1. To find the equilibrium price and quantity of beef, we need to set the quantity demanded equal to the quantity supplied and solve for Pb.
Equating Qd and Qs:
22 + 0.1Y - 10Pb + 5Pc = -400 + 500Pb - 200Pf
Given:
Y = $10,000
Pc = $2.00
Pf = $1.00
Substituting the given values:
22 + 0.1(10,000) - 10Pb + 5(2) = -400 + 500Pb - 200(1)
Simplifying the equation:
22 + 1,000 - 10Pb + 10 = -400 + 500Pb - 200
1,032 - 10Pb = 300 + 500Pb
1,332 = 510Pb
Pb = 1,332 / 510
Pb = $2.61 (approximately)
Substituting the equilibrium price in either the demand or supply equation to find the equilibrium quantity:
Qd = 22 + 0.1(10,000) - 10(2.61) + 5(2)
Qd = 22 + 1,000 - 26.1 + 10
Qd = 1,006.9 (approximately)
Therefore, the equilibrium price of beef is approximately $2.61 per unit, and the equilibrium quantity of beef is approximately 1,006.9 units.
2. To find the point price elasticity of demand for beef when its price is equal to $4.00, we need to calculate the percentage change in quantity demanded divided by the percentage change in price.
Given:
Price of beef (Pb) = $4.00
Demand equation: Qd = 22 + 0.1(10,000) - 10Pb + 5(2)
Calculating the quantity demanded at Pb = $4.00:
Qd = 22 + 1,000 - 10(4) + 10
Qd = 1,008
Now, calculating the quantity demanded at Pb = $3.60:
Qd2 = 22 + 1,000 - 10(3.6) + 10
Qd2 = 1,012
Calculating the percentage change in quantity demanded:
% Change in quantity demanded = (Qd2 - Qd) / Qd
% Change in quantity demanded = (1,012 - 1,008) / 1,008
% Change in quantity demanded = 0.00397 or 0.397% (approximately)
Calculating the percentage change in price:
% Change in price = (4 - 3.6) / 3.6
% Change in price = 0.1111 or 11.11%
Calculating the point price elasticity of demand:
Price elasticity of demand = % Change in quantity demanded / % Change in price
Price elasticity of demand = 0.00397 / 0.1111
Price elasticity of demand = 0.0357 (approximately)
Therefore, the point price elasticity of demand for beef when its price is equal to $4.00 is approximately 0.0357.
3. a. Average fixed costs are calculated by dividing total fixed costs by the quantity produced. In this case, when the firm produces 50 widgets per day, the average variable cost is given as $600 per widget. Therefore, average fixed costs would be:
Average fixed costs = Average total costs - Average variable costs
Average fixed costs = $1,000 - $600
Average fixed costs = $400 per widget
b. To determine the average total and variable costs for producing 49 widgets, we need more information. Given the information provided, the average total and variable costs cannot be determined specifically for producing 49 widgets.
c. The rising or falling nature of the average variable and average total cost curves between the production of 49 and 50 widgets per day cannot be determined without more information.
d. To determine whether the firm will maintain production at its current rate, increase it, decrease it, or shut down in the short run, we need to compare the firm's price per widget with its average variable cost. If the price per widget is greater than the average variable cost, the firm will continue production in the short run. However, if the price per widget is less than the average variable cost, the firm may shut down to minimize losses.