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Difficulty: HardConcept and Calculation of Consumer Surplus

A consumer's inverse demand function for bags of fertilizer in an agricultural zone is given by the equation P=200−10QP = 200 - 10Q, where PP is the price per bag in Naira (₦\text{₦}) and QQ is the quantity of bags demanded. If the prevailing market price of fertilizer is ₦80\text{₦}80 per bag, what is the consumer surplus in Naira (₦\text{₦})?

Answer: 720 ₦

Answer

The consumer surplus derived from purchasing fertilizer at the market price is ₦720.
Consumer surplus is calculated as the area of the region under the linear demand curve and above the market price line. Given P=200−10QP = 200 - 10Q, substituting the market price P=80P = 80 yields Q=12Q = 12. The maximum price PmaxP_{max} at Q=0Q = 0 is ₦200₦200. Using the area formula 12×base×height=12×12×(200−80)=₦720\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times (200 - 80) = ₦720.

Step-by-Step Solution

1
Determine quantity demanded (QQ) at the prevailing market price (P=80P = 80)
Q=12Q = 12 units
Setting P=80P = 80 in the demand equation 80=200−10Q80 = 200 - 10Q yields 10Q=12010Q = 120, giving Q=12Q = 12.
2
Find maximum willingness to pay (PmaxP_{max}) when quantity demanded is zero (Q=0Q = 0)
Pmax=₦200P_{max} = ₦200
The vertical intercept of the linear demand curve represents the maximum price the consumer would consider paying.
3
Calculate the height of the consumer surplus triangle (Pmax−PmarketP_{max} - P_{market})
200−80=₦120200 - 80 = ₦120
This measures the net benefit per unit between maximum willingness to pay and the market price.
4
Calculate the total consumer surplus using the triangular area formula CS=12×Base×HeightCS = \frac{1}{2} \times \text{Base} \times \text{Height}
CS=12×12×120=₦720CS = \frac{1}{2} \times 12 \times 120 = ₦720
For a linear demand curve, consumer surplus is visually and mathematically represented by the triangular area below the demand curve and above the market price.

Key Concept

Calculation of Consumer Surplus using Linear Demand Equations
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Topics

Concept and Calculation of Consumer Surplus

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