1. Suppose that the firm operates in a perfectly competitive market. The market price of his product is $10. The firm estimates its cost of production with the following cost function
TC=10q-4q2+q3
A. What level of output should the firm produce to maximize its profit.?
B. Determine the level of profit at equilibrium.
C. What minimum price is required by the firm to stay in the market.?
A. To maximize profit, the firm should produce at the level of output where MR=MC. In a perfectly competitive market, MR = P = $10.
So, first we need to find the marginal cost (MC) function:
MC = dTC/dq = 10 - 8q + 3q^2
Setting MR = MC:
10 = 10 - 8q + 3q^2
Rearranging and solving for q:
3q^2 - 8q = 0
q(3q - 8) = 0
q = 0 or q = 8/3
Since q cannot be zero (in a competitive market, the firm produces positive output), the firm should produce 8/3 units of output.
B. To find the level of profit at equilibrium, we need to calculate total revenue (TR) and total cost (TC) at the output level of 8/3 units:
TR = Pq = $10 * 8/3 = $26.67
TC = 10q - 4q^2 + q^3 = (10 * 8/3) - (4 * 64/9) + (8/27)
TC = 197.33/27 ≈ $7.32
Profit = TR - TC = $26.67 - $7.32 = $19.35
C. To stay in the market, the firm must cover its variable cost (VC). Variable cost is the part of the total cost that varies with the level of output, i.e. the cost of production excluding fixed costs. In this case, the firm's variable cost is:
VC = 10q - 4q^2 = q(10 - 4q)
The minimum price required to cover VC is the point where VC = P:
q(10 - 4q) = 10
4q^2 - 10q + 10 = 0
Dividing by 2:
2q^2 - 5q + 5 = 0
Using the quadratic formula:
q = (5 ± sqrt(5))/4
Since q cannot be negative (in a competitive market, the firm produces positive output), we take the positive root:
q = (5 + sqrt(5))/4 ≈ 1.56
Therefore, the minimum price required to cover the firm's variable cost is $1.56.
The Dean of Students Office of samara University reported that because of strict control over chat smuggling in to the campus, the demand for coffee in swale Café has dramatically increased. How do you justify this report?
There are a few ways to justify this report:
1. Coffee and cigarettes are often seen as complementary goods - that is, goods that tend to be consumed together or have a similar consumer base. If cigarette smuggling is being strictly controlled on campus, it is likely that students who used to smoke cigarettes on campus are now looking for other ways to satisfy their cravings or take a break from studying. Coffee could be a natural substitute for cigarettes in this scenario, leading to an increase in demand for coffee at the Swale Café.
2. It is also possible that the strict control over cigarette smuggling has led to a reduction in smoking overall, leading students who used to take cigarette breaks to now take coffee breaks instead. This would also increase the demand for coffee at the Swale Café.
3. Another possibility is that the strict control over cigarette smuggling has led to a broader cultural shift on campus, with students becoming more health-conscious and looking for ways to improve their well-being. Coffee is often seen as a healthier alternative to smoking, leading more students to choose it as a beverage option. This could also contribute to an increase in demand for coffee at the Swale Café.
Overall, there are several plausible explanations for why the Dean of Students Office at Samara University may have reported an increase in demand for coffee at the Swale Café as a result of strict control over cigarette smuggling.
To find the level of output that maximizes profit, we can use the concept of marginal revenue (MR) and marginal cost (MC). When a firm operates in a perfectly competitive market, the price is equal to the marginal revenue. Additionally, profit is maximized when marginal revenue equals marginal cost.
A. To find the level of output that maximizes profit, we need to determine the marginal cost and marginal revenue functions from the given cost function. The total cost function (TC) is given as:
TC = 10q - 4q^2 + q^3
To find the marginal cost (MC), we take the derivative of the total cost function with respect to q:
MC = d(TC)/dq = 10 - 8q + 3q^2
The marginal revenue (MR) is equal to the market price in a perfectly competitive market, which is $10.
Setting MR equal to MC, we have:
MR = MC
10 = 10 - 8q + 3q^2
Simplifying the equation, we get:
8q - 3q^2 = 0
Factoring out q, we have:
q(8 - 3q) = 0
This equation is satisfied when either q = 0 or 8 - 3q = 0.
If q = 0, it implies that the firm should not produce anything as the output level is zero. This is not an optimal solution since the firm would not be able to generate any profit.
If 8 - 3q = 0, we can solve for the value of q:
8 - 3q = 0
3q = 8
q = 8/3
The firm should produce an output of 8/3 to maximize its profit.
B. To determine the level of profit at equilibrium, we need to calculate the total revenue (TR) and total cost at the output level determined in part A. The total revenue is given by:
TR = Price × Quantity
TR = $10 × (8/3)
TR = $80/3
Next, we need to calculate the total cost (TC) at this output level. Using the given cost function:
TC = 10q - 4q^2 + q^3
TC = 10(8/3) - 4(8/3)^2 + (8/3)^3
Evaluating the equation:
TC = 80/3 - 256/9 + 512/27
TC = 240/9 - 256/9 + 512/27
TC = 496/27
Finally, we can determine profit (π) using the equation:
π = TR - TC
π = (80/3) - (496/27)
Simplifying the equation, we get:
π = (2160/81) - (496/27)
π = 2160/81 - 1936/81
π = 224/81
Therefore, the level of profit at equilibrium is 224/81.
C. To find the minimum price required by the firm to stay in the market, we need to determine the break-even point where profit (π) is equal to zero. Using the profit equation:
0 = TR - TC
0 = 80/3 - TC
TC = 80/3
Setting the total cost equal to 80/3 and substituting it into the cost function:
80/3 = 10q - 4q^2 + q^3
Simplifying the equation, we get:
q^3 - 4q^2 + 10q - 80/3 = 0
To find the minimum price, we need to solve this cubic equation for the value of q. This can be done using numerical methods or software like MATLAB, Excel, etc. Once we find the value of q, we can substitute it into the price equation:
Price = 10
This will give us the minimum price required for the firm to stay in the market.