8. A rational consumer spends all of her income on two goods: Apple and Banana.
Suppose the last dollar spent on Apple increased her total utility from 60 utils to 68 utils
and the last dollar spent on Banana increased her total utility from 25 utils to 29 utils. If
the price of a unit of Apple is 2 Birr, what is the price of a unit of Banana at equilibrium?
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
�
The consumer spends all of her income on Apple and Banana, so we can assume that her total income is equal to her total spending on these two goods. Let's say her income is Y.
If the price of a unit of Apple is 2 Birr, then the amount of Apple she buys is X (in units), and the amount of Banana she buys is Y/2 (in units).
We know that the last dollar spent on Apple increased her total utility from 60 utils to 68 utils. This means that the marginal utility of the last unit of Apple purchased is 8 utils.
Similarly, the last dollar spent on Banana increased her total utility from 25 utils to 29 utils. This means that the marginal utility of the last unit of Banana purchased is 4 utils.
Since the consumer is maximizing her utility, she will equate the marginal utility per Birr spent on Apple to the marginal utility per Birr spent on Banana. This can be represented as:
MUa/price of Apple = MUb/price of Banana
8/2 = 4/price of Banana
4 = 4/price of Banana
price of Banana = 1 Birr
Therefore, the price of a unit of Banana at equilibrium is 1 Birr.
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Assuming capital is fixed, the production function is Q(L,K) = L^(3/4) * K^(1/4).
The Average Product of Labor (APL) is given by APL = Q/L. Substituting the production function, we have:
APL = (L^(3/4) * K^(1/4)) / L
APL = L^(-1/4) * K^(1/4)
The Marginal Product of Labor (MPL) is given by MPL = ∂Q/∂L. Taking the derivative of the production function with respect to L, we have:
MPL = (3/4) * L^(-1/4) * K^(1/4) * K^(1/4)
MPL = (3/4) * K^(1/4) * L^(-1/4) * L^(3/4)
MPL = (3/4) * K^(1/4) * L^(1/2)
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The total cost function is TC = (1/3)Q^3 -2Q^2 + 60Q + 100.
The Average Variable Cost (AVC) is given by AVC = VC/Q, where VC is the total variable cost. The variable cost is the cost that varies with the level of output.
To find the minimum value of AVC, we need to find the minimum value of VC. This occurs where Marginal Cost (MC) is equal to Average Variable Cost (AVC).
MC = ∂TC/∂Q = Q^2 - 4Q + 60
AVC = VC/Q = (TC - FC)/Q
AVC = (1/3)Q^2 - 2Q + 60
Setting MC = AVC, we have:
Q^2 - 4Q + 60 = (1/3)Q^2 - 2Q + 60
(2/3)Q^2 - 2Q = 0
(2/3)Q(Q - 3) = 0
Q = 0 or Q = 3
Since Q cannot be 0, the minimum value of AVC occurs at Q = 3. Substituting this into the average variable cost equation, we get:
AVC = (1/3)(3)^2 - 2(3) + 60
AVC = 11
Therefore, the minimum value of AVC is 11.
To find the minimum value of MC, we can calculate MC at Q = 3:
MC = (3)^2 - 4(3) + 60
MC = 51
Therefore, the minimum value of MC is 51.
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Oligopoly and monopolistically competitive market structures are both imperfect competition market structures, meaning they deviate from the ideal of perfect competition.
Similarities:
1. Both market structures have a small number of sellers in the market.
2. Entry and exit barriers exist in both market structures, although they may be higher in oligopoly compared to monopolistic competition.
3. In both market structures, firms have some degree of market power and can influence prices.
4. Both market structures involve product differentiation to some extent, although it may be greater in monopolistic competition compared to oligopoly.
Differences:
1. Oligopoly involves a few large firms dominating the market, whereas monopolistic competition has many small firms competing with each other.
2. In oligopoly, firms may engage in strategic behavior and engage in collusion or form cartels. In monopolistic competition, firms compete independently and may engage in non-price competition.
3. Oligopoly tends to have higher concentration ratios, meaning a few firms control a large share of the market, whereas monopolistic competition has lower concentration ratios.
4. In oligopoly, there is a high level of interdependence between firms, as they must take into account the actions and reactions of other firms in the market. In monopolistic competition, firms are relatively independent and do not need to consider the actions of other firms as much.
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In a perfectly competitive market, the market price of the product is determined by the intersection of the market demand and supply curves. In this case, the market price is given as 4 birr.
The total cost function is TC = (1/3)Q^3 - 5Q^2 + 20Q + 50.
To find the profit-maximizing level of output for the firm, we need to find the level of output where Marginal Cost (MC) is equal to the market price.
MC = ∂TC/∂Q = Q^2 - 10Q + 20
4 = Q^2 - 10Q + 20
Q^2 - 10Q + 16 = 0
Using the quadratic formula, we can solve for Q:
Q = (-(-10) ± √((-10)^2 - 4(1)(16)))/(2(1))
Q = (10 ± √(100 - 64))/2
Q = (10 ± √36)/2
Q = (10 ± 6)/2
Q = 8 or Q = 2
Since the firm's output level cannot be negative or zero, the profit-maximizing level of output is Q = 8.
To find the Minimum Cost (MC), we substitute Q = 8 into the total cost function:
TC = (1/3)(8)^3 - 5(8)^2 + 20(8) + 50
MC = 100
Therefore, the minimum value of MC for this firm is 100.
A8. To find the price of a unit of Banana at equilibrium, we need to compare the marginal utility per dollar spent on Apple and Banana.
- The marginal utility per dollar spent on Apple is the change in total utility divided by the change in the quantity of Apple purchased. In this case, it is (68 - 60) utils / 1 Apple = 8 utils per Apple.
- The marginal utility per dollar spent on Banana is the change in total utility divided by the change in the quantity of Banana purchased. In this case, it is (29 - 25) utils / (2 Birr) = 2 utils per Birr.
To be at equilibrium, a rational consumer should allocate their spending in such a way that the marginal utility per dollar spent on each good is equal. Therefore, the price of a unit of Banana at equilibrium should be the price at which the marginal utility per dollar spent on Banana is equal to the marginal utility per dollar spent on Apple.
Let's assume the price per unit of Banana is P Birr. So, the marginal utility per dollar spent on Banana would be (2 utils per Birr) / P.
Since we know that the marginal utility per dollar spent on Apple is 8 utils per Apple, we can set up the equation:
(2 utils per Birr) / P = 8 utils per Apple
Simplifying the equation, we have:
2 / P = 8
Cross-multiplying, we get:
P = 2 / 8
Therefore, the price of a unit of Banana at equilibrium is 0.25 Birr.
A9. To find Average Product of Labor (APL) and Marginal Product of Labor (MPL) given the production function Q(L,K) = L^(3/4) * K^(1/4), where capital (K) is fixed:
- Average Product of Labor (APL) is the total quantity of output produced (Q) divided by the quantity of labor used (L).
APL = Q / L
- Marginal Product of Labor (MPL) is the change in total output (Q) resulting from employing one additional unit of labor (L), while holding the amount of capital (K) constant.
MPL = ∂Q / ∂L
To find APL and MPL, we need to differentiate the production function with respect to labor (L).
∂Q / ∂L = (3/4) * L^(-1/4) * K^(1/4)
Simplifying the expression, we have:
∂Q / ∂L = 3/4 * K^(1/4) / L^(1/4)
Now, we can substitute this into the equations for APL and MPL:
APL = Q / L = (L^(3/4) * K^(1/4)) / L = K^(1/4) * L^(-1/4)
MPL = ∂Q / ∂L = 3/4 * K^(1/4) / L^(1/4)
A10. To find the minimum value of Average Variable Cost (AVC) and Marginal Cost (MC), we need to differentiate the short-run cost function with respect to quantity (Q) and find the points where AVC and MC are equal to zero.
The given short-run cost function is TC = (1/3)Q^3 - 2Q^2 + 60Q + 100.
To find AVC, we divide the total variable cost (TVC) by the quantity (Q):
AVC = TVC / Q
Since we know that TC = TVC + TFC, where TFC is the total fixed cost and it is constant, we can find AVC as follows:
AVC = (TVC + TFC) / Q = (TC - TFC) / Q
Now, let's differentiate the cost function with respect to quantity (Q) to find MC:
MC = ∂TC / ∂Q
Differentiating the cost function, we get:
MC = d/dQ [(1/3)Q^3 - 2Q^2 + 60Q + 100]
Simplifying the expression, we have:
MC = Q^2 - 4Q + 60
To find the minimum value of AVC, we set AVC equal to zero and solve:
AVC = 0
(QVC + TFC) / Q = 0
TVC + TFC = 0
Since TFC is constant, we can ignore it in this calculation. Therefore, we need to find the quantity (Q) that makes TVC equal to zero.
To find the minimum value of MC, we set the derivative MC equal to zero and solve:
MC = 0
Q^2 - 4Q + 60 = 0
Using the quadratic formula, we can solve for Q:
Q = (-(-4) ± sqrt((-4)^2 - 4(1)(60))) / (2(1))
Q = (4 ± sqrt(16 - 240)) / 2
Q = (4 ± sqrt(-224)) / 2
Since the square root of a negative number is undefined, there is no real solution for Q in this case. Therefore, we cannot find the minimum value of MC in this scenario.
A11. The similarities between oligopoly and monopolistically competitive market structures are as follows:
1. Both market structures have a high degree of market concentration, meaning that a few firms dominate the market in terms of producing and selling goods or services.
2. Both market structures involve significant barriers to entry, restricting new firms from easily entering the market and competing with existing firms.
3. In both market structures, firms are profit-maximizing and engage in strategic decision-making, such as pricing, advertising, and product differentiation, to gain a competitive advantage.
4. Both market structures are characterized by imperfect competition, meaning that firms have some control over product prices due to differentiation or market power.
The differences between oligopoly and monopolistically competitive market structures are as follows:
1. Oligopoly market structure consists of a small number of large firms dominating the market, while monopolistically competitive market structure consists of a large number of small firms.
2. In oligopoly, firms may engage in collusion or price-fixing to maximize profits, while in monopolistically competitive market structure, firms are independent and do not collude.
3. Oligopolistic firms produce either homogeneous or differentiated products, while monopolistically competitive firms produce differentiated products.
4. Oligopolistic firms often face interdependence and strategic behavior among themselves, while monopolistically competitive firms have limited interdependence and little strategic behavior due to the large number of firms in the market.
A12. To find the profit-maximizing level of output for a perfectly competitive firm, we compare the market price of the product to the marginal cost (MC). The profit-maximizing level of output occurs when marginal cost equals the market price.
In this case, the market price is 4 birr and the total cost function is TC = (1/3)Q^3 - 5Q^2 + 20Q + 50.
To find the marginal cost (MC), we differentiate the total cost function with respect to quantity (Q):
MC = ∂TC / ∂Q
Differentiating the cost function, we get:
MC = d/dQ [(1/3)Q^3 - 5Q^2 + 20Q + 50]
Simplifying the expression, we have:
MC = Q^2 - 10Q + 20
To find the profit-maximizing level of output, we set MC equal to the market price and solve:
MC = 4
Q^2 - 10Q + 20 = 4
Q^2 - 10Q + 16 = 0
Using the quadratic formula, we can solve for Q:
Q = (-(-10) ± sqrt((-10)^2 - 4(1)(16))) / (2(1))
Q = (10 ± sqrt(100 - 64)) / 2
Q = (10 ± sqrt(36)) / 2
Q = (10 ± 6) / 2
Q = 8 or Q = 2
Since the firm operates in a perfectly competitive market, it is a price taker and cannot influence the market price. Therefore, the profit-maximizing level of output for the firm is Q = 2 units, where MC = 4.