Question

Consider the following utility function u(x₁,x₂)=(xᵖ₁ + xᵖ₂)¹/ᵖ where rho<1. Assuming that demands are determined by maximization of utility, derive the Walrasian demand curves for both goods as functions of the prices of the two goods (p1 and p2), the level of wealth w and r= rho /(rho-1), where =1/(1-rho) is the elasticity of substitution. Show that CES utility function above is homothetic (there are different ways of showing this, just try one). Then, show that the wealth elasticity of the Walrasian demand is equal to 1 for both goods. Then, derive an expression for the indirect utility function and verify that the latter is homogenous of degree zero in prices and wealth and that it is increasing in wealth and decreasing in prices. Then, apply Roy’s identity to the indirect utility function to obtain the Walrasian demand functions and show that they are equivalent to those that have been derived above. Use expenditure minimization to derive the expenditure function and derive the Hicksian demand curves.Finally, show that the Walrasian and Hicksian demand functions coincide at the point where the consumer can achieve a utility level u at the prices p and wealth level w. Last, show that =1/(rho-1) is actually the elasticity of substitution

Answer 1

To derive the Walrasian demand curves, we compare the ratio of prices to the ratio of marginal utilities. We show that the CES utility function is homothetic and derive the Walrasian demand functions using Roy's identity. The Walrasian and Hicksian demand functions coincide when the desired utility level is achieved.

Step-by-step explanation:

To derive the Walrasian demand curves for goods 1 and 2, we first need to consider the utility-maximizing rule. The ratio of the prices of the two goods should be equal to the ratio of the marginal utilities. Mathematically, this can be expressed as the price of good 1 divided by the price of good 2 being equal to the marginal utility of good 1 divided by the marginal utility of good 2.

In the case of the CES utility function u(x₁,x₂)=(xᵖ₁ + xᵖ₂)¹/ᵖ, we can use the homogeneity property of the function to show that it is homothetic. The utility function is homothetic if the marginal rate of substitution (MRS) depends only on the ratio of the quantities consumed, not on the actual quantities. In this case, the MRS is equal to (x₁/x₂)^(rho-1).

Using Roy's identity, we differentiate the indirect utility function concerning the prices and set it equal to zero to obtain the Walrasian demand functions. By minimizing expenditure, we can derive the expenditure function and then use it to find the Hicksian demand curves.

Finally, we can show that the Walrasian and Hicksian demand functions coincide at the point where the consumer can achieve the desired utility level u at the given prices and wealth level w. Additionally, we can verify that the elasticity of substitution is equal to 1/(rho-1).