its price changes, holding income and the price of the other good constant.
Hicksian Demand Curve
The Hicksian demand curve, or the compensated demand curve, focuses on a different approach. Here, the consumer is assumed to remain at a constant utility level, and income is adjusted (compensated) to ensure this. The consumer’s goal is to minimize expenditure subject to achieving a target utility level ( U_0 ). The expenditure minimization problem can be written as:
[
\min_{x_1, x_2} p_1x_1 + p_2x_2 \quad \text{subject to} \quad U(x_1, x_2) = U_0
]
The Hicksian demand function for good 1, ( h_1(p_1, p_2, U_0) ), tells us the quantity of good 1 the consumer demands at a given price, with income adjusted to maintain a fixed level of utility.
Slutsky Demand Curve
The Slutsky demand curve combines both the Marshallian and Hicksian approaches. It breaks the total change in demand due to a price change into two effects:
- The substitution effect (change in demand due to the change in relative prices, holding utility constant).
- The income effect (change in demand due to the change in real income, keeping utility constant).
Mathematically, the Slutsky equation can be written as:
[
\frac{\partial x_1}{\partial p_1} = \frac{\partial h_1}{\partial p_1} – \frac{\partial x_1}{\partial I} x_1
]
This equation shows the relationship between changes in demand and price, accounting for both substitution and income effects.
Which Curve is Steepest?
The Hicksian demand curve is typically the steepest. This is because the income effect is eliminated, and only the substitution effect is taken into account. In contrast, the Marshallian curve includes both substitution and income effects, meaning that the demand is more responsive to price changes, often resulting in a less steep slope. The Slutsky demand curve typically lies between the Marshallian and Hicksian curves in terms of steepness, as it accounts for both effects but in a different way.