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.