20. Supply and Demand Let the supply and demand functions
for butter pecan ice cream be given by
p=s (q) = 2/5q and p= d (q) = 100 – 2/5 q
where p is the price in dollars and q is the number of 10-
gallon tubs.
a. Graph these on the same axes.
b. Find the equilibrium quantity and the equilibrium
price.
a. To graph the supply and demand functions for butter pecan ice cream, we need to plot the price (p) on the vertical axis and the quantity (q) on the horizontal axis.
For the supply function, p = s(q) = 2/5q, we can choose some values for q and calculate the corresponding values of p:
- When q = 0, p = s(0) = 2/5(0) = 0
- When q = 10, p = s(10) = 2/5(10) = 4
- When q = 20, p = s(20) = 2/5(20) = 8
Plotting these points on the graph, we can draw a line connecting them. This represents the supply function.
For the demand function, p = d(q) = 100 - 2/5q, we can do the same:
- When q = 0, p = d(0) = 100 - 2/5(0) = 100
- When q = 10, p = d(10) = 100 - 2/5(10) = 96
- When q = 20, p = d(20) = 100 - 2/5(20) = 92
Plotting these points and connecting them with a line will represent the demand function.
b. The equilibrium quantity is the quantity at which the supply and demand functions intersect. To find this, we set the supply and demand functions equal to each other:
2/5q = 100 - 2/5q
Simplifying this equation, we have:
4/5q = 100
Multiplying both sides by 5/4 to isolate q, we get:
q = (5/4) * 100 = 125
So the equilibrium quantity is 125 10-gallon tubs.
To find the equilibrium price, we substitute this value of q back into either the supply or demand function. Let's use the demand function:
p = d(125) = 100 - 2/5(125) = 100 - 50 = 50
Therefore, the equilibrium price is $50.