Write me an exercice similar to this one: Consider the Cobb-Douglas utility function U(x,y) = √xy where x measures con-
sumption of oil and y consumption of salt, and a budget constraint p1 + p2 = w,
where p1 = 4, p2 = 2 and w= 16.
(i) write the equation of the level curves of U;
(ii) derive the slope of the level curves in a generic point (x,y);
(iii) find the optimal consumption amount of oil x and salt y by exploiting the
tangency condition of level curves with utility;
(iv) double check the optimal consumption solution by using the Lagrange multi-
plier method.

Here's an exercise similar to the one you provided:

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**Consider the Cobb-Douglas utility function \( U(x, y) = x^{\alpha} y^{\beta} \), where \( x \) measures consumption of food and \( y \) consumption of clothing, with \( \alpha = 0.5 \) and \( \beta = 0.5 \). The consumer faces a budget constraint \( p_1 x + p_2 y = w \), where the price of food is \( p_1 = 5 \), the price of clothing is \( p_2 = 3 \), and total income is \( w = 30 \).**

(i) Write the equation of the level curves of \( U(x, y) \).

(ii) Derive the slope of the level curves in a generic point \( (x, y) \).

(iii) Find the optimal consumption amount of food \( x \) and clothing \( y \) by exploiting the tangency condition of the level curves with the budget constraint.

(iv) Verify the optimal consumption solution by using the Lagrange multiplier method.

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### Additional Exercise Ideas for Practice:

1. **Budget Constraint Analysis**: If the prices of food and clothing were \( p_1 = 6 \) and \( p_2 = 4 \), but the income remains at \( w = 30 \), how would the optimal consumption change?
2. **Income Variation**: What happens to the consumption amounts if the income is doubled to \( w = 60 \), keeping prices constant?
3. **Utility Function Variations**: Consider the utility function \( U(x, y) = x^{0.4} y^{0.6} \). How does this change the optimal consumption allocation between food and clothing?
4. **Non-linear Prices**: Suppose the price of clothing increases with consumption, such that \( p_2 = 3 + 0.1y \). How does this affect the consumer's choice?
5. **Price Elasticity of Demand**: Calculate the price elasticity of demand for food when the price of food changes from \( p_1 = 5 \) to \( p_1 = 7 \).

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### Tip:
When optimizing consumption using Cobb-Douglas utility functions, the ratio \( \frac{\alpha}{\beta} \) often directly influences the ratio in which goods are consumed. Keep this in mind when tackling similar exercises.

Write me an exercise similar to this one: Consider the Cobb-Douglas utility function U(x,y) = √xy where x measures consumption of oil and y consumption of salt, and a budget constraint p1 + p2 = w, where p1 = 4, p2 = 2 and w= 16. (i) write the equation of the level curves of U; (ii) derive the slope of the level curves in a generic point (x,y); (iii) find the optimal consumption amount of oil x and salt y by exploiting the tangency condition of level curves with utility; (iv) double check the optimal consumption solution by using the Lagrange multiplier method.

Explore the Cobb-Douglas utility function U(x, y) = √xy for optimal consumption of oil and salt under a budget constraint. This exercise demonstrates how to derive the level curves, slope, and find the optimal consumption using the tangency condition and the Lagrange multiplier method. Perfect for undergraduate students studying microeconomics or calculus.

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