The Preferences Of A Consumer Are Represented By The Utility

The Preferences Of A Consumer Are Represented By The Utility Functionu The preferences of a consumer are represented by the utility function \( U = X + 2\sqrt{Y} \). a) In the initial situation, the prices of the commodities and the income of the consumer are \( P_X = 5 \), \( P_Y = 1 \), and \( I = 150 \). Determine the optimal consumption vector \((X^*, Y^*)\) and compute the maximum utility \( U^* \) of the consumer. b) Assume the government implements a commodity tax \( t \). Then, the prices become \( P_X = 5 + t \), \( P_Y = 1 \), with income \( I = 150 \) and \( t = 5 \). i. Determine the optimal consumption vector, the maximum utility of the consumer, and the government revenue from the tax. ii. Given the new prices, determine the additional income the consumer would need to reach the same utility \( U^* \) as in the initial situation. iii. Explain how this additional income compares to the government revenue from the tax.

Paper For Above instruction The consumer's utility maximization problem is foundational to understanding consumer choice theory in microeconomics. The utility function \( U = X + 2\sqrt{Y} \) implies that the consumer derives utility additively from goods \( X \) and \( Y \), with diminishing marginal utility from \( Y \) due to the square root component. The goal is to identify the consumer's optimal basket of goods given prices and income constraints, and later analyze the effects of taxation. **Part A: Initial Optimal Consumption** The initial scenario involves prices \( P_X = 5 \), \( P_Y = 1 \), and income \( I = 150 \). The problem is to maximize utility: \[ \max_{X,Y} U = X + 2\sqrt{Y} \] subject to the budget constraint: \[

5X + Y = 150 \] The method involves setting up the Lagrangian: \[ \mathcal{L} = X + 2\sqrt{Y} + \lambda (150 - 5X - Y) \] Calculating the partial derivatives: \[ \frac{\partial \mathcal{L}}{\partial X} = 1 - 5\lambda = 0 \Rightarrow \lambda = \frac{1}{5} \] \[ \frac{\partial \mathcal{L}}{\partial \frac{1}{\sqrt{Y}} = \lambda

Y}

=

\frac{2}{2\sqrt{Y}}

-

\lambda

\] Substitute \( \lambda = \frac{1}{5} \): \[ \frac{1}{\sqrt{Y}} = \frac{1}{5} \Rightarrow \sqrt{Y} = 5 \Rightarrow Y = 25 \] Using the budget constraint to find \( X \): \[ 5X + 25 = 150 \Rightarrow 5X = 125 \Rightarrow X = 25 \] The optimal consumption vector:

=

0

\Rightarrow

\[ (X^*, Y^*) = (25, 25) \] Maximum utility: \[ U^* = 25 + 2 \times 5 = 25 + 10 = 35 \] This solution indicates the consumer chooses 25 units of \( X \) and 25 units of \( Y \), achieving the highest utility under the given prices and income constraints. **Part B: Effects of a Commodity Tax** The government introduces a specific commodity tax \( t=5 \) on good \( X \), raising its price from 5 to \( 5 + 5= 10 \). *Part B.i: New Optimal Consumption and Utility* With \( P_X= 10 \) and \( P_Y= 1 \), the new budget constraint: \[ 10X + Y = 150 \] Reformulate the utility maximization: \[ \max_{X,Y} U = X + 2\sqrt{Y} \] subject to: \[ 10X + Y = 150

\] Set up the Lagrangian: \[ \mathcal{L} = X + 2\sqrt{Y} + \mu (150 - 10X - Y) \] Partial derivatives: \[ \frac{\partial \mathcal{L}}{\partial X} = 1 - 10 \mu = 0 \Rightarrow \mu = \frac{1}{10} \] \[ \frac{\partial \mathcal{L}}{\partial Y} = \frac{1}{\sqrt{Y}} - \mu = 0 \Rightarrow \frac{1}{\sqrt{Y}} = \frac{1}{10} \Rightarrow \sqrt{Y} = 10 \Rightarrow Y= 100 \] Determine \( X \): \[ 10X + 100 = 150 \Rightarrow 10X=50 \Rightarrow X=5 \] Maximum utility given the tax: \[ U_{tax} = 5 + 2 \times 10 = 5 + 20= 25 \] The government revenue is: \[

\text{Tax Revenue} = t \times X^* = 5 \times 5 = 25 \] The consumer's optimal bundle shifts notably due to the price increase, indicating a significant reduction in consumption of \( X \) and \( Y \), and utility declines from 35 to 25. *Part B.ii: Additional Income Needed to Maintain Original Utility* To reach the original utility level \( U^* = 35 \) with the new prices: \[ U = X + 2\sqrt{Y} = 35 \] and the budget constraint: \[ 10X + Y = I_{new} \] Express \( Y \): \[ Y = ( (U - X)/2 )^2 \] Substitute into the budget constraint: \[ 10X + ( \frac{U - X}{2} )^2 = I_{new} \] Set \( U= 35 \): \[

10X + \left( \frac{35 - X}{2} \right)^2= I_{new} \] Compute the square term: \[ \left(\frac{35 - X}{2}\right)^2 = \frac{(35 - X)^2}{4} \] The total: \[ I_{new} = 10X + \frac{(35 - X)^2}{4} \] To find the minimal additional income, optimize \( I_{new} \) with respect to \( X \). Take the derivative: \[ \frac{d I_{new}}{d X} = 10 - \frac{2(35 - X)}{4} = 10 - \frac{35 - X}{2} \] Set to zero: \[ 10 = \frac{35 - X}{2} \Rightarrow 20= 35 -X \Rightarrow X= 15 \] Calculate \( Y \): \[ Y= \left(\frac{35 - 15}{2}\right)^2 = (10)^2= 100 \] Calculate \( I_{new} \):

\[ I_{new} = 10 \times 15 + 100= 150 + 100= 250 \] The initial income was 150, thus the additional income needed: \[ \Delta I= 250 - 150= 100 \] **Comparison of Additional Income and Tax Revenue:** The consumer needs an extra 100 income units to attain the original utility after the tax, whereas the government revenue from the tax is only 25. This disparity underscores the consumer's significant loss in utility due to the price increase, which isn't fully compensated by the tax revenue. The difference reflects the welfare loss imposed on consumers by taxation, often termed as deadweight loss in economic terms. **Conclusion:** The analysis highlights the trade-offs involved in taxation policy. While the government gains revenue, it imposes a utility loss on consumers that exceeds this revenue, illustrating the importance of considering consumer welfare effects in policy design. Additionally, this demonstrates the concept of compensating variations indicative of how much income a consumer would need to offset the effects of price changes, which plays a vital role in welfare economics.

References Varian, H. R. (2014). Microeconomics (9th ed.). W.W. Norton & Company.