𝑄𝐷 = 20 − 2𝑃
𝑄𝑆 = 4𝑃 − 10
a. Graph the supply and demand curves. Find the equilibrium price and quantity (Please
work out the equilibrium price and quantity).
b. Unlike the scenario we discussed in class, now the government legislates a $1 tax to
be collected from producers. Find a new market quantity. Also, find the price
consumers pay, and the price producers actually receive.
c. How much the tax revenue did the government raise?
d. Does the price consumers pay or producers receive depend on whether the seller or
buyer pays the tax in this example? If they are the same, what affects the tax
incidence?


Let's break down the problem step by step.

### Part a: Graph the supply and demand curves, and find the equilibrium price and quantity.

The demand function (Qᴰ) is given as:
\[
Q_D = 20 - 2P
\]
The supply function (Qˢ) is given as:
\[
Q_S = 4P - 10
\]

#### Equilibrium Price and Quantity:
At equilibrium, the quantity demanded equals the quantity supplied, meaning $Q_D = Q_S$. Let's solve for the equilibrium price $P_{eq}$.

\[
20 - 2P = 4P - 10
\]

1. Combine like terms:
\[
20 + 10 = 4P + 2P
\]
\[
30 = 6P
\]

2. Solve for $P$:
\[
P = \frac{30}{6} = 5
\]

The equilibrium price is $P_{eq} = 5$.

Now, substitute $P = 5$ into either the demand or supply equation to find the equilibrium quantity $Q_{eq}$.

Using the demand equation:
\[
Q_D = 20 - 2(5) = 20 - 10 = 10
\]

Thus, the equilibrium quantity is $Q_{eq} = 10$.

#### Graph:
- The demand curve is downward sloping and can be plotted as a straight line from $P = 0, Q = 20$ to $P = 10, Q = 0$.
- The supply curve is upward sloping and can be plotted as a straight line from $P = 0, Q = -10$ to a positive intercept at higher prices.

### Part b: Tax imposition and its effect on price and quantity

Now, a $1 tax is imposed on producers, meaning the effective price producers receive is $P_S = P - 1$, where $P$ is the price consumers pay. This modifies the supply curve.

New supply function:
\[
Q_S = 4(P_S) - 10 = 4(P - 1) - 10 = 4P - 4 - 10 = 4P - 14
\]

Now, find the new equilibrium by setting the modified supply curve equal to the demand curve:
\[
20 - 2P = 4P - 14
\]

1. Combine like terms:
\[
20 + 14 = 4P + 2P
\]
\[
34 = 6P
\]

2. Solve for $P$:
\[
P = \frac{34}{6} = 5.67
\]

So, the new price consumers pay is $P_C = 5.67$.

To find the price producers actually receive after tax:
\[
P_S = P_C - 1 = 5.67 - 1 = 4.67
\]

Now substitute the consumer price back into the demand or supply function to find the new equilibrium quantity.

Using the demand equation:
\[
Q_D = 20 - 2(5.67) = 20 - 11.34 = 8.66
\]

Thus, the new market quantity is $Q_{new} = 8.66$.

### Part c: Tax Revenue

Tax revenue is calculated as the tax per unit multiplied by the quantity sold.

Tax per unit = $1  
Quantity sold = 8.66 units

Tax revenue = $1 \times 8.66 = 8.66$

So, the government raised $8.66 in tax revenue.

### Part d: Tax Incidence

Does the price consumers pay or producers receive depend on who pays the tax? 

No, in this case, the price consumers pay ($P_C = 5.67$) and the price producers receive ($P_S = 4.67$) do not depend on whether the tax is collected from consumers or producers. The total tax burden is shared between consumers and producers. The **tax incidence** is determined by the relative elasticities of demand and supply. The less elastic side of the market bears more of the tax burden. In this case:

- Consumers bear the burden of the increase in price from $5 to $5.67.
- Producers bear the burden of receiving a price reduction from $5 to $4.67.

### Do you have any questions or want more details on a specific part?

### 5 Related Questions:
1. How does elasticity affect tax incidence?
2. What happens to consumer surplus after the imposition of the tax?
3. How would the graph look with a different tax amount, say $2?
4. Can you compare the effect of the tax on producer surplus vs. consumer surplus?
5. What would happen if the tax was imposed on consumers instead of producers?

### Tip:
The side of the market (demand or supply) that is more inelastic tends to bear more of the tax burden.

𝑄𝐷 = 20 − 2𝑃
𝑄𝑆 = 4𝑃 − 10
a. Graph the supply and demand curves. Find the equilibrium price and quantity (Please work out the equilibrium price and quantity).
b. Unlike the scenario we discussed in class, now the government legislates a $1 tax to be collected from producers. Find a new market quantity. Also, find the price consumers pay, and the price producers actually receive.
c. How much the tax revenue did the government raise?
d. Does the price consumers pay or producers receive depend on whether the seller or buyer pays the tax in this example? If they are the same, what affects the tax incidence?

Graph the supply and demand curves and find the equilibrium price and quantity for the functions Q_D = 20 - 2P and Q_S = 4P - 10. Learn how the $1 tax affects market quantity, consumer price, producer price, and government tax revenue. Explore tax incidence and its relation to elasticity.

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