The supply function ps(x) for a commodity gives the relation between the selling price and the number of units that manufacturers will produce at that price. For a higher price, manufacturers will produce more units, so ps is an increasing function of x. Let x0 be the amount of the commodity currently produced and let P = ps(x0) be the current price. Some producers would be willing to make and sell the commodity for a lower selling price and are therefore receiving more than their minimal price. The excess is called the producer surplus. An argument similar to that for consumer surplus shows that the surplus is given by the integral x0 P − ps(x) 0 dx.

Question is incomplete

The supply function ps(x) for a commodity gives the relation between the selling price and the number of units that manufacturers will produce at that price. For a higher price, manufacturers will produce more units, so ps is an increasing function of x. Let x0 be the amount of the commodity currently produced and let P = ps(x0) be the current price. Some producers would be willing to make and sell the commodity for a lower selling price and are therefore receiving more than their minimal price. The excess is called the producer surplus. An argument similar to that for consumer surplus shows that the surplus is given by the integral ∫[P − ps(x)]dx

Calculate the producer surplus for the supply function ps(x) = 3 + 0.01x² at the sales level X=10

The producer surplus is = 6.67

Explanation:

Given

Sales level, x = 10.

PS(x) = 3 + 0.01x²

Consumer Surplus = x∫0[P − ps(x)]dx

Calculating PS(x) where x = 10

PS(10) = 3 + 0.01*10²

PS(10) = 3 + 0.01 * 100

PS(10) = 3 + 1

PS(10) = 4

To calculate the producer surplus, we have to integrate x∫0[P − ps(x)]dx

∫[P − ps(x)]dx {0,10} where p = PS(10) = 4

= ∫[4 − (3 + 0.01x²)]dx {0,10}

= ∫[4 − 3 - 0.01x²]dx {0,10}

= ∫[1 - 0.01x²]dx {0,10}

= (x - 0.01x³/3) {0,10}

= (10 - 0.01(10)³/3)

= (10 - 0.01 * 1000/3)

= 10 - 10/3

= 20/3

= 6.67

Hence, the producer surplus is = 6.67

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