The demand and supply curves for titanium are given by
P = 100 - Q/2
and
P = 2Q
respectively.
The government, fearful that a titanium shortage could jeopardize national security, imposes a tax of $20/oz. on the retail price of this rare metal. It collects the tax from titanium 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?
Yes, Q = 40.
To find the equilibrium quantity, first, you need to find the equilibrium price before the tax was imposed. To find the equilibrium price, you need to set the demand equation equal to the supply equation:
100 - Q/2 = 2Q
Now, solve for Q.
Add Q/2 to both sides:
100 = 2Q + Q/2
Multiply both sides by 2 to get rid of the fraction:
200 = 4Q + Q
Combine like terms:
200 = 5Q
Now, solve for Q:
Q = 40
So, the equilibrium quantity before the tax is 40.
Now let's find the price paid by the buyer, which should include the tax. Since the government imposes a tax of $20/oz, we need to add the tax to the initial price. So, we take the demand equation and add the tax:
P = (100 - Q/2) + 20
Now we plug in the equilibrium quantity, Q = 40:
P = (100 - 40/2) + 20
P = (100 - 20) + 20
P = 100
So, the price paid by the buyer is $100/oz.
Now, let's find the price received by the seller net of the tax. To do this, we take the initial supply equation and subtract the tax:
P = 2Q - 20
Now we plug in the equilibrium quantity, Q = 40:
P = 2(40) - 20
P = 80 - 20
P = 60
So, the price received by the seller net of the tax is $60/oz.