The demand and supply curves for bricks are given by

Question

The demand and supply curves for bricks are given by

P = 100 - Q/2

and

P = 2Q

respectively.

The government, fearful that a bricks 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?

Answer 1

First, we need to find the equilibrium quantity and price without the tax. To do this, set the demand and supply equations equal to each other:

100 - Q/2 = 2Q

Now, solve for Q:

100 = Q/2 + 2Q
100 = (1/2 Q) + (4/2 Q)
100 = (5/2 Q)
Q = (2/5)*100
Q = 40

Now that we know the equilibrium quantity is 40, we can find the equilibrium price without the tax by plugging Q back into either the demand or supply equation (I'll use the supply equation):

P = 2Q
P = 2 * 40
P = 80

Now, let's consider the tax. The government collects a $20 tax from the sellers, so the price the sellers receive will be the price paid by the buyer minus the tax. We can represent this in a new supply equation:

P_seller = 2Q - 20

Now we set the demand equation equal to the new supply equation and solve for the new equilibrium quantity:

100 - Q/2 = 2Q - 20

Solve for Q:

120 - Q/2 = 2Q
120 = Q/2 + 4Q
120 = (1/2 Q) + (8/2 Q)
120 = (9/2 Q)
Q = (2/9)*120
Q = 40/3 ≈ 13.33

Now we can find the price paid by the buyer and the price received by the seller. Plug the new equilibrium quantity into the demand and modified supply equations:

Price paid by the buyer (demand equation):
P_buyer = 100 - Q/2
P_buyer = 100 - (13.33/2)
P_buyer ≈ 93.33

Price received by the seller net of tax (modified supply equation):
P_seller = 2Q - 20
P_seller = 2*13.33 - 20
P_seller ≈ 6.67

So, with the tax, the equilibrium quantity in the market is approximately 13.33. The price paid by the buyer is approximately $93.33, and the price received by the seller net of the tax is approximately $6.67.