Go through step by step how the answers got out

Here's a step-by-step breakdown of how the Marginal Utility (MU) and Marginal Utility per dollar (MU/P) values were calculated for Ozzie's utility maximization problem.

Calculating Marginal Utility (MU):
Marginal Utility (MU) is the additional satisfaction gained from consuming one more unit of a good. It is calculated as the change in Total Utility (TU) divided by the change in the quantity consumed. Since we are looking at discrete units (visits), it's simply the difference in TU from the previous unit.

$$\text{MU}_n = \text{TU}_n - \text{TU}_{n-1}$$

Calculating Marginal Utility per dollar (MU/P):
Marginal Utility per dollar is the additional satisfaction gained per dollar spent on a good. It is calculated by dividing the Marginal Utility (MU) by the price (P) of that good.

$$\frac{\text{MU}}{\text{P}}$$

Let's apply these formulas to the given data:

For Nightclub Visits (P_N = \20$):

• 1st Visit:

  • TU = 100
  • MU = $100 - 0 = 100$ (assuming TU is 0 for 0 visits)
  • MU/P = $\frac{100}{20} = 5.0$

• 2nd Visit:

  • TU = 180
  • MU = $180 - 100 = 80$
  • MU/P = $\frac{80}{20} = 4.0$

• 3rd Visit:

  • TU = 240
  • MU = $240 - 180 = 60$
  • MU/P = $\frac{60}{20} = 3.0$

• 4th Visit:

  • TU = 280
  • MU = $280 - 240 = 40$
  • MU/P = $\frac{40}{20} = 2.0$

• 5th Visit:

  • TU = 300
  • MU = $300 - 280 = 20$
  • MU/P = $\frac{20}{20} = 1.0$

• 6th Visit:

  • TU = 310
  • MU = $310 - 300 = 10$
  • MU/P = $\frac{10}{20} = 0.5$

For Cinema Visits (P_C = \10$):

• 1st Visit:

  • TU = 60
  • MU = $60 - 0 = 60$ (assuming TU is 0 for 0 visits)
  • MU/P = $\frac{60}{10} = 6.0$

• 2nd Visit:

  • TU = 110
  • MU = $110 - 60 = 50$
  • MU/P = $\frac{50}{10} = 5.0$

• 3rd Visit:

  • TU = 150
  • MU = $150 - 110 = 40$
  • MU/P = $\frac{40}{10} = 4.0$

• 4th Visit:

  • TU = 180
  • MU = $180 - 150 = 30$
  • MU/P = $\frac{30}{10} = 3.0$

• 5th Visit:

  • TU = 200
  • MU = $200 - 180 = 20$
  • MU/P = $\frac{20}{10} = 2.0$

• 6th Visit:

  • TU = 210
  • MU = $210 - 200 = 10$
  • MU/P = $\frac{10}{10} = 1.0$

These calculations fill out the table, which is then used to find the utility-maximizing combination by comparing the MU/P values and allocating the budget.