Suppose there are 1000 identical firms producing diamonds. Diamond miners receive the wage rate w. Assume that the short-run cost function for each diamond-producing firm is C(q) = wq + q2.
a. If w = 10, what is the supply curve of an individual firm? What is the market supply curve?
b. How many diamonds will the 1000 firms produce (in total) if the market equilibrium price is $25 and w = 10?
c. Assume w = 10 and that the market demand for diamonds is Q = 7000 – 100P. What is the short-run equilibrium price and (market) quantity?
d. Suppose public pressure for improving diamond mining conditions increases w to 13. How does this affect individual and market supply? How does this affect the equilibrium price and quantity of diamonds?
I need to know the answer how or show me calculate the firm,s loss or is the firm making a profit or a loss?
a. To find the supply curve of an individual firm, we need to determine the quantity of diamonds it will produce at different prices. The cost function C(q) = wq + q^2 represents the firm's short-run cost, where w is the wage rate and q is the quantity of diamonds produced. To find the quantity, we need to minimize the cost function with respect to q.
Taking the derivative of C(q) with respect to q gives us: C'(q) = w + 2q.
Setting C'(q) equal to zero, we have: w + 2q = 0. Solving for q, we get: q = -w/2.
Since we are assuming w = 10, the quantity produced by an individual firm is q = -10/2 = -5.
However, a negative quantity does not make sense in this context, so we need to consider the range of possible quantities. In the short run, firms can choose to produce zero output, which means when prices are too low, a firm may choose not to produce any diamonds.
Therefore, the supply curve of an individual firm is zero quantity at prices lower than their minimum average variable cost and a positive quantity (q > 0) at prices above that threshold.
To find the market supply curve, we need to sum up the quantities supplied by all 1000 firms at different prices. So, if the quantity supplied by each firm is q, then the market supply curve is simply 1000*q.
b. If the market equilibrium price is $25 and w = 10, we need to find the quantity of diamonds produced by the 1000 firms. The market equilibrium occurs where the demand curve of diamonds intersects with the market supply curve.
Given the market demand function Q = 7000 – 100P, we substitute P = 25 to find the quantity demanded: Q = 7000 – 100*25 = 7000 – 2500 = 4500.
Since the market supply curve is 1000*q, the quantity supplied by the 1000 firms is q = 4500/1000 = 4.5.
Therefore, the 1000 firms will produce a total of 4.5 * 1000 = 4500 diamonds.
c. To find the short-run equilibrium price and quantity when w = 10 and the market demand is Q = 7000 – 100P, we need to find the intersection point of the market demand and market supply curves.
The market supply curve is still 1000*q, and we can set 1000*q equal to the market demand function: 1000*q = 7000 – 100P.
To find the equilibrium price, we substitute q with 7000 – 100P and solve for P: 1000*(7000 – 100P) = 7000 – 100P.
Expanding and simplifying, we get: 7,000,000 – 100,000P = 7000 – 100P.
Bringing like terms to one side, we have: 99,900P = 6,993,000.
Dividing both sides by 99,900, we find: P = 69.7.
Substituting this equilibrium price into the market demand function, we find the equilibrium quantity: Q = 7000 – 100 * 69.7 = 7000 – 6970 = 30.
Therefore, the short-run equilibrium price is $69.7 and the quantity is 30.
d. If the public pressure for improving diamond mining conditions increases w to 13, individual and market supply curves will be affected.
For an individual firm, the increase in the wage rate w from 10 to 13 will increase the cost of production. As a result, the cost function C(q) = wq + q^2 will increase for the same quantity of production q. Thus, the supply curve of an individual firm will shift upward.
Similarly, the market supply curve, which is the summation of the individual firms' supply curves, will shift upward due to the increase in production costs for all firms.
The equilibrium price and quantity of diamonds will be affected by these shifts in supply. As the supply curve shifts upward, the equilibrium price will increase, and the equilibrium quantity will decrease.
To determine the new equilibrium price and quantity, we would need information about the demand function for diamonds after the wage rate increase. Unfortunately, this information is not provided in the given context, so we cannot explicitly determine the new equilibrium price and quantity.