(b). Suppose a consumer has 900,000 shs. To be allocated between commodities x and y. The price of commodity y is 2000.find the demand function for x and y if the utility function is
The correct answer and explanation is:
To find the demand function for commodities xx and yy given a consumer’s utility function and budget, we proceed as follows:
Problem Setup
Let:
- U(x,y)U(x, y): Utility function.
- PxP_x: Price of commodity xx.
- Py=2000P_y = 2000: Price of commodity yy.
- M=900,000M = 900,000: Total income or budget.
Suppose the utility function is of the form U(x,y)=xaybU(x, y) = x^a y^b, where aa and bb are positive constants that define the relative preference for xx and yy. Without a specific utility function provided, we assume this general Cobb-Douglas utility function.
Demand Function Derivation
- Budget Constraint: The total expenditure must equal the budget: Pxx+Pyy=MP_x x + P_y y = M
- Optimization Problem: The consumer maximizes utility U(x,y)=xaybU(x, y) = x^a y^b subject to the budget constraint Pxx+Pyy=MP_x x + P_y y = M.
- Lagrangian Method: Define the Lagrangian function: L=xayb+λ(M−Pxx−Pyy)\mathcal{L} = x^a y^b + \lambda (M – P_x x – P_y y) Take partial derivatives with respect to xx, yy, and λ\lambda, and set them equal to zero: ∂L∂x=axa−1yb−λPx=0\frac{\partial \mathcal{L}}{\partial x} = a x^{a-1} y^b – \lambda P_x = 0 ∂L∂y=bxayb−1−λPy=0\frac{\partial \mathcal{L}}{\partial y} = b x^a y^{b-1} – \lambda P_y = 0 ∂L∂λ=M−Pxx−Pyy=0\frac{\partial \mathcal{L}}{\partial \lambda} = M – P_x x – P_y y = 0
- Solving for Demand Functions: Divide the first two equations to eliminate λ\lambda: axa−1ybbxayb−1=PxPy\frac{a x^{a-1} y^b}{b x^a y^{b-1}} = \frac{P_x}{P_y} Simplify: ab⋅yx=PxPy\frac{a}{b} \cdot \frac{y}{x} = \frac{P_x}{P_y} Rearrange: yx=bPxaPy\frac{y}{x} = \frac{b P_x}{a P_y} Solve for yy in terms of xx: y=bPxaPyxy = \frac{b P_x}{a P_y} x Substitute yy into the budget constraint: Pxx+Py(bPxaPyx)=MP_x x + P_y \left(\frac{b P_x}{a P_y} x\right) = M Simplify: Pxx(1+ba)=MP_x x \left(1 + \frac{b}{a}\right) = M x=MPx(1+ba)x = \frac{M}{P_x \left(1 + \frac{b}{a}\right)} Similarly, find yy by substituting xx back: y=bPxaPy⋅MPx(1+ba)y = \frac{b P_x}{a P_y} \cdot \frac{M}{P_x \left(1 + \frac{b}{a}\right)} y=bMPy(a+b)y = \frac{b M}{P_y (a + b)}
Final Demand Functions:
The demand functions for xx and yy are: x=aMPx(a+b)x = \frac{a M}{P_x (a + b)} y=bMPy(a+b)y = \frac{b M}{P_y (a + b)}
Explanation
The demand functions for xx and yy are derived based on the consumer’s objective to maximize utility given a fixed income and prices. The utility function U(x,y)=xaybU(x, y) = x^a y^b reflects the consumer’s preferences, with aa and bb indicating the relative importance of each commodity. The budget constraint ensures the consumer does not spend more than their income, M=900,000M = 900,000.
The Lagrangian optimization method is used to solve this constrained maximization problem. By setting up and solving the first-order conditions, we derive relationships between the quantities of xx and yy and their prices. The ratio y/x=bPxaPyy/x = \frac{b P_x}{a P_y} indicates that the consumer’s allocation between xx and yy depends on their relative prices and preferences.
The demand function for xx shows that the quantity demanded decreases as the price of xx increases (law of demand). Similarly, the demand for yy decreases with higher PyP_y. The income MM is allocated proportionally based on the weights aa and bb, ensuring optimal satisfaction.
These functions provide a foundational tool in consumer theory, helping to predict behavior under different price and income scenarios.