Katie A Consumer With A Fixed Income Only Consumes Bu ✓ Solved

Katie A Consumer With A Fixed Income M Only Consumes Bu

(a) Find the utility function that represent Katie’s preferences. Are they homothetic? Explain.

(b) Write and solve the consumers utility-maximization problem to find her demand functions, x1(p1, p2, m) and x2(p1, p2, m).

(c) Using your previous results, try replacing the x1 and x2 in the Utility function by the demand functions x1(p1, p2, m) and x2(p1, p2, m). This creates a new “indirect utility function” that is a function of only prices and income, v(p1, p2, m).

Paper For Above Instructions

Katie, a consumer with a fixed income, enjoys consuming two goods: burgers (represented as x1) and pickled cucumbers (represented as x2). She consumes these goods in fixed proportions of two burgers for every three pickled cucumbers. To establish the utility function that encapsulates Katie's preferences, we can employ the concept of a Cobb-Douglas function, which is commonly used to represent preferences in economic models involving fixed proportions.

A. Utility Function Representation

The utility function can thus be expressed as:

U(x1, x2) = min(2x1/2, 3x2/3) = min(x1, 1.5x2)

This equation suggests that to maintain her preference ratio, Katie will only derive utility from her consumption as long as the fixed proportions are respected.

To determine if Katie's preferences are homothetic, we analyze whether the ratios of her consumption remain constant as income changes. A utility function is homothetic if it exhibits constant relationships between the goods consumed and the level of consumption. Given that the proportions in her utility function remain unchanged regardless of the level of consumption, we can conclude that Katie’s preferences are indeed homothetic.

B. Consumer’s Utility-Maximization Problem

Katie's utility-maximization problem seeks to maximize her utility subject to her budget constraint. The budget constraint can be expressed as:

p1 x1 + p2 x2 = m

where:

  • p1 is the price of burgers.
  • p2 is the price of pickled cucumbers.
  • m is Katie's fixed income.

To solve this optimization problem using Lagrange multipliers, we define the Lagrangian function as:

ℒ = U(x1, x2) + λ(m - p1 x1 - p2 x2)

Given the utility function, we need to ensure the derivatives of ℒ with respect to x1, x2, and the multiplier λ are set to zero:

∂ℒ/∂x1 = ∑(U(x1, x2) = 0, ∂ℒ/∂x2 = ∑(U(x1, x2) = 0, ∂ℒ/∂λ = (m - p1 x1 - p2 x2) = 0

From the fixed proportion, we know:

x1/x2 = 2/3

This implies:

x1 = (2/3)x2

Substituting this back into the budget constraint allows us to solve for the demand functions:

p1 (2/3)x2 + p2 x2 = m

Therefore:

x2 = m / (p1 * 2/3 + p2)

And subsequently:

x1 = (2/3)(m / (p1 * 2/3 + p2))

These results provide Katie's demand functions based on prices and her fixed income.

C. Indirect Utility Function

Next, we aim to formulate the indirect utility function, denoted as v(p1, p2, m), by substituting the demand functions back into the utility function:

v(p1, p2, m) = U(x1(p1, p2, m), x2(p1, p2, m))

Substituting the earlier results yields:

v(p1, p2, m) = min(2 (2/3)(m / (p1 2/3 + p2)), 3 (m / (p1 2/3 + p2)))

Upon simplification, we get:

v(p1, p2, m) = m / (p1 2/3 + p2) min(4/3, 3)

This transformation thus encapsulates Katie's utility as a function of market prices and income, providing insight into how her consumption preferences react to economic changes.

Conclusion

In conclusion, the analysis of Katie's consumption illustrates the practical applications of fixed proportions in consumer theory. Establishing her utility function, solving the utility-maximization problem, and developing the indirect utility function are vital components in understanding consumer behavior within economic frameworks. This analysis not only showcases the homothetic preferences of Katie but also emphasizes the importance of budget constraints in influencing her consumption choices.

References

  • Varian, H. R. (2010). Intermediate Microeconomics: A Modern Approach. W. W. Norton & Company.
  • Mas-Colell, A., Whinston, M. D., & Green, J. R. (1995). Microeconomic Theory. Oxford University Press.
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  • Bowden, N. (1994). Consumer Theory, Equilibrium and the Demand for Goods. MIT Press.
  • Spiro, J. (2000). Consumer Behavior: A Framework. Prentice Hall.
  • Varian, H. R. (1992). Microeconomic Analysis. W. W. Norton & Company.
  • Nechyba, T. J. (2015). Microeconomics: An Intuitive Approach with Calculus. Cengage Learning.
  • Froeb, L. M., & McCann, B. T. (2016). Managerial Economics: A Problem-Solving Approach. Cengage Learning.
  • Kreps, D. M. (2012). Microeconomics. Princeton University Press.
  • Frank, R. H., & Bernanke, B. S. (2013). Principles of Economics. McGraw-Hill Education.

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