3) Given that; $$ \mathrm{U}=\mathrm{x}^{0.5} \mathrm{y}^{0.5} $$ and $$ \mathrm{I}=\mathrm{xPx}+\mathrm{yPy} $$ Where: U denotes Utility function x denotes Quantity of good x y denotes Quantity of good y Px denotes Price per unit of x Py denotes Price per unit of y Using the above information and the Lagrange method, derive: a) the Hicksian demand of x

Question

UpStudy AI · Accepted Answer

We begin with the expenditure minimization problem. The consumer seeks to minimize expenditure

$$
E = P_x x + P_y y
$$

subject to achieving a given utility $$ u $$ as defined by

$$
x^{0.5} y^{0.5} = u.
$$

We construct the Lagrangian function as

$$
\mathcal{L}(x,y,\lambda)=P_x x+P_y y+\lambda \left( u - x^{0.5}y^{0.5}\right).
$$

**Step 1. First-order conditions**

Differentiate $$ \mathcal{L} $$ with respect to $$ x $$, $$ y $$, and $$ \lambda $$:

1. With respect to $$ x $$:

$$
\frac{\partial \mathcal{L}}{\partial x}=P_x-\lambda \cdot \frac{1}{2}x^{-0.5}y^{0.5}=0.
$$

This implies

$$
P_x=\lambda \frac{1}{2}x^{-0.5}y^{0.5}\quad \Longrightarrow \quad \lambda=\frac{2P_xx^{0.5}}{y^{0.5}}.
$$

2. With respect to $$ y $$:

$$
\frac{\partial \mathcal{L}}{\partial y}=P_y-\lambda \cdot \frac{1}{2}x^{0.5}y^{-0.5}=0.
$$

This implies

$$
P_y=\lambda \frac{1}{2}x^{0.5}y^{-0.5}\quad \Longrightarrow \quad \lambda=\frac{2P_yy^{0.5}}{x^{0.5}}.
$$

3. The constraint:

$$
x^{0.5}y^{0.5}=u.
$$

**Step 2. Equate the expressions for $$\lambda$$**

From the first two conditions, we have

$$
\frac{2P_xx^{0.5}}{y^{0.5}}=\frac{2P_yy^{0.5}}{x^{0.5}}.
$$

Canceling the common factor $$2$$ and cross-multiplying gives

$$
P_x x^{0.5} \cdot x^{0.5} = P_y y^{0.5} \cdot y^{0.5}.
$$

Since $$x^{0.5} \cdot x^{0.5}=x$$ and $$y^{0.5} \cdot y^{0.5}=y$$, we obtain

$$
P_x x = P_y y.
$$

This can be rearranged to express $$x$$ in terms of $$y$$:

$$
x=\frac{P_y}{P_x}y.
$$

**Step 3. Substitute into the utility constraint**

Substitute $$x=\frac{P_y}{P_x}y$$ into the constraint

$$
\left(\frac{P_y}{P_x}y\right)^{0.5}y^{0.5}=u.
$$

Combine the terms:

$$
y^{0.5} \left(\frac{P_y}{P_x}\right)^{0.5}y^{0.5}= y\left(\frac{P_y}{P_x}\right)^{0.5}= u.
$$

Solve for $$y$$:

$$
y = u\left(\frac{P_x}{P_y}\right)^{0.5}.
$$

Now substitute back to solve for $$x$$:

$$
x=\frac{P_y}{P_x}y = \frac{P_y}{P_x}\; u\left(\frac{P_x}{P_y}\right)^{0.5}.
$$

Simplify the expression:

$$
x = u \left(\frac{P_y}{P_x}\right) \left(\frac{P_x}{P_y}\right)^{0.5} = u \left(\frac{P_y}{P_x}\right)^{0.5}.
$$

**Final Answer**

The Hicksian demand for $$ x $$ is

$$
x^h(u,P_x,P_y)= u\sqrt{\frac{P_y}{P_x}}.
$$
The Hicksian demand for $$ x $$ is $$ x^h(u, P_x, P_y) = u \sqrt{\frac{P_y}{P_x}} $$.

Given that; and
Where:
U denotes Utility function
x denotes Quantity of good x
y denotes Quantity of good y
Px denotes Price per unit of x
Py denotes Price per unit of y
Using the above information and the Lagrange method, derive:
a) the Hicksian demand of x

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Economics Questions

Given that; and Where: U denotes Utility function x denotes Quantity of good x y denotes Quantity of good y Px denotes Price per unit of x Py denotes Price per unit of y Using the above information and the Lagrange method, derive: a) the Hicksian demand of x