The market equilibrium point for a product occurs when 13500 units are produced at a price of $45 per unit. The producer will supply no units at $10 and the consumers will demand no units at $200. Find the supply and demand equations if they are both linear.
demand
d = k p + b
13500 = k (45) + b
0 = k (200) + b
b = -200 k
13500 = 45 k - 200 k = - 155
k = - 87.1
so the demand function is
d = -87.1 p +17400
now you do the supply function
s = j p + c
0 = 10 j + c
so c = -10 j
13500 = 45 j - 10 j = 35 j
etc, all yours
Un-understandable
To find the supply and demand equations, we can use the given points on the supply and demand curves.
Given:
- At an equilibrium price of $45, the quantity demanded is 13,500 units.
- At a price of $10, the quantity supplied is zero.
- At a price of $200, the quantity demanded is zero.
To find the supply equation, we need to find the equation of a straight line that passes through the points ($10, 0) and ($45, 13,500). Let's assign (x, y) as (price, quantity supplied).
Using the slope-intercept form of a linear equation (y = mx + b), we can determine the equation for supply.
First, let's find the slope (m):
m = (change in y) / (change in x)
= (13,500 - 0) / (45 - 10)
= 13,500 / 35
= 386
Now, write the equation using the slope (m) and one point (x, y):
y = mx + b
0 = 386(10) + b
0 = 3860 + b
b = -3860
Therefore, the supply equation is:
Supply: y = 386x - 3860
To find the demand equation, we need to find the equation of a straight line that passes through the points ($45, 13,500) and ($200, 0). Let's assign (x, y) as (price, quantity demanded).
Using the slope-intercept form of a linear equation (y = mx + b), we can determine the equation for demand.
First, let's find the slope (m):
m = (change in y) / (change in x)
= (0 - 13,500) / (200 - 45)
= -13,500 / 155
= -87.097
Now, write the equation using the slope (m) and one point (x, y):
y = mx + b
0 = -87.097(200) + b
0 = -17,419.4 + b
b = 17,419.4
Therefore, the demand equation is:
Demand: y = -87.097x + 17,419.4
To find the supply and demand equations, we need to determine the slope and intercept for each equation. Let's work through the steps:
1. Determine the slope of the demand equation:
We know that the consumers demand no units at $200 and 13,500 units at $45. To find the slope, we use the formula: slope = (change in quantity demanded) / (change in price).
The change in quantity demanded is: 13,500 - 0 = 13,500
The change in price is: $45 - $200 = -$155 (note the negative sign indicates a decrease in price)
So, the slope of the demand equation is: 13,500 / -155 = -87.0968 (rounded to four decimal places).
2. Determine the intercept of the demand equation:
We have the price ($45) and the quantity demanded (13,500) at the market equilibrium point. We can use these values in the demand equation y = mx + b, where y is the quantity demanded and x is the price. Plugging in the values, we have:
13,500 = -87.0968 * 45 + b
13,500 = -3919.3544 + b
b = 13,500 + 3919.3544
b = 17,419.3544
Therefore, the demand equation is: quantity demanded = -87.0968 * price + 17,419.3544.
3. Determine the slope of the supply equation:
We know that the producer supplies no units at $10 and 13,500 units at $45. To find the slope, we use the formula: slope = (change in quantity supplied) / (change in price).
The change in quantity supplied is: 13,500 - 0 = 13,500
The change in price is: $45 - $10 = $35
So, the slope of the supply equation is: 13,500 / 35 = 385.7143 (rounded to four decimal places).
4. Determine the intercept of the supply equation:
We have the price ($45) and the quantity supplied (13,500) at the market equilibrium point. We can use these values in the supply equation y = mx + b, where y is the quantity supplied and x is the price. Plugging in the values, we have:
13,500 = 385.7143 * 45 + b
13,500 = 17,357.1429 + b
b = 13,500 - 17,357.1429
b = -3,857.1429
Therefore, the supply equation is: quantity supplied = 385.7143 * price - 3,857.1429.
To summarize, the demand equation is: quantity demanded = -87.0968 * price + 17,419.3544, and the supply equation is: quantity supplied = 385.7143 * price - 3,857.1429.