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?
Sure, let's solve it step by step.
1. First, let's find the equilibrium quantity and price without the tax. To do this, we set the demand and supply curves equal to each other:
100 - Q/2 = 2Q
2. Now, let's solve for Q:
Add Q/2 to both sides: 100 = Q/2 + 2Q
Multiply both sides by 2 to get rid of the fraction: 200 = Q + 4Q
Combine like terms: 200 = 5Q
Divide by 5: Q = 40
We found that the equilibrium quantity (Q) is 40. Now, let's find the equilibrium price (P) by plugging Q back into either the demand or supply curve. Let's use the demand curve:
P = 100 - Q/2 = 100 - 40/2 = 100 - 20 = 80
So the equilibrium price (P) without the tax is $80/oz.
3. Now let's consider the tax, which is $20/oz. This tax is collected from the sellers, so it affects the supply curve. The new supply curve will be:
P = 2Q + Tax = 2Q + 20
4. Find the new equilibrium quantity and price after the tax. Set the demand curve equal to the new supply curve:
100 - Q/2 = 2Q + 20
5. Solve for Q (with tax):
Add Q/2 to both sides: 100 = Q/2 + 2Q + 20
Subtract 20 from both sides: 80 = Q/2 + 2Q
Multiply both sides by 2 to get rid of the fraction: 160 = Q + 4Q
Combine like terms: 160 = 5Q
Divide by 5: Q = 32
So the new equilibrium quantity (Q) with the tax is 32. Now, let's find the price paid by the buyer (P) by plugging Q back into the demand curve:
P = 100 - Q/2 = 100 - 32/2 = 100 - 16 = 84
The price paid by the buyer is $84/oz.
6. Finally, let's find the price received by the seller net of the tax. We can use the new supply curve for this:
P = 2Q + 20 => P - 20 = 2Q => (84 - 20) = 2(32) => 64 = 2(32)
So the price received by the seller net of the tax is $64/oz.
To summarize:
- The equilibrium quantity with the tax is 32 oz
- The price paid by the buyer is $84/oz
- The price received by the seller net of the tax is $64/oz