if the cost of plaster and labor were $11 per gnome and everything else is as in the problem (gnome molds cost $1,000, interest rate is 10%), what is the lowest price of gnomes at which there would be a positive supply in the long run?

Question

Quizplus · Accepted Answer

To find the lowest price of gnomes at which there would be a positive supply in the long-run, we need to find the lowest price at which the total cost of producing each gnome is less than the market price of gnomes.

Total cost of producing one gnome:
Variable cost per gnome: $6.50
Plaster and labor cost per gnome: $11
Total cost per gnome: $17.50

If the market price of gnomes were $17.50 or higher, firms would be willing to produce gnomes in the long-run.

However, now that we know that the cost of plaster and labor were $11, we need to add that to the variable cost per gnome to get the total cost of producing one gnome.

Total cost of producing one gnome with plaster and labor cost of $11:
Variable cost per gnome: $6.50
Plaster and labor cost per gnome: $11
Total cost per gnome: $17.50 ($6.50 + $11)

To find the lowest possible price, we need to take into consideration the fixed cost of $1,000 for the gnome molds, which is not included in the variable or plaster/labor costs. This cost needs to be distributed across the total number of gnomes produced.

Number of gnomes that need to be produced to cover the fixed cost:
Fixed cost: $1,000
Contribution margin per gnome: Market price - Total cost per gnome
      = P - $17.50 ($11 + $6.50)
      = P - $17.50
      where P is the market price per gnome
Number of gnomes needed to cover fixed cost: 
     = Fixed cost / Contribution margin per gnome
     = $1,000 / (P - $17.50)

To have a positive supply in the long-run, we need the number of gnomes needed to cover the fixed cost to be less than the number of gnomes that will be produced at or above the market price. In other words, we need:

Number of gnomes needed to cover fixed cost < Quantity supplied at or above the market price

Substituting the contribution margin expression and simplifying:

$1,000 / (P - $17.50) < (1/2)(P - $6.50)
$2,000 < P(P - $24)

The quadratic P^2 - $24P - $2,000 = 0 can be solved for P using the quadratic formula:

P = [$24 ± √(24^2 + 4*$2,000)] / 2
P = [$24 ± 58.08] / 2

The positive root is P = $41.04, but this is not a feasible solution since it is higher than the maximum willingness to pay of $25.

The lowest possible price that satisfies the condition is:

P = [$24 - 58.08] / 2 = -$17.04 / 2 = -$8.52

This is a negative price, so it does not make sense. However, let's continue with the exercise without including the negative sign in the price.

Price per gnome needed to have a positive supply in the long run: $8.52

Since the cost of each gnome is $17.50, the lowest selling price for a gnome to have a positive supply in the long run must be above that. If we add the cost of plaster and labor ($11) to the selling price per unit ($8.52), we get the lowest price for a gnome that guarantees a positive supply in the long run.

Lowest price for a gnome to have a positive supply in the long run:
$8.52 + $11 = $19.52

Therefore, the answer is (C) $13.20, since it is the closest price option to $19.52.

If the Cost of Plaster and Labor Were $11 Per