Find the consumer surplus at the equilibrium point.(I keep using the formula listed below but get a completely different answer from what the book says. I am a visual person so please show me step by step so I can see where I went wrong.
D(x)=(x-6)^2 ; x=3
Formula: D(x)-QP
The Answer according to the book is $36
To find the consumer surplus at the equilibrium point, we need to first understand what consumer surplus represents. Consumer surplus is the difference between what consumers are willing to pay for a product and what they actually pay.
In this case, the demand function is given by D(x) = (x-6)^2, and the equilibrium point is x = 3.
To visualize the problem, let's plot the demand curve and indicate the equilibrium point:
1. Plot the demand curve:
To plot the demand curve, we need to evaluate the demand function D(x) for different values of x. We'll start by evaluating it for x = 0, 1, 2, 3, and 4:
D(0) = (0-6)^2 = 36
D(1) = (1-6)^2 = 25
D(2) = (2-6)^2 = 16
D(3) = (3-6)^2 = 9
D(4) = (4-6)^2 = 4
Plotting these points on a graph, we get a downward-sloping parabolic curve.
2. Indicate the equilibrium point:
The equilibrium point is given as x = 3. Mark this point on the graph.
3. Calculate the consumer surplus:
Consumer surplus can be calculated as the area under the demand curve and above the price line (QP).
In this case, the price is represented by the x-coordinate at the equilibrium point, which is x = 3. Thus, the price line is a horizontal line at y = 3.
To calculate the consumer surplus, we need to find the area under the demand curve and above the price line between the equilibrium point (x = 3) and the y-axis (demand function starts at x = 6, so the x-intercept is 6).
To find the area, we can use the formula for the area of a triangle:
Area = (base * height) / 2
- Base: The distance between x = 3 and x = 6 is 6 - 3 = 3.
- Height: The height is the difference between the demand function and the price line at any given x value. At x = 3, the demand function is D(3) = (3-6)^2 = 9, and the price line is y = 3. Thus, the height is 9 - 3 = 6.
Plugging these values into the formula, we get:
Area = (3 * 6) / 2 = 9
Therefore, the consumer surplus at the equilibrium point is $9.
It appears that the answer provided in the book ($36) may be incorrect or there may be some misunderstanding. Make sure to follow the steps explained above to double-check your calculations.