Suppose the demand and supply of firearms in Utopia are given as follows:

-----------------------------------

Demand: *P* = 93 − 53*Q*

Supply: *P* = 63

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in which *Q* is the quantity of firearms in million units and *P* is the price per each firearm in Utopian dollars. Suppose the government raises an amount of 126 million Utopian dollars to buy the firearms back from the market. After the government enters the market, on the new demand curve, when the price is 95 Utopian dollars, the total quantity demanded is **[ Answer ]** million units of firearms.

We are given the following information:

- The initial **demand** function for firearms in Utopia:  
  \[
  P = 93 - 53Q
  \]
- The **supply** function for firearms in Utopia:  
  \[
  P = 63
  \]
  
- The government raises 126 million Utopian dollars to buy back firearms from the market.
- We are tasked to find the **new quantity demanded** when the price is 95 Utopian dollars after the government intervention.

### Step 1: Find the initial market equilibrium.

At equilibrium, the **demand** equals the **supply**, so we can set the demand and supply equations equal to each other.

\[
93 - 53Q = 63
\]

Solving for \( Q \):

\[
93 - 63 = 53Q
\]
\[
30 = 53Q
\]
\[
Q = \frac{30}{53} \approx 0.566
\]

Thus, the initial **equilibrium quantity** is approximately 0.566 million units of firearms, and the equilibrium price is 63 Utopian dollars (from the supply equation).

### Step 2: Analyze the government's buyback intervention.

The government spends 126 million Utopian dollars to buy firearms. To find out how many firearms the government buys back, we need to determine how many firearms the government can purchase at the equilibrium price of 63 Utopian dollars per firearm:

\[
\text{Firearms bought by the government} = \frac{126 \text{ million dollars}}{63 \text{ dollars per firearm}} = 2 \text{ million firearms}
\]

This means the government buys 2 million firearms from the market.

### Step 3: Adjust the demand curve.

Since the government has bought back 2 million firearms, the **new demand curve** will have 2 million fewer firearms in the market. Thus, the new quantity \( Q' \) demanded is:

\[
Q' = Q - 2
\]

Therefore, the new demand equation is:

\[
P = 93 - 53(Q' + 2)
\]
Simplifying this:

\[
P = 93 - 53Q' - 106
\]
\[
P = -53Q' - 13
\]

Thus, the **new demand equation** after the government's buyback is:

\[
P = -53Q' - 13
\]

### Step 4: Find the new quantity demanded when the price is 95.

We are now asked to find the quantity demanded \( Q' \) when the price \( P \) is 95 Utopian dollars under the new demand curve. Substituting \( P = 95 \) into the new demand equation:

\[
95 = -53Q' - 13
\]
Solving for \( Q' \):

\[
95 + 13 = -53Q'
\]
\[
108 = -53Q'
\]
\[
Q' = \frac{108}{-53} \approx -2.04
\]

The **new quantity demanded** is approximately **−2.04 million units**. Since this result is negative, it implies that at a price of 95 Utopian dollars, there is no demand for firearms (i.e., the demand becomes zero).

---

Would you like further clarification or have any additional questions?

### 5 Related Questions:
1. How would the market equilibrium change if the supply function were also altered by government intervention?
2. What would happen if the government implemented a price floor at 95 Utopian dollars?
3. How does a government buyback affect consumer and producer surplus in this context?
4. Could you explain how taxation or subsidies would shift the demand or supply curves?
5. How is price elasticity of demand affected by such government interventions?

**Tip:** When solving supply and demand problems, always check for equilibrium conditions before and after market interventions for a clearer comparison.

Suppose the demand and supply of firearms in Utopia are given as follows: Demand: P = 93 − 53Q Supply: P = 63, in which Q is the quantity of firearms in million units and P is the price per each firearm in Utopian dollars. Suppose the government raises an amount of 126 million Utopian dollars to buy the firearms back from the market. After the government enters the market, on the new demand curve, when the price is 95 Utopian dollars, the total quantity demanded is [ Answer ] million units of firearms.

Explore the solution to a demand and supply problem where the government intervenes by buying back firearms. Learn how to adjust the demand curve and calculate the new quantity demanded at a price of 95 Utopian dollars after the buyback. This problem covers algebraic manipulation of linear demand and supply functions.

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