Understanding Hicksian Demand with Utility Maximization

In economics and consumer choice theory, an important concept is Hicksian demand. This refers to the quantity of a good that a consumer would demand in order to achieve a certain utility level at the least possible cost. We will solve a specific example to understand Hicksian demand using the utility function, given prices, and the Lagrange multiplier method. Let's delve into the detailed steps.

Utility Function and Given Parameters

The utility function is given by:

U(x, y) 4x^2y^2

We are given the utility level as 64 and the prices of goods x and y are 3 and 2 respectively.

Step 1: Set Up the Utility Constraint

The utility constraint is expressed as:

4x^2y^2 64

Step 2: Set Up the Expenditure Function

The objective is to minimize the expenditure required to achieve a given level of utility. The expenditure function E is defined as:

E 3x 2y

Step 3: Solve the Problem Using the Lagrange Multiplier

We introduce a Lagrange multiplier λ to find the minimum expenditure. The Lagrangian is defined as:

mathcal{L}(x, y, lambda) 3x 2y - lambda(64 - 4x^2y^2)

We then take the partial derivatives with respect to x, y, and λ, and set them to zero.

Partial Derivatives

(frac{partial mathcal{L}}{partial x} 3 - 8lambda x 0) implies (8lambda x 3) implies (lambda frac{3}{8x})

(frac{partial mathcal{L}}{partial y} 2 - 2lambda y 0) implies (2lambda y 2) implies (lambda frac{1}{y})

(frac{partial mathcal{L}}{partial lambda} 64 - 4x^2y^2 0) implies (4x^2y^2 64)

Step 4: Equate the Two Expressions for λ

From the first two equations, we have:

(frac{3}{8x} frac{1}{y})

Cross-multiplying gives:

3y 8x) implies (y frac{8}{3}x)

Step 5: Substitute y Back into the Utility Constraint

Substituting y frac{8}{3}x into the utility constraint:

4x^2 left(frac{8}{3}xright)^2 64

This simplifies to:

4x^2 frac{64}{9}x^2 64

Combining terms:

left(4 frac{64}{9}right)x^2 64

Converting 4 to a fraction:

(frac{36}{9} frac{64}{9} frac{100}{9}) implies (frac{100}{9} x^2 64)

Step 6: Solve for x

Multiplying both sides by (frac{9}{100}):

x^2 frac{64 cdot 9}{100} frac{576}{100} frac{144}{25})

Since x is an expenditure on goods, it must be positive:

x sqrt{frac{144}{25}} frac{12}{5}

Step 7: Find y

Now, substitute x frac{12}{5}) to find y:

y frac{8}{3} cdot frac{12}{5} frac{96}{15} frac{32}{5})

Conclusion

The Hicksian demand for goods x and y are:

Hicksian demand for x: (x^* frac{12}{5} approx 2.4)

Hicksian demand for y: (y^* frac{32}{5} approx 6.4)

Thus, the Hicksian demands for the goods x and y are (x^* frac{12}{5}) and (y^* frac{32}{5}).