Final answer:
The associated linear demand and supply functions can be found by using the given information about the quantity demanded and supplied at different prices. The linear demand function is Qd = (-1/30)P + 31/30, and the linear supply function is Qs = (1/70)P + 19/70. The price at which there is neither a surplus nor a shortage of chias is $16.8 each.
Step-by-step explanation:
The linear demand function can be written as Qd = mP + b, where Qd is the quantity demanded, P is the price, m is the slope of the demand curve, and b is the y-intercept. From the given information, we know that at $1 per chia, 100 chias are sold per week, and at $2 per chia, 70 chias are sold per week. We can use this information to find the slope and y-intercept of the demand function.
At $1 per chia, 100 chias are sold. This gives us the point (100, 1). At $2 per chia, 70 chias are sold. This gives us the point (70, 2). Using these two points, we can find the slope (m) and y-intercept (b) of the demand function using the slope formula:
m = (2 - 1) / (70 - 100) = -1/30
Using the point (100, 1) and the slope (-1/30), we can find the y-intercept (b):
1 = (-1/30)(100) + b
b = 31/30
Therefore, the demand function is Qd = (-1/30)P + 31/30.
The linear supply function can be written as Qs = mP + b, where Qs is the quantity supplied, P is the price, m is the slope of the supply curve, and b is the y-intercept. From the given information, we know that at $1 per chia, 20 chias are supplied per week, and at $2 per chia, 90 chias are supplied per week. We can use this information to find the slope and y-intercept of the supply function.
At $1 per chia, 20 chias are supplied. This gives us the point (20, 1). At $2 per chia, 90 chias are supplied. This gives us the point (90, 2). Using these two points, we can find the slope (m) and y-intercept (b) of the supply function using the slope formula:
m = (2 - 1) / (90 - 20) = 1/70
Using the point (20, 1) and the slope (1/70), we can find the y-intercept (b):
1 = (1/70)(20) + b
b = 19/70
Therefore, the supply function is Qs = (1/70)P + 19/70.
To find the price at which there is neither a surplus nor a shortage of chias, we need to find the equilibrium price. At equilibrium, the quantity demanded is equal to the quantity supplied. Therefore, we can set Qd equal to Qs and solve for P:
Qd = Qs
(-1/30)P + 31/30 = (1/70)P + 19/70
(1/70)P + (1/30)P = 19/70 - 31/30
(5/210)P = -12/30
P = (-12/30) * (210/5)
P = -84/5
P = -16.8
Therefore, the chias should be marked at $16.8 each in order to have neither a surplus nor a shortage of chias.