To calculate the profit from the black market under a price ceiling, we need to follow these steps:

1. Identify the equilibrium price and quantity by equating the demand and supply functions.
2. Check if the price ceiling is effective (i.e., lower than the equilibrium price).
3. If the ceiling is effective, determine the black market price by using the demand curve at the quantity supplied under the ceiling.
4. Calculate the profit in the black market by finding the difference in prices and multiplying by the quantity traded.

Step 1: Equilibrium Price and Quantity

The demand function is: $P_d = 16.5 - 0.025Q$ The supply function is: $P_s = 2.4 + 0.018Q$ At equilibrium, $P_d = P_s$, so: $16.5 - 0.025Q = 2.4 + 0.018Q$ Solving for $Q$, we get: $16.5 - 2.4 = 0.018Q + 0.025Q$ $14.1 = 0.043Q$ $Q = \frac{14.1}{0.043} = 327.91 \, \text{kg}$

Now, substitute $Q = 327.91 \, \text{kg}$ back into either the demand or supply function to find the equilibrium price. Using the supply function: $P_s = 2.4 + 0.018 \times 327.91 = 2.4 + 5.90 = 8.30 \, \text{\$/gram}$

Step 2: Check if the Price Ceiling is Effective

The ceiling price is $P_{\text{ceiling}} = 7.74 \, \text{\$/gram}$, which is lower than the equilibrium price $P_e = 8.30 \, \text{\$/gram}$. Hence, the price ceiling is effective.

Step 3: Determine Quantity Supplied and Quantity Demanded at the Ceiling Price

• At the ceiling price, the quantity supplied is found by using the supply function: $7.74 = 2.4 + 0.018Q_s$ Solving for $Q_s$: $7.74 - 2.4 = 0.018Q_s$ $5.34 = 0.018Q_s$ $Q_s = \frac{5.34}{0.018} = 296.67 \, \text{kg}$

• At the ceiling price, the quantity demanded is found using the demand function: $7.74 = 16.5 - 0.025Q_d$ Solving for $Q_d$: $7.74 - 16.5 = -0.025Q_d$ $-8.76 = -0.025Q_d$ $Q_d = \frac{8.76}{0.025} = 350.40 \, \text{kg}$

Step 4: Determine the Black Market Profit

The quantity traded on the black market will be the difference between the quantity demanded and the quantity supplied: $Q_{\text{black}} = Q_d - Q_s = 350.40 - 296.67 = 53.73 \, \text{kg}$

The black market price will correspond to the price consumers are willing to pay at the quantity supplied, which we find using the demand curve for $Q_s = 296.67 \, \text{kg}$: $P_{\text{black}} = 16.5 - 0.025 \times 296.67 = 16.5 - 7.42 = 9.08 \, \text{\$/gram}$

The profit per gram in the black market is: $P_{\text{profit}} = P_{\text{black}} - P_{\text{ceiling}} = 9.08 - 7.74 = 1.34 \, \text{\$/gram}$

Since $Q_{\text{black}} = 53.73 \, \text{kg}$, the total profit in grams is $53.73 \times 1000 = 53730 \, \text{grams}$. Thus, the total profit is: $\text{Profit} = 53730 \times 1.34 = 72098.20 \, \text{\$}$ In terms of 000$, this is: $\text{Profit} = \frac{72098.20}{1000} = 72.10 \, \text{000\$}$

Conclusion

The profit in the black market is 72.10 thousand dollars.

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Would you like further details on any of these steps? Here are five related questions you could explore:

1. What happens if the price ceiling is above the equilibrium price?
2. How do black market profits change with different price ceilings?
3. What role does elasticity play in determining black market quantities?
4. How would government interventions like rationing affect this market?
5. Can we predict black market behavior with nonlinear demand and supply functions?

Tip: Always check if a price ceiling is binding (i.e., lower than the equilibrium price) before analyzing market effects.