The hand made snuffbox industry is composed of 100 identical firms, each having short-run total costs given by
STC=0.5q^2+10q+5 and short-run marginal costs by SMC=q+10 where q is the output of snuffboxes per day.
a. What is the short-run supply curve for each snuffbox maker? What is the short-run supply curve for the market as a whole?
b. Suppose the demand for total snuffbox production is given by Q=110-50P What is the equilibrium in this marketplace? What is eachfirms total shortrun profit?
a) the short-run supply for each maker is their MC curve, The supply curve for the industry is the sum of the supply curves from each firm dis-regarding the constant, so... supply = 100q+10
b) given your demand equation, there is no equilibrium value with a positive production level.
a. To find the short-run supply curve for each snuffbox maker, we need to compare the marginal cost (SMC) to the market price.
Since the firms are identical, the marginal cost for each firm is SMC = q + 10.
To find the short-run supply curve for the market as a whole, we need to sum up the individual firms' supply at each price level.
Let's assume the market price is P. To find the individual firm's supply, we equate the marginal cost to the market price:
q + 10 = P
Solving for q, we get:
q = P - 10
This equation represents the short-run supply curve for each snuffbox maker.
To find the short-run supply curve for the market as a whole, we sum up the supply of all 100 identical firms:
Q = 100(q)
Substituting q from the previous equation, we get:
Q = 100(P - 10)
This equation represents the short-run supply curve for the market as a whole.
b. The demand for total snuffbox production is given by Q = 110 - 50P. To find the equilibrium in this marketplace, we need to set the quantity supplied (Q) equal to the quantity demanded (Q).
So, we equate the supply curve (Q = 100(P - 10)) to the demand curve (Q = 110 - 50P):
100(P - 10) = 110 - 50P
Simplifying, we get:
100P - 1000 = 110 - 50P
Bringing like terms to one side, we get:
150P = 1110
Dividing both sides by 150, we get:
P = 1110 / 150
P ≈ 7.4
So, the equilibrium price in this marketplace is approximately $7.4 per snuffbox.
To find each firm's total short-run profit, we need to subtract the total cost (STC) from the total revenue, which is given by price (P) multiplied by the quantity produced (q).
Total revenue = P * q = P * (P - 10)
Total cost = STC = 0.5q^2 + 10q + 5
Total profit = Total revenue - Total cost
Substituting the expressions, we get:
Profit = P * (P - 10) - (0.5q^2 + 10q + 5)
Calculating this expression with P ≈ 7.4 for each firm will give us the total short-run profit for each snuffbox maker.
To determine the short-run supply curve for each snuffbox maker, we need to consider the marginal cost (MC) and the minimum average variable cost (AVC). In the short run, firms will only produce if their price is above the minimum AVC.
a. Short-run supply curve for each snuffbox maker:
Since the short-run marginal cost (SMC) is given as q + 10, we can equate it to the minimum average variable cost (AVC) to find the output level where firms will start producing.
AVC = STC/q
AVC = (0.5q^2 + 10q + 5) / q
To find the minimum AVC, we can take the derivative with respect to q and set it equal to zero:
dAVC/dq = (q + 10 - 0.5q^2 - 10q - 5) / q^2 = 0
Simplifying the equation, we get:
-0.5q^2 - 9q + 5 = 0
Using the quadratic formula, we can solve for q:
q = [-(-9) ± √((-9)^2 - 4(-0.5)(5))]/(2*(-0.5))
q = [9 ± √(81 + 10)] / 1
Using the quadratic formula, we get two possible solutions for q:
q1 = [9 + √91] ≈ 12.55
q2 = [9 - √91] ≈ -3.55 (Ignoring since it's not possible in this case)
The short-run supply curve for each snuffbox maker is given by q = 12.55.
b. Short-run supply curve for the market as a whole:
Since there are 100 identical firms, we can multiply the individual firm's output level by the number of firms to obtain the market supply curve.
Market supply curve = 100 * q = 100 * 12.55 = 1255
To find the equilibrium in this marketplace, we need to equate the quantity demanded (Q) with the quantity supplied (QS):
Q = 110 - 50P
QS = 1255
Setting these two equations equal to each other and solving for P, we get:
110 - 50P = 1255
-50P = 1145
P = -22.9 (Ignoring since it's not possible in this case)
Therefore, there is no equilibrium in this marketplace given the demand and supply conditions specified.
To calculate each firm's total short-run profit, we need to consider the market price (P) and the total cost (TC) of producing q units.
Total cost (TC) = STC = 0.5q^2 + 10q + 5
Total profit (π) = Total revenue (TR) - Total cost (TC)
Total revenue (TR) = P * q
Since we haven't found a market equilibrium, we can't calculate the total short-run profit for each firm.