Consider a market where the demand is pD = 200 – 2Q while the supply is pS = 50 + Q. Calculate the consumers' surplus if the government introduces a price cap at p = 80.
Question
Consider a market where the demand is pD = 200 – 2Q while the supply is pS = 50 + Q. Calculate the consumers' surplus if the government introduces a price cap at p = 80.
Solution
To calculate the consumers' surplus with a price cap, we first need to find the quantity demanded and supplied at the price cap.
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Set the price cap (p=80) equal to the demand equation and solve for Q:
80 = 200 - 2Q 2Q = 200 - 80 2Q = 120 Q = 60
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Set the price cap (p=80) equal to the supply equation and solve for Q:
80 = 50 + Q Q = 80 - 50 Q = 30
Since the quantity supplied (30) is less than the quantity demanded (60), the market is not in equilibrium and there is a shortage. The actual quantity traded will be the lower of the two, which is 30.
- Calculate the consumers' surplus:
The consumers' surplus is the area of the triangle formed by the demand curve, the price axis, and the price line. The base of the triangle is the quantity traded (30), and the height is the difference between the maximum price consumers are willing to pay (found on the demand curve when Q=30) and the price cap.
Find the maximum price consumers are willing to pay:
pD = 200 - 2Q pD = 200 - 2(30) pD = 200 - 60 pD = 140
Calculate the consumers' surplus:
CS = 0.5 * base * height CS = 0.5 * 30 * (140 - 80) CS = 0.5 * 30 * 60 CS = 900
So, the consumers' surplus with a price cap at p = 80 is 900.
Similar Questions
Consider a perfectly competitive market for frozen meals. The demand for frozen meals is given by Pd = 137 - 4Q and the supply of frozen meals is given by Ps = 8 + Q. What is the value of Consumer Surplus?
To find the consumer surplus given the demand function \( P = \frac{100}{Q + 2} \) and the market price \( P = 20 \), follow these steps: 1. **Find the quantity \( Q \) at the market price \( P = 20 \)**: \[ 20 = \frac{100}{Q + 2} \] Solve for \( Q \): \[ 20(Q + 2) = 100 \] \[ 20Q + 40 = 100 \] \[ 20Q = 60 \] \[ Q = 3 \] 2. **Set up the integral for consumer surplus**: Consumer surplus is the area between the demand curve and the market price, from \( Q = 0 \) to \( Q = 3 \): \[ \text{Consumer Surplus} = \int_0^3 \left( \frac{100}{Q + 2} - 20 \right) \, dQ \] 3. **Evaluate the integral**: \[ \int_0^3 \left( \frac{100}{Q + 2} - 20 \right) \, dQ = \int_0^3 \frac{100}{Q + 2} \, dQ - \int_0^3 20 \, dQ \] 4. **Find the antiderivatives**: \[ \int \frac{100}{Q + 2} \, dQ = 100 \ln|Q + 2| + C \] \[ \int 20 \, dQ = 20Q + C \] 5. **Apply the limits of integration**: \[ \left[ 100 \ln|Q + 2| \right]_0^3 - \left[ 20Q \right]_0^3 \] 6. **Evaluate the definite integrals**: \[ \left[ 100 \ln|Q + 2| \right]_0^3 = 100 \ln(5) - 100 \ln(2) \] \[ \left[ 20Q \right]_0^3 = 20(3) - 20(0) = 60 \] 7. **Combine the results**: \[ 100 \ln(5) - 100 \ln(2) - 60 \] \[ 100 (\ln(5) - \ln(2)) - 60 \] \[ 100 \ln\left(\frac{5}{2}\right) - 60 \] 8. **Calculate the numerical value**: \[ 100 \ln\left(\frac{5}{2}\right) - 60 \approx 100 \times 0.9163 - 60 \approx 91.63 - 60 \approx 31.63 \] So, the consumer surplus is approximately \( 31.6291 \). The correct answer is: - \( 31.6291 \)
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