Questions: Dillon DeMarco is considering opening a small Italian bakery in the nearby mall, close to the Italian section of the city. He has chosen a good location where he believes there will be interest in the bakery. However, Dillon is unsure how much interest there will be and is trying to decide whether to open a small, medium or large shop. It really depends on what the economy is like in the next year whether people are able to spend their money on his Italian delicacies. Based on the latest financial reports, there is a 40% probability for a strong economy, 40% probability for an average economy and 20% probability for a bad economy. The potential payoffs for a small, medium or large shop for a given year are shown in the decision table. Decision Table - Alternatives: Strong Economy, Average Economy, Bad Economy - Small Shop: 30,000, 19,000, 30,000 - Medium Shop: 55,000, 49,000, 44,000 - Large Shop: 90,000, 67,000, 49,000 - Probabilities: 0.40, 0.40, 0.20 Calculate the Expected Monetary Values (EMV) for each alternative.

Dillon DeMarco is considering opening a small Italian bakery in the nearby mall, close to the Italian section of the city. He has chosen a good location where he believes there will be interest in the bakery. However, Dillon is unsure how much interest there will be and is trying to decide whether to open a small, medium or large shop. It really depends on what the economy is like in the next year whether people are able to spend their money on his Italian delicacies. Based on the latest financial reports, there is a 40% probability for a strong economy, 40% probability for an average economy and 20% probability for a bad economy. The potential payoffs for a small, medium or large shop for a given year are shown in the decision table. Decision Table - Alternatives: Strong Economy, Average Economy, Bad Economy - Small Shop: 30,000, 19,000, 30,000 - Medium Shop: 55,000, 49,000, 44,000 - Large Shop: 90,000, 67,000, 49,000 - Probabilities: 0.40, 0.40, 0.20 Calculate the Expected Monetary Values (EMV) for each alternative.
Transcript text: Dillon DeMarco is considering opening a small Italian bakery in the nearby mall, close to the Italian section of the city. He has chosen a good location where he believes there will be interest in the bakery. However, Dillon is unsure how much interest there will be and is trying to decide whether to open a small, medium or large shop. It really depends on what the economy is like in the next year whether people are able to spend their money on his Italian delicacies. Based on the latest financial reports, there is a $40 \%$ probability for a strong economy, $40 \%$ probability for an average economy and $20 \%$ probability for a bad economy. The potential payoffs for a small, medium or large shop for a given year are shown in the decision table. \begin{tabular}{|c|c|c|c|} \hline \multicolumn{4}{|c|}{ Decision Table } \\ \hline & \multicolumn{3}{|c|}{ State of Nature } \\ \hline Alternatives & Strong Economy & Average Economy & Bad Economy \\ \hline Small Shop & 30,000 & 19,000 & 30,000 \\ \hline Medium Shop & 55,000 & 49,000 & 44,000 \\ \hline Large Shop & 90,000 & 67,000 & 49,000 \\ \hline Probabilities & 0.40 & 0.40 & 0.20 \\ \hline \end{tabular} Calculate the Expected Monetary Values (EMV) for each alternative.
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Solution

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Solution Steps

Step 1: Identify the Probabilities and Payoffs

The probabilities for each state of nature are:

  • Strong Economy: \( P = 0.40 \)
  • Average Economy: \( P = 0.40 \)
  • Bad Economy: \( P = 0.20 \)

The payoffs for each alternative are:

  • Small Shop:
    • Strong Economy: \$30,000
    • Average Economy: \$19,000
    • Bad Economy: \$30,000
  • Medium Shop:
    • Strong Economy: \$55,000
    • Average Economy: \$49,000
    • Bad Economy: \$44,000
  • Large Shop:
    • Strong Economy: \$90,000
    • Average Economy: \$67,000
    • Bad Economy: \$49,000
Step 2: Calculate the Expected Monetary Value (EMV) for Each Alternative

The EMV is calculated as the sum of the products of each payoff and its corresponding probability.

EMV for Small Shop:

\[ \text{EMV}_{\text{Small}} = (30,000 \times 0.40) + (19,000 \times 0.40) + (30,000 \times 0.20) \] \[ \text{EMV}_{\text{Small}} = 12,000 + 7,600 + 6,000 = 25,600 \]

EMV for Medium Shop:

\[ \text{EMV}_{\text{Medium}} = (55,000 \times 0.40) + (49,000 \times 0.40) + (44,000 \times 0.20) \] \[ \text{EMV}_{\text{Medium}} = 22,000 + 19,600 + 8,800 = 50,400 \]

EMV for Large Shop:

\[ \text{EMV}_{\text{Large}} = (90,000 \times 0.40) + (67,000 \times 0.40) + (49,000 \times 0.20) \] \[ \text{EMV}_{\text{Large}} = 36,000 + 26,800 + 9,800 = 72,600 \]

Final Answer

The Expected Monetary Values (EMV) for each alternative are: \[ \boxed{ \begin{aligned} \text{EMV}_{\text{Small}} &= \$25,600 \\ \text{EMV}_{\text{Medium}} &= \$50,400 \\ \text{EMV}_{\text{Large}} &= \$72,600 \end{aligned} } \]

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