The handmade snuffbox industry is composed of 100 identical firms, each having short – run total costs given by STC = 0.5q2 + 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 = 1,100 – 50P. What will be the equilibrium in this marketplace? What will each firm’s total short – run profits be?
c) Graph the market equilibrium and compute total short – run producer surplus in this case.
d) Show that the total producer surplus you calculated in part c is equal to industry profits plus industry short – run total fixed costs.
e) Suppose the government imposed a 3$ tax on snuffboxes in the industry described above.
i) How would this tax change the market equilibrium.
ii) How would the burden of this tax be shared between snuffbox buyers and sellers?
iii) Calculate the total loss of producer surplus as a result of the taxation of snuffboxes. Show that this loss equals the change in total short – run profits in the snuffbox industry. Why do not fixed costs enter into computation of the change in the short-run producer surplus?
Special
a) The short-run supply curve for each snuffbox maker is determined by the firms' marginal costs. Since the short-run marginal cost (SMC) for each firm is given by SMC = q + 10, the supply curve for each firm is given by the equation q = SMC - 10.
To find the short-run supply curve for the market as a whole, we sum up the quantities supplied by each firm at each price level. Since there are 100 identical firms, the market supply curve would be the sum of the individual firm supply curves. Therefore, the market supply curve is given by the equation Q = 100(q) = 100(SMC - 10).
b) To find the equilibrium in this marketplace, we need to set the quantity supplied equal to the quantity demanded and solve for the price. The quantity demanded is given by Q = 1,100 - 50P. Equating the quantity demanded and supplied, we have:
100(SMC - 10) = 1,100 - 50P
Simplifying the equation, we get:
100q = 1,100 - 50P + 1,000
100q = 2,100 - 50P
Dividing both sides by 100, we have:
q = 21 - 0.5P
Substituting this into the equation for SMC, we get:
q = SMC - 10
21 - 0.5P = SMC - 10
Solving for P, we find:
0.5P = 21 - SMC
P = 42 - 2SMC
To find each firm's total short-run profits, we need to subtract the total costs from the total revenue. Total costs are given by STC = 0.5q^2 + 10q + 5, and total revenue is given by TR = Pq.
Total profits = TR - TC
= Pq - (0.5q^2 + 10q + 5)
= Pq - 0.5q^2 - 10q - 5
c) To graph the market equilibrium, we plot the demand curve Q = 1,100 - 50P and the market supply curve Q = 100(SMC - 10). The equilibrium is where the two curves intersect. The total short-run producer surplus is the area between the supply curve and the equilibrium price.
d) To show that the total producer surplus is equal to industry profits plus industry short-run total fixed costs, we need to calculate the producer surplus. The producer surplus is the sum of the individual firm profits, which can be calculated as follows:
Individual firm profit = Pq - (0.5q^2 + 10q + 5)
Industry profits = 100 * (Pq - 0.5q^2 - 10q - 5)
Industry short-run total fixed costs = 100 * (0.5q^2 + 10q + 5)
Total producer surplus = Industry profits + Industry short-run total fixed costs
e) i) If the government imposed a $3 tax on snuffboxes, the new market equilibrium would be affected. The tax is a cost that sellers have to bear, so it would increase the short-run total costs for each firm. This would shift the supply curve upward by $3.
ii) The burden of the tax would be shared between snuffbox buyers and sellers. The buyers would have to pay a higher price due to the tax, while the sellers would receive a lower price after paying the tax. The extent to which each side bears the burden depends on the elasticity of demand and supply.
iii) To calculate the total loss of producer surplus as a result of the taxation of snuffboxes, we need to calculate the change in short-run total profits. This can be calculated as the difference between the new equilibrium industry profits and the original industry profits without the tax. The change in total short-run profits is equal to the total loss of producer surplus. Fixed costs do not enter into the computation of the change in short-run producer surplus because they do not change with the level of output. The tax only affects the variable costs, such as marginal and average costs.
a) The short-run supply curve for each snuffbox maker can be found by equating marginal cost (SMC) with the market price. In this case, the market price is the same as the price each firm can sell their snuffboxes for, so we can use the demand equation given to determine the market price. Therefore, the short-run supply curve for each snuffbox maker is given by setting SMC equal to the price.
The short-run supply curve for the market as a whole can be obtained by summing up the quantities supplied by each individual firm at different price levels. Since all firms are identical, they will have the same SMC and thus the same supply curve. Therefore, the short-run supply curve for the market as a whole is the horizontal summation of the individual firm's supply curves.
b) To find the equilibrium in this marketplace, we need to find the price at which the quantity demanded equals the quantity supplied. By setting the demand equation equal to the total quantity supplied by all firms, we can solve for the equilibrium price (P).
Q (Total quantity supplied) = q (quantity supplied by one firm) * Number of firms
Q = (q + 10) * 100
Equating the quantity demanded (Q) with the quantity supplied, we have:
Q = 1,100 - 50P
Setting these two equations equal to each other, we can solve for the equilibrium price:
(q + 10) * 100 = 1,100 - 50P
100q + 1000 = 1,100 - 50P
100q = 100 - 50P
q = 1 - 0.5P
To find each firm's total short-run profit, we need to subtract the total variable cost (TVC) from the total revenue (TR). Total variable cost is equal to the short-run total cost (STC) minus the short-run total fixed cost (TFC). In this case, the total fixed cost is given as 5.
TR = P * q
TVC = STC - TFC = 0.5q^2 + 10q
Total short-run profit = TR - TVC
c) To graph the market equilibrium, we plot the quantity supplied by all firms (Q) on the x-axis and the price (P) on the y-axis. The market equilibrium occurs where the quantity supplied (Q) is equal to the quantity demanded.
To calculate the total short-run producer surplus, we first find the individual producer surplus for each firm by subtracting the average variable cost (AVC) from the price. Producer surplus for each firm is given by:
Producer surplus = P - AVC = P - (STC - TFC)/q
Then, we sum up the individual producer surpluses of all firms to get the total short-run producer surplus.
d) To show that the total producer surplus calculated in part c is equal to industry profits plus industry short-run total fixed costs, we need to add up the individual short-run profits of all firms, which we calculated in part b.
Total producer surplus = Total short-run profits of all firms + Industry short-run total fixed costs
e) To analyze the effects of a $3 tax on snuffboxes, we need to consider how it would change the market equilibrium.
i) The tax would increase the cost of production for each firm. Since the short-run marginal cost (SMC) formula includes the cost of production, it would shift the supply curve upward. This means that firms would have to charge a higher price to cover the additional cost imposed by the tax. Therefore, the market equilibrium price would increase.
ii) The burden of the tax would be shared between snuffbox buyers and sellers. The tax can be thought of as an additional cost for the sellers, which means they would be willing to sell less at a given price. The quantity supplied would decrease, leading to a higher equilibrium price. At the new equilibrium price, buyers would be willing to purchase less, resulting in a decrease in the quantity demanded.
iii) To calculate the total loss of producer surplus as a result of the taxation, we compare the total short-run producer surplus before and after the tax. The loss in producer surplus is the difference between the two.
The change in total short-run profits in the snuffbox industry can be calculated by subtracting the new total variable cost (including the tax) from the total revenue. This change in total profits should be equal to the total loss of producer surplus.
Fixed costs do not enter into the computation of the change in short-run producer surplus because they are not affected by changes in quantity or price. They are costs that the firms have to incur regardless of production levels. Therefore, only the variable costs are considered in calculating the change in short-run producer surplus.