Using the MU theory to derive the marginal utility for gobe which is a giffen good

To derive the marginal utility for Gobe, a Giffen good, we will employ the principles of marginal utility theory. Here's how we can approach this problem:

1. Break Down the Problem

1. Define the concept of marginal utility.
2. Understand what a Giffen good is and how it behaves.
3. Formulate the relationship between the price of Gobe, its quantity, and the consumer's budget constraint.

2. Relevant Concepts

• Marginal Utility (MU): The additional satisfaction a consumer gains from consuming one more unit of a good. It is calculated as the change in total utility divided by the change in quantity consumed:

  $MU = \frac{\Delta U}{\Delta Q}$

• Giffen Good: A Giffen good is an inferior good for which demand increases as its price increases, violating the law of demand. This phenomenon occurs because the income effect outweighs the substitution effect.

3. Analysis and Detail

1. Assume a Utility Function: Let $U(Q)$ represent the total utility derived from consuming Gobe. The utility function can take different forms, but for simplicity, we can assume a linear utility function such as:

   $U(Q) = aQ - bQ^2$

   where $a$ and $b$ are constants.

2. Marginal Utility Calculation: The marginal utility is derived from the utility function by taking the derivative:

   $MU = \frac{dU}{dQ} = a - 2bQ$

3. Income and Substitution Effects: For Gobe, an increase in price leads to:

   • Income Effect: Reduces real income, leading consumers to consume more Gobe (since it's an inferior good).
   • Substitution Effect: Consumers switch to Gobe due to relatively higher priced alternatives.

4. Price Increase Analysis: Let the price of Gobe increase from $P_1$ to $P_2$. The budget constraint adjusts, and consumers will reassess their consumption choice based on the new price:

   $\text{New Budget Constraint: } I = P_2 \cdot Q_\text{Gobe} + P_X \cdot Q_X$

4. Verify and Summarize

1. As per the conditions of Giffen goods, while the substitution effect would suggest a decrease in quantity demanded (as price rises), the income effect (due to Gobe being inferior) suggests an increase in quantity demanded.
2. The derived marginal utility $MU = a - 2bQ$ indicates how utilities change with different levels of consumption.

Final Answer

The marginal utility for Gobe, as a Giffen good, is represented by the equation

$MU = a - 2bQ$

This equation is critical for understanding consumer behavior in relation to Gobe's changing price, reaffirming that despite the price increase, consumers may demand more due to the unique nature of Giffen goods.