Question

A consumer has preferences for two goods, x and y, expressed by the following utility function U(z,y) = 1/3y2/3 The price of xis P. the price of yis P, and her budget is $1000. The consumer wishes to choose the quantities x andy that maximize her utility subject to her budget constraint

1. Write down the consumer's budget constraint.
2 Write down the Lagrangian for this constraint optimisation problem,
3. Using the principles of constraint optimisation, find out the optimal demand function for both goods and y
4. Find the price elasticity of demand for good s at the equilibrium and interpret the meaning of the elasticity.

Answer 1

The consumer's budget constraint can be represented as Px * x + Py * y = Budget. The Lagrangian for this constraint optimization problem is L = 1/3y^(2/3) - λ(Px * x + Py * y - Budget). Using the principles of constraint optimization, we can find the optimal demand function for both goods x and y.

Step-by-step explanation:

Budget Constraint

The consumer's budget constraint can be represented as:

Px * x + Py * y = Budget

Lagrangian

The Lagrangian for this constraint optimization problem is:

L = 1/3y^(2/3) - λ(Px * x + Py * y - Budget)

Optimal Demand Function

Using the principles of constraint optimization, we can find the optimal demand function for both goods x and y by taking the partial derivatives of the Lagrangian with respect to x and y:

dL/dx = -λ * Px = 0

dL/dy = 2/9 * y^(-1/3) - λ * Py = 0

Price Elasticity of Demand

The price elasticity of demand for good s at the equilibrium can be calculated using the formula:

E = (ΔQ/Q) / (ΔP/P)

Where ΔQ is the change in quantity demanded and ΔP is the change in the price of the good. The interpretation of the price elasticity of demand is that it measures the responsiveness of the quantity demanded to a change in price.