Determining Relative Price of Public Good X in Consumer Choice

Introduction

Consumer choice theory is a fundamental concept in economics that examines how individuals make decisions when allocating their limited resources among various goods and services. In this scenario, we will explore how a consumer chooses between a public good X and a private good Y, using a specific utility function. Furthermore, we will calculate the relative price of X given a budget constraint, providing a clear and concise explanation throughout this essay.

Utility Function and Consumer Preferences

The consumer in question evaluates their preferences using a utility function, which quantifies the satisfaction or happiness derived from consuming different goods. In this case, the utility function is expressed as U = X^(3/4) * Y^(1/4), where X represents the quantity of the public good and Y represents the quantity of the private good. The exponent values of 3/4 and 1/4 indicate the consumer’s preference for X and Y, respectively.

Budget Constraint

To make rational consumption choices, the consumer must consider their budget constraint. The budget constraint represents the total amount of resources or income available for spending on goods and services. In this scenario, the budget constraint is given as 10X + 15Y = 3000. This equation implies that the consumer’s total spending must not exceed $3000, and it includes the prices of both public good X and private good Y.

Determining Relative Price of X

To find the relative price of X, we need to rearrange the budget constraint equation to isolate the price of X in terms of Y.

10X + 15Y = 3000

First, subtract 15Y from both sides:

10X = 3000 – 15Y

Next, divide both sides by 10:

X = (3000 – 15Y)/10

Now, we have an expression for X in terms of Y. To calculate the relative price, we need to compare the change in X with a one-unit change in Y. This is represented by the marginal rate of substitution (MRS), which is the negative ratio of the marginal utility of X to the marginal utility of Y:

MRS = – (∂U/∂X) / (∂U/∂Y)

To calculate the MRS, we first find the partial derivatives of the utility function:

∂U/∂X = (3/4) * X^(-1/4) * Y^(1/4)

∂U/∂Y = (1/4) * X^(3/4) * Y^(-3/4)

Now, we can calculate the MRS:

MRS = – [(3/4) * X^(-1/4) * Y^(1/4)] / [(1/4) * X^(3/4) * Y^(-3/4)]

Simplify the expression:

MRS = – (3/4) * (Y/X)

Now, we can substitute the expression for X in terms of Y from the budget constraint:

MRS = – (3/4) * [Y / ((3000 – 15Y)/10)]

Simplify further:

MRS = – (3/4) * [10Y / (3000 – 15Y)]

MRS = – (30Y / (3000 – 15Y))

Now, we have the expression for MRS, which represents the relative price of X in terms of Y. To find the numerical value of this relative price, we can substitute various values of Y into the equation. However, for brevity, we will provide the relative price as a decimal fraction rounded to the second digit after the decimal point.

Relative Price of X (rounded to two decimal places): -0.75

Conclusion

In this essay, we have examined how a consumer chooses between a public good X and a private good Y using a utility function and a budget constraint. We calculated the relative price of X in terms of Y, taking into account the consumer’s preferences and budgetary limitations. The relative price of X was determined to be approximately -0.75, indicating that X is relatively less preferred compared to Y. This information is crucial for understanding consumer behavior and decision-making in the allocation of resources.