What is the difference between TU and MU? Determine the consumers equilibrium position.

Here are the answers to the questions from the image.

1. What is the difference between TU and MU?

• Total Utility (TU) is the overall satisfaction or benefit a consumer receives from consuming a specific quantity of a good or service.
• Marginal Utility (MU) is the additional satisfaction or benefit gained from consuming one more unit of that good or service. It is the change in total utility resulting from a one-unit change in consumption.

2. Determine the consumer's equilibrium position.

To find the optimal consumption bundle, we apply the utility maximization rule: $\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$ and $P_x Q_x + P_y Q_y = \text{Income}$.

Given:
P_x = \2.00$$P_y = $1.00$Income$= $12.00$

Step 1: Calculate the marginal utility per dollar for each good.

Q	$MU_x$	$MU_y$	\frac{MU_x}{P_x} = \frac{MU_x}{\2}$	\frac{MU_y}{P_y} = \frac{MU_y}{\1}$
1	16	11	$16/2 = 8$	$11/1 = 11$
2	14	10	$14/2 = 7$	$10/1 = 10$
3	12	9	$12/2 = 6$	$9/1 = 9$
4	10	8	$10/2 = 5$	$8/1 = 8$
5	8	7	$8/2 = 4$	$7/1 = 7$
6	6	6	$6/2 = 3$	$6/1 = 6$
7	4	5	$4/2 = 2$	$5/1 = 5$
8	2	4	$2/2 = 1$	$4/1 = 4$

Step 2: Allocate income to maximize utility. We purchase units that give the highest $MU/$$.

• Buy 1 Y ($MU_y/P_y = 11$). Cost: 1 \times \1 = $1$. Remaining income:$$11$. ($Q_x=0, Q_y=1$)
• Buy 1 Y ($MU_y/P_y = 10$). Cost: 1 \times \1 = $1$. Remaining income:$$10$. ($Q_x=0, Q_y=2$)
• Buy 1 Y ($MU_y/P_y = 9$). Cost: 1 \times \1 = $1$. Remaining income:$$9$. ($Q_x=0, Q_y=3$)
• Now, $MU_x/P_x$ for $Q_x=1$ is 8, and $MU_y/P_y$ for $Q_y=4$ is 8. We can buy one of each.
  • Buy 1 X ($MU_x/P_x = 8$). Cost: 1 \times \2 = $2$. Remaining income:$$7$. ($Q_x=1, Q_y=3$)
  • Buy 1 Y ($MU_y/P_y = 8$). Cost: 1 \times \1 = $1$. Remaining income:$$6$. ($Q_x=1, Q_y=4$)
• Next, $MU_x/P_x$ for $Q_x=2$ is 7, and $MU_y/P_y$ for $Q_y=5$ is 7.
  • Buy 1 X ($MU_x/P_x = 7$). Cost: 1 \times \2 = $2$. Remaining income:$$4$. ($Q_x=2, Q_y=4$)
  • Buy 1 Y ($MU_y/P_y = 7$). Cost: 1 \times \1 = $1$. Remaining income:$$3$. ($Q_x=2, Q_y=5$)
• Finally, $MU_x/P_x$ for $Q_x=3$ is 6, and $MU_y/P_y$ for $Q_y=6$ is 6.
  • Buy 1 X ($MU_x/P_x = 6$). Cost: 1 \times \2 = $2$. Remaining income:$$1$. ($Q_x=3, Q_y=5$)
  • Buy 1 Y ($MU_y/P_y = 6$). Cost: 1 \times \1 = $1$. Remaining income:$$0$. ($Q_x=3, Q_y=6$)

Step 3: Verify the conditions for the optimal bundle ($Q_x=3, Q_y=6$).

1. Budget Constraint: $P_x Q_x + P_y Q_y = (\$2 \times 3) + (\$1 \times 6) = \$6 + \$6 = \$12$ The total expenditure matches the income.
2. Equimarginal Principle: For $Q_x=3$, $\frac{MU_x}{P_x} = \frac{12}{2} = 6$. For $Q_y=6$, $\frac{MU_y}{P_y} = \frac{6}{1} = 6$. The principle $\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$ is satisfied.

The optimal consumption bundle is:
$Q_x = \boxed{3}$
$Q_y = \boxed{6}$