Problem Breakdown and Solution

1. Efficient Number of Bales of Hay

The efficient number of bales is determined by maximizing the total surplus, which is the total value of the hay produced minus the cost to Jack of producing it. Let $x$ represent the number of bales produced (or hours worked):

• Total value of hay: $x$
• Cost of production: $c(x) = \frac{x^2}{10}$

The surplus $S(x)$ is given by: $S(x) = \text{Value of hay} - \text{Cost to Jack}$ $S(x) = x - \frac{x^2}{10}$

To find the efficient level of production, maximize $S(x)$ by setting its derivative to zero: $\frac{dS}{dx} = 1 - \frac{2x}{10} = 0$ $1 = \frac{x}{5} \quad \Rightarrow \quad x = 5$

Thus, the efficient number of bales is 5.

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2. Payment to Jack if His Outside Earning is Zero

If Jack’s alternative earning is zero, he will work for MacDonald as long as his cost is covered. His cost of producing $x$ bales is: $c(x) = \frac{x^2}{10}$

At the efficient production level $x = 5$: $c(5) = \frac{5^2}{10} = \frac{25}{10} = 2.5$

Thus, MacDonald would have to pay Jack at least $2.50 to work efficiently.

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3. MacDonald’s Net Profit

MacDonald’s profit is the total value of hay minus the payment to Jack: $\text{Profit} = \text{Value of hay} - \text{Payment to Jack}$

The value of hay at $x = 5$ is $5$, and Jack is paid $2.5$. Thus: $\text{Profit} = 5 - 2.5 = 2.5$

MacDonald’s net profit is $2.50.

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4. Payment to Jack if He Earns $1 for Leafleting

If Jack earns $1 from leafleting with no effort, MacDonald must pay Jack enough to ensure he prefers working in the hayfield. The total cost of working for $x = 5$ is still $2.5$, but now Jack also needs to be compensated for the lost $1 opportunity. Thus, the payment must be: $\text{Payment} = \text{Cost of working} + \text{Opportunity cost}$ $\text{Payment} = 2.5 + 1 = 3.5$

MacDonald must pay Jack $3.50 for him to produce efficiently.

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5. Rental Fee for the Hayfield

If MacDonald rents the hayfield to Jack, the fee must equal the maximum surplus Jack could generate, since Jack would not pay more than this. At $x = 5$, the surplus is: $\text{Surplus} = \text{Value of hay} - \text{Cost to Jack}$ $\text{Surplus} = 5 - 2.5 = 2.5$

Thus, MacDonald could rent the hayfield for $2.50.

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Summary of Results:

1. Efficient number of bales of hay: $5$
2. Payment to Jack if outside earning is zero: $\$2.50$
3. MacDonald’s net profit: $\$2.50$
4. Payment to Jack if he earns $1 from leafleting: $\$3.50$
5. Rental fee for the hayfield: $\$2.50$

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Expanding Questions:

1. How does the efficiency calculation change if the cost function is different, say $c(x) = \frac{x^3}{20}$?
2. What happens to MacDonald’s profit if the selling price of hay increases to $2 per bale?
3. If Jack’s opportunity cost rises to $2 per hour, how would that affect the efficient number of bales and payments?
4. Could MacDonald use a performance-based payment system to incentivize Jack to produce efficiently? If so, what would it look like?
5. What other economic factors might influence the profitability of this arrangement?

Tip:

When calculating opportunity costs, always include all forms of foregone benefits to ensure accurate decision-making.