Question at position 6 [table]
Number of Paintings | Profits to Painter | Total Cost to Gardener 0 | $0 | $0 1 | $18 | $8 2 | $32 | $16 3 | $42 | $24 4 | $48 | $32 5 | $50 | $40
[/table] A painter likes to paint pictures of flowers to sell. The table above shows the profits earned by a painter based on the number of paintings that he creates. However, when painting the painter drips paint which damages the flowers and causes a cost to the gardener who would sell those flowers. Suppose the gardener sues the painter in a common law court. The judge finds in favor of the painter. If the gardener wants fewer paintings painted she will have to pay the painter to reduce the number. The gardener is willing to pay up to $8 while the painter needs to receive at least $2 to be convinced not to paint the 5th painting. The painter and the gardener will consider more deals of this type. What is the number of paintings that will be painted? (Hint: compare the change in the gardeners' costs to the change in the painter's profit. If the painting adds more to the gardener's costs than it does to the painter's profits, they will be able to make a deal to eliminate that painting.) Answer[table]
Number of Paintings | Profits to Painter | Total Cost to Gardener 0 | $0 | $0 1 | $18 | $8 2 | $32 | $16 3 | $42 | $24 4 | $48 | $32 5 | $50 | $40
[/table] A painter likes to paint pictures of flowers to sell. The table above shows the profits earned by a painter based on the number of paintings that he creates. However, when painting the painter drips paint which damages the flowers and causes a cost to the gardener who would sell those flowers. Suppose the gardener sues the painter in a common law court. The judge finds in favor of the painter. If the gardener wants fewer paintings painted she will have to pay the painter to reduce the number. The gardener is willing to pay up to $8 while the painter needs to receive at least $2 to be convinced not to paint the 5th painting. The painter and the gardener will consider more deals of this type. What is the number of paintings that will be painted? (Hint: compare the change in the gardeners' costs to the change in the painter's profit. If the painting adds more to the gardener's costs than it does to the painter's profits, they will be able to make a deal to eliminate that painting.) [input]

Question

Question at position 6 [table]
Number of Paintings | Profits to Painter | Total Cost to Gardener 0 | $0 | $0 1 | $18 | $8 2 | $32 | $16 3 | $42 | $24 4 | $48 | $32 5 | $50 | $40
[/table] A painter likes to paint pictures of flowers to sell. The table above shows the profits earned by a painter based on the number of paintings that he creates. However, when painting the painter drips paint which damages the flowers and causes a cost to the gardener who would sell those flowers. Suppose the gardener sues the painter in a common law court. The judge finds in favor of the painter. If the gardener wants fewer paintings painted she will have to pay the painter to reduce the number. The gardener is willing to pay up to $8 while the painter needs to receive at least $2 to be convinced not to paint the 5th painting. The painter and the gardener will consider more deals of this type. What is the number of paintings that will be painted? (Hint: compare the change in the gardeners' costs to the change in the painter's profit. If the painting adds more to the gardener's costs than it does to the painter's profits, they will be able to make a deal to eliminate that painting.) Answer[table]
Number of Paintings | Profits to Painter | Total Cost to Gardener 0 | $0 | $0 1 | $18 | $8 2 | $32 | $16 3 | $42 | $24 4 | $48 | $32 5 | $50 | $40
[/table] A painter likes to paint pictures of flowers to sell. The table above shows the profits earned by a painter based on the number of paintings that he creates. However, when painting the painter drips paint which damages the flowers and causes a cost to the gardener who would sell those flowers. Suppose the gardener sues the painter in a common law court. The judge finds in favor of the painter. If the gardener wants fewer paintings painted she will have to pay the painter to reduce the number. The gardener is willing to pay up to $8 while the painter needs to receive at least $2 to be convinced not to paint the 5th painting. The painter and the gardener will consider more deals of this type. What is the number of paintings that will be painted? (Hint: compare the change in the gardeners' costs to the change in the painter's profit. If the painting adds more to the gardener's costs than it does to the painter's profits, they will be able to make a deal to eliminate that painting.) [input]

BlackTom AI · Accepted Answer

Here's how to reason through the bargaining scenario step by step, using the data in the table and the bargaining conditions.
First, note the numbers from the table: when there are 0 paintings, profits to the painter are 0 and the gardener's total cost is 0. At 1 painting, painter earns 18 and gardener's cost is 8. At 2 paintings, painter earns 32 and gardener's cost is 16. At 3 paintings, painter earns 42 and gardener's cost is 24. At 4 paintings, painter earns 48 and gardener's cost is 32. At 5 paintings, painter earns 50 and gardener's cost is 40.
Next, focus on the incremental changes between successive numbers of paintings. From 0 to 1 painting, the painter’s profit rises by 18 and the gardener’s cost rises by 8. From 1 to 2 paintings, the painter gains 14 more (32 − 18) while the gardener’s cost rises by 8 (16 − 8). From 2 to 3 paintings, the painter gains 10 more (42 − 32) and the gardener’s cost rises by 8 (24 − 16). From 3 to 4 paintings, the painter gains 6 more (48 − 42) while the gardener’s cost rises by 8 (32 − 24). From 4 to 5 paintings, the painter gains 2 more (50 − 48) and the gardener’s cost rises by 8 (40 − 32).
Now apply the hint about bargaining: if adding another painting increases the gardener’s costs by more than it increases the painter’s profits, there is room for a deal to eliminate that painting. Compare gardener’s incremental cost to painter’s incremental profit for each step.
- For 0→1: gardener +8 vs painter +18. Gardener’s increase is smaller, so no automatic case to eliminate yet.
- For 1→2: gardener +8 vs painter +14. Again, gardener’s increase is smaller, so eliminating the 2nd painting isn’t forced by this rule.
- For 2→3: gardener +8 vs painter +10. Gardener’s increase is smaller, still not a forcing case.
- For 3→4: gardener +8 vs painter +6. Here, the gardener’s increase (8) exceeds the painter’s increase (6). This makes it feasible for the two to bargain to eliminate the 4th painting, since the gardener would save 8 in costs while the painter would lose only 6 in profit, creating an incentive for a trade that benefits both.
- For 4→5: gardener +8 vs painter +2. The gardener’s increase again exceeds the painter’s, suggesting another potential bargaining point to eliminate the 5th painting as well.
Consider the stated bargaining constraints: the gardener is willing to pay up to 8 to reduce the number of paintings, while the painter requires at least 2 to be convinced not to paint the 5th painting. If they agree to eliminate the 5th painting, the gardener would save 8 in costs and would pay the painter an amount between 2 and 8 to secure the deal. The painter, in turn, would accept since he would lose only 2 (his minimal acceptable payment) relative to the 0→5 outcome, but he would receive compensation from the gardener reaching at least his threshold. The net effect is favorable to both sides when eliminating the 5th painting.
Since the analysis shows a valid mutual gain path by eliminating the 5th painting (and the same mutual-gain logic applies to eliminating earlier paintings but the given constraints specify the 5th as the focal point with explicit payoffs), the likely negotiated outcome is to settle at 4 paintings.
In short, the bargaining condition is satisfied: the extra gardener cost from adding the 5th painting (and similarly for the 4th, in the broader view) can be offset by a payment from the gardener to the painter that meets the painter’s minimum requirement, producing a deal to reduce the number to 4 paintings.
Therefore, the number of paintings that will be painted in the negotiated outcome is 4.

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