9. Suppose the production function is given by Q(L,K) = L3/4K
1/4 Assuming capital is
fixed, find APL and MPL.
10. Given a short run cost function as TC = 1/3Q3
-2Q2
+60Q+100 , find the minimum
value of AVC and MC.
11. What are the similarities and differences between oligopoly and monopolistically
competitive market structure?
12. A firm operates in a perfectly competitive market. The market price of its product is
4 birr and the total cost function is given by TC= 1/3Q3
-5Q2
+20Q+50, where TC is the
�
Total Cost function.
To find APL (Average Product of Labor), we need to divide the total product (Q) by the quantity of labor (L). APL = Q / L.
To find MPL (Marginal Product of Labor), we need to take the derivative of the production function with respect to labor (L) and evaluate it at a given quantity of labor (L). MPL = ∂Q / ∂L.
Using the given production function Q(L,K) = L^(3/4) * K^(1/4), and assuming capital (K) is fixed, we can ignore it for now.
1. APL = Q / L = (L^(3/4) * K^(1/4)) / L
Canceling out L,
APL = K^(1/4) * L^(-1/4)
2. MPL = ∂Q / ∂L = (3/4) * L^(-1/4) * K^(1/4)
Now, let's move on to the second question.
To find the minimum value of AVC (Average Variable Cost) and MC (Marginal Cost), we need to differentiate the Total Cost (TC) function with respect to quantity (Q) and equate it to zero.
Given TC = (1/3)Q^3 - 2Q^2 + 60Q + 100,
1. AVC = VC / Q, where VC is the Variable Cost.
As VC = TC - FC (Fixed Cost), and FC is not given, we cannot directly calculate AVC. We need more information to determine FC.
2. MC = ∂TC / ∂Q = Q^2 - 4Q + 60
Moving on to the third question,
Oligopoly and monopolistically competitive market structures have similarities and differences.
Similarities:
- Both market structures are characterized by imperfect competition, meaning that there are some barriers to entry and firms have some degree of market power.
- In both structures, firms can differentiate their products and engage in non-price competition.
- Both structures can lead to a wide range of prices and outputs.
Differences:
- Oligopoly has a smaller number of firms compared to monopolistically competitive markets, where there are many firms.
- In oligopoly, there is a high degree of interdependence among firms, as their actions directly affect each other's profits. In monopolistic competition, firms are relatively independent.
- Oligopoly often experiences collusion or strategic behavior, with firms working together to maximize joint profits. Monopolistic competition does not usually involve collusion.
- Oligopolistic firms may engage in price leadership, where one firm sets the price and other firms follow. This is not observed in monopolistically competitive markets.
Finally, for the fourth question,
To find the profit-maximizing level of output for a firm operating in a perfectly competitive market, we need to equate marginal cost (MC) to the market price (P).
Given the market price P = 4 and the total cost function TC = (1/3)Q^3 - 5Q^2 + 20Q + 50,
1. MC = dTC / dQ = Q^2 - 10Q + 20
2. Set MC equal to the market price P:
Q^2 - 10Q + 20 = 4
Q^2 - 10Q + 16 = 0
Solving this quadratic equation will give us the profit-maximizing level of output for the firm.
1. To find the average product of labor (APL), we divide the total product (Q) by the amount of labor (L). APL = Q/L.
Since capital is fixed, the total product (Q) remains constant. Therefore, the average product of labor (APL) is represented by the same function as the production function: APL = Q(L,K) = L^(3/4)K^(1/4).
To find the marginal product of labor (MPL), we differentiate the production function with respect to labor (L). MPL is the partial derivative of Q(L,K) with respect to L.
MPL = partial derivative of Q(L,K)/partial derivative of L = (3/4)L^(-1/4)K^(1/4).
10. To find the minimum value of average variable cost (AVC), we divide the variable cost (VC) by the quantity (Q). AVC = VC/Q.
The variable cost (VC) can be obtained by subtracting the fixed cost (FC) from the total cost (TC). VC = TC - FC.
The marginal cost (MC) can be obtained by differentiating the total cost (TC) with respect to quantity (Q).
Given the short-run cost function as TC = (1/3)Q^3 - 2Q^2 + 60Q + 100, we can calculate VC and MC.
VC = TC - FC = (1/3)Q^3 - 2Q^2 + 60Q + 100 - FC.
MC = partial derivative of TC/partial derivative of Q = d(TC)/dQ = d/dQ[(1/3)Q^3 - 2Q^2 + 60Q + 100].
11. Similarities between oligopoly and monopolistically competitive market structures:
- Both involve a limited number of firms operating in the market.
- In both market structures, firms have some degree of market power and are able to influence prices.
- Both market structures allow for product differentiation, meaning firms can offer slightly different products to attract customers.
- In both market structures, firms compete and strive for profitability.
Differences between oligopoly and monopolistically competitive market structures:
- Oligopoly consists of a few large firms dominating the market, while monopolistically competitive market structures have many small firms.
- Oligopoly can result in collusion and cooperation between firms, leading to interdependence, while monopolistically competitive firms tend to be more independent.
- In an oligopoly, barriers to entry can be high, making it difficult for new firms to enter the market, while monopolistically competitive markets typically have low barriers to entry.
12. In a perfectly competitive market, the firm is a price taker, meaning it has no control over the market price. The market price is determined by the overall supply and demand in the market. In this case, the market price is 4 birr.
The total cost function is given as TC = (1/3)Q^3 - 5Q^2 + 20Q + 50.
To find the profit-maximizing level of output, we need to determine the level of output (Q) that minimizes the average total cost (ATC).
ATC = TC/Q = [(1/3)Q^3 - 5Q^2 + 20Q + 50]/Q.
To find the minimum value of AVC, we first need to determine the level of output (Q) at which AVC is minimized.
AVC = VC/Q = (TC - FC)/Q = [(1/3)Q^3 - 5Q^2 + 20Q + 50 - FC]/Q.
To find the minimum value of MC, we differentiate the total cost (TC) function with respect to output (Q).
MC = d(TC)/dQ = d/dQ[(1/3)Q^3 - 5Q^2 + 20Q + 50].