At the start of December 2001, the retail price of a 25kg bag of cornmeal was $10 in Zambia, while by the end of the month, the price had fallen to $6. The result was that one retailer reported an increase in sales from 3 bags/day to 5 bags/day. Assume that the retailer is prepared to sell 18 bags/day at $8 and 12 bags/day at $6. Obtain linear demand supply equations, and compute the retailer's equilibrium price.
To obtain the linear demand and supply equations and compute the retailer's equilibrium price, we need to gather data and analyze the information given. Let's break down the problem step by step:
1. Identify the demand equation:
- The demand equation represents the relationship between the price and the quantity demanded.
- We know that at the start of December, the price of a 25kg bag of cornmeal was $10, and the retailer sold 3 bags per day.
- This data point gives us one point on the demand curve: (10, 3).
2. Identify the supply equation:
- The supply equation represents the relationship between the price and the quantity supplied.
- We know that the retailer was prepared to sell 18 bags per day at $8 and 12 bags per day at $6.
- This data gives us two points on the supply curve: (8, 18) and (6, 12).
3. Find the equations of the demand and supply curves:
- To find the equations, we need to determine the slope and y-intercept of each line using the given points.
- Let's start with the demand curve:
- The slope of the demand curve can be calculated as (change in quantity demanded) / (change in price).
- Using the points (10, 3) and (6, 5), the change in quantity is 5 - 3 = 2, and the change in price is 6 - 10 = -4.
- Therefore, the slope of the demand curve is (2 / -4) = -0.5.
- Now we can use the slope-intercept form of a linear equation: y = mx + b, where y is the quantity demanded, x is the price, m is the slope, and b is the y-intercept.
- Using the point (10, 3) in the equation, we can solve for the y-intercept:
- 3 = -0.5 * 10 + b
- 3 = -5 + b
- b = 3 + 5
- b = 8
- Therefore, the demand equation is: Quantity Demanded = -0.5 * Price + 8.
- Now let's find the supply curve equation:
- Using the points (8, 18) and (6, 12), calculate the slope:
- The change in quantity is 12 - 18 = -6, and the change in price is 6 - 8 = -2.
- The slope is (-6 / -2) = 3.
- Using the point (6, 12) in the equation, solve for the y-intercept:
- 12 = 3 * 6 + b
- 12 = 18 + b
- b = 12 - 18
- b = -6
- Therefore, the supply equation is: Quantity Supplied = 3 * Price - 6.
4. Find the equilibrium price:
- The equilibrium price is the price at which the quantity demanded equals the quantity supplied.
- To find this, we set the demand equation equal to the supply equation and solve for the price.
- -0.5 * Price + 8 = 3 * Price - 6
- -0.5 * Price - 3 * Price = -6 - 8
- -3.5 * Price = -14
- Price = -14 / -3.5
- Price = 4
The retailer's equilibrium price is $4.