Final answer:
To calculate the consumer surplus for the demand function d(x)=800-0.8x at x=400, we find the maximum willingness to pay for the 400th unit using the demand function, then compute the surplus as the area between this price and the market price for 400 units. Assuming a market price equal to the willingness to pay yields zero surplus, but in a realistic scenario with a lower market price, the surplus would be positive.
Step-by-step explanation:
To calculate the consumer surplus, we first need to evaluate the maximum price consumers are willing to pay for the 400th unit, which can be found using the demand function d(x). The demand function given is d(x) = 800 - 0.8x. Plugging in x = 400, we have d(400) = 800 - 0.8(400) = 800 - 320 = 480. This is the maximum price that consumers are willing to pay for the 400th unit.
The consumer surplus is the area above the market price and below the demand curve up to the given quantity. To find this area, we calculate the area under the demand curve from x = 0 to x = 400, and then subtract the area under the price they actually pay, which is a rectangle with the height equal to the market price and the width equal to the quantity.
Assume that the market price is P, and consumers purchase Q units at this price. The consumer surplus can be found with:
Consumer Surplus = ½ x Q x (Max Willingness to Pay - P)
Since the demand function is linear, the maximum willingness to pay for the 400th unit is $480, and we assume that the market price is $480 as well for simplicity (since we're not given the actual market price), the consumer surplus for purchasing 400 units would be:
Consumer Surplus = ½ x 400 x (480 - 480) = 0
However, if the market price was lower than $480, there would be a positive consumer surplus. Without the actual market price, we cannot calculate the exact surplus.