3) Given that; $$ U=x^{0.5} y^{0.5} $$ and $$ I=x P x+y P y $$ 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 $$ x $$ b) the Hicksian demand of $$ y $$

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We want to minimize expenditure
$$
E = P_x x + P_y y
$$
subject to achieving a given utility level
$$
x^{0.5} y^{0.5} = U.
$$

We form the Lagrangian
$$
\mathcal{L} = P_x x + P_y y + \lambda \Bigl( U - x^{0.5} y^{0.5} \Bigr).
$$

### Step 1. First-Order Conditions

Differentiate with respect to $$ x $$ and $$ y $$:

1. With respect to $$ x $$:
   $$
   \frac{\partial \mathcal{L}}{\partial x} = P_x - \lambda \frac{1}{2} x^{-0.5} y^{0.5} = 0.
   $$
   
2. With respect to $$ y $$:
   $$
   \frac{\partial \mathcal{L}}{\partial y} = P_y - \lambda \frac{1}{2} x^{0.5} y^{-0.5} = 0.
   $$
   
### Step 2. Solve for the Relationship Between $$ x $$ and $$ y $$

Divide the first condition by the second:
$$
\frac{P_x}{P_y} = \frac{x^{-0.5} y^{0.5}}{x^{0.5} y^{-0.5}}.
$$

Simplify the right-hand side:
$$
\frac{x^{-0.5} y^{0.5}}{x^{0.5} y^{-0.5}} = \frac{y^{0.5+0.5}}{x^{0.5+0.5}} = \frac{y}{x}.
$$

Thus, we have:
$$
\frac{P_x}{P_y} = \frac{y}{x} \quad \Longrightarrow \quad y = \frac{P_x}{P_y}\, x.
$$

### Step 3. Substitute Back Into the Utility Constraint

The utility constraint is:
$$
x^{0.5} y^{0.5} = U.
$$

Substitute $$ y = \frac{P_x}{P_y}\, x $$:
$$
x^{0.5} \left(\frac{P_x}{P_y}\, x \right)^{0.5} = U.
$$

Since $$\left(\frac{P_x}{P_y}\, x \right)^{0.5} = \left(\frac{P_x}{P_y}\right)^{0.5} x^{0.5}$$, we have:
$$
x^{0.5} \cdot \left(\frac{P_x}{P_y}\right)^{0.5} x^{0.5} = x \left(\frac{P_x}{P_y}\right)^{0.5} = U.
$$

Solving for $$ x $$:
$$
x = U \left(\frac{P_y}{P_x}\right)^{0.5}.
$$

### Step 4. Find the Hicksian Demand for $$ y $$

Using the relation $$ y = \frac{P_x}{P_y} x $$:
$$
y = \frac{P_x}{P_y} \, U \left(\frac{P_y}{P_x}\right)^{0.5}.
$$

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

### Final Answers

a) The Hicksian demand for $$ x $$ is
$$
x^H = U \sqrt{\frac{P_y}{P_x}}.
$$

b) The Hicksian demand for $$ y $$ is
$$
y^H = U \sqrt{\frac{P_x}{P_y}}.
$$
a) The Hicksian demand for $$ x $$ is $$ x^H = U \sqrt{\frac{P_y}{P_x}} $$. b) The Hicksian demand for $$ y $$ is $$ y^H = U \sqrt{\frac{P_x}{P_y}} $$.

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

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