The demand and supply curves for bricks are given by
P = 100 - Q/2
and
P = 2Q
respectively.
The government, fearful that a bricks have shortage could jeopardize national security, imposes a tax of $20/oz. on the retail price of this rare metal. It collects the tax from bricks sellers. What is the equilibrium quantity in the market? What is the price paid by the buyer? The price received by the seller net of the tax?
I do not understand the equations, how to manipulate then, and how I end with the tax. Can someone please show the work?
Is Q = 40?
To find the equilibrium quantity in the market, we need to set the demand curve equal to the supply curve before the tax is implemented. That is:
100 - Q/2 = 2Q
Now we need to solve for Q:
100 = Q/2 + 2Q
100 = Q/2 + 4Q/2
100 = 5Q/2
Q = 100 * 2/5 = 40
So yes, the equilibrium quantity in the market is Q = 40.
Now we need to find the equilibrium price before the tax is implemented. Plug the equilibrium quantity into either the demand or supply equation:
P = 100 - Q/2
P = 100 - 40/2
P = 100 - 20 = $80
Now let's factor in the $20 tax. Since the government collects the tax from brick sellers, we will add it to the supply curve (since it represents the cost for sellers):
P = 2Q + 20
Set the new supply curve equal to the demand curve to find the new equilibrium:
100 - Q/2 = 2Q + 20
Solve for Q:
80 = Q/2 + 2Q
80 = Q/2 + 4Q/2
80 = 5Q/2
Q = 80 * 2/5 = 32
The new equilibrium quantity in the market is Q = 32.
Now let's find the price paid by the buyer by plugging the new equilibrium quantity into the new supply equation:
P = 2Q + 20
P = 2 * 32 + 20
P = 64 + 20
P = $84
So the price paid by the buyer is $84.
Finally, let's find the price received by the seller net of the tax. We need to calculate the price without the tax:
P = 2Q
P = 2 * 32
P = $64
Since the tax is $20, we subtract this from the price the seller receives:
$64 - $20 = $44
So the price received by the seller net of the tax is $44.