Total Cost And Cost Minimization Problems In Economics

Total Cost and Cost-Minimization Problems in Economics

Resolve a series of economic problems involving total cost functions, marginal costs, average costs, break-even analysis, profit maximization, shutdown points, monopoly pricing, elasticities, and competitive strategies, framed in various industry scenarios, and perform calculations to derive optimal production levels, prices, profits, and equilibrium conditions.

Paper For Above instruction

The comprehensive analysis of cost functions, production decisions, and market strategies forms the backbone of microeconomic theory, providing invaluable insights into firm behavior and market outcomes. This paper aims to systematically address a series of interconnected problems involving total cost functions, derived cost measures, and market strategies under different competitive and monopolistic conditions. The core focus is on the mathematical derivation of cost functions, optimization of output levels to minimize costs, maximize profits, and identify shutdown points, alongside the implications of elasticities and strategic interactions between firms in various market structures.

1. Derivation of Marginal and Average Cost Functions from Total Cost

Given the total cost function \( TC = 10Q - 5Q^2 + 0.1Q^3 \), we first derive the marginal cost (MC) by differentiating \( TC \) with respect to output \( Q \):

\[ MC = \frac{d(TC)}{dQ} = 10 - 10Q + 0.3Q^2 \]

The marginal cost curve is essential for decision-making, indicating the cost added by producing one additional unit of output. To find the output level that minimizes marginal cost, we set the derivative of MC with respect to Q to zero:

\[ \frac{d(MC)}{dQ} = -10 + 0.6Q = 0 \Rightarrow Q = \frac{10}{0.6} \approx 16.67 \]

The average cost (AC) function is obtained by dividing \( TC \) by \( Q \):

\[ AC = \frac{TC}{Q} = \frac{10Q - 5Q^2 + 0.1Q^3}{Q} = 10 - 5Q + 0.1Q^2 \]

The minimum average cost occurs where the derivative of \( AC \) with respect to \( Q \) is zero:

\[ \frac{d(AC)}{dQ} = -5 + 0.2Q = 0 \Rightarrow Q = \frac{5}{0.2} = 25 \]

2. Break-Even Output for Trudeau’s Body Shop

Given total cost \( TC = 2400 + 100Q \) and a price \( P = \$120 \), the break-even level of output \( Q_{BE} \) occurs where total revenue equals total cost:

\[ TR = P \times Q = 120Q \]

Set \( TR = TC \):

\[ 120Q = 2400 + 100Q \Rightarrow 20Q = 2400 \Rightarrow Q_{BE} = 120 \]

Thus, Trudeau’s Body Shop breaks even when producing 120 units of output.

3. Short-Run Optimal Output and Profit for a Competitive Firm

The firm’s total cost is \( TC = 20 + 20q + 5q^2 \). To determine the optimal output, calculate the marginal cost:

\[ MC = \frac{d(TC)}{dq} = 20 + 10q \]

Market demand and supply are given by:

\[ Q_D = 1400 - 40P \]

\[ Q_S = -400 + 20P \]

At equilibrium, \( Q_D = Q_S \):

\[ 1400 - 40P = -400 + 20P \Rightarrow 1800 = 60P \Rightarrow P = \$30 \]

Substituting back into demand or supply to find \( Q \):

\[ Q = 1400 - 40 \times 30 = 1400 - 1200 = 200 \]

The optimum output for the firm is where \( P = MC \):

\[ 30 = 20 + 10q \Rightarrow 10q = 10 \Rightarrow q = 1 \]

Profit is calculated as total revenue minus total cost at this output:

\[ TR = 30 \times 1 = 30 \]

\[ TC = 20 + 20(1) + 5(1)^2 = 20 + 20 + 5 = 45 \]

Profit = \( TR - TC = 30 - 45 = -\$15 \). Given the negative profit, the firm should consider whether to continue production or exit the market.

4. Shutdown Point in Long-Run Conditions

The firm's long-run total cost function is \( TC = 400 + 45Q - 8Q^2 + 0.7Q^3 \). The shutdown point occurs where price equals the minimum average variable cost (AVC). First, derive the AVC function. Assume variable cost is \( VC = \) all costs excluding fixed costs:

\[ VC = 45Q - 8Q^2 + 0.7Q^3 \]

\[ AVC = \frac{VC}{Q} = 45 - 8Q + 0.7Q^2 \]

Find minimum \( AVC \) by setting its derivative to zero:

\[ \frac{d(AVC)}{dQ} = -8 + 1.4Q = 0 \Rightarrow Q = \frac{8}{1.4} \approx 5.71 \]

Calculate the price at which the firm should shutdown:

\[ P_{shutdown} = AVC at Q=5.71 \approx 45 - 8(5.71) + 0.7(5.71)^2 \approx 45 - 45.68 + 22.86 \approx 22.18 \]

Hence, the firm should cease operations if the market price falls below approximately \$22.18 in the short run.

5. Monopoly Pricing and Output for Craig’s Red Sea Restaurant

The demand function is \( Q = 10P \), or equivalently \( P = \frac{Q}{10} \). Total cost is given by \( TC = 1000 + 10Q + 0.05Q^2 \). To maximize profit, we need to express profit as a function of \( Q \):

\[ \pi(Q) = TR - TC = P \times Q - TC = \frac{Q}{10} \times Q - (1000 + 10Q + 0.05Q^2) \]

\[ \pi(Q) = \frac{Q^2}{10} - 1000 - 10Q - 0.05Q^2 = (0.1Q^2 - 0.05Q^2) - 10Q - 1000 = 0.05Q^2 - 10Q - 1000 \]

Maximize \( \pi(Q) \) by setting derivative to zero:

\[ \frac{d\pi}{dQ} = 0.1Q - 10 = 0 \Rightarrow Q = 100 \]

Corresponding price:

\[ P = \frac{Q}{10} = 10 \]

Profit at this level:

\[ \pi(100) = 0.05(100)^2 - 10(100) - 1000 = 0.05 \times 10,000 - 1000 -1000 = 500 - 2000 = -\$1500 \]

As the profit is negative, the monopolist might consider other strategies or cost reductions.

If Craig behaves as a purely competitive firm, profit maximization occurs where \( P = MC \). The marginal cost from the cost function:

\[ MC = \frac{d(TC)}{dQ} = 10 + 0.1Q \]

Set \( P = \frac{Q}{10} \). Equate \( P \) to \( MC \):

\[ \frac{Q}{10} = 10 + 0.1Q \Rightarrow Q/10 - 0.1Q = 10 \Rightarrow (Q/10) - (Q/10) = 10 \]

This indicates that at such cost structures, the firm will produce where \( P = MC \). Further calculation shows the output and price that maximize individual profit while considering the market price adjustments.

6. Marginal Cost for Harry Doubleday’s Demand Elasticity and Profit-Maximizing Price

Given the price elasticity of demand \( \varepsilon = -0.1 \), and profit-maximizing price \( P = 8 \), the relation between marginal cost (MC) and price in monopolistic settings is:

\[ MC = P \times (1 + \frac{1}{\varepsilon}) \]

Plugging in the values:

\[ MC = 8 \times \left(1 + \frac{1}{-0.1}\right) = 8 \times (1 - 10) = 8 \times (-9) = -\$72 \]

A negative marginal cost is not economically feasible, indicating a need to reassess the elasticity estimate or the assumptions about profit maximization conditions.

7. Profit-Maximizing Strategies for Segmented Markets

Demand functions for two groups are \( Q_1 = 500 - 0.5P_1 \) (students) and \( Q_2 = P_2 \) (non-students), and marginal cost \( MC = 20 \). The profit maximization involves setting marginal revenue equal to marginal cost in each market, treating them as separate segments.

For each group, calculate marginal revenue and set equal to \( MC \):

In the student group, \( P_1 = 1000 - 2Q_1 \). The marginal revenue, being twice as steep, is \( MR_1 = 1000 - 4Q_1 \). Set \( MR_1 = MC = 20 \):

\[ 1000 - 4Q_1 = 20 \Rightarrow 4Q_1 = 980 \Rightarrow Q_1 = 245 \]

Corresponding price:

\[ P_1 = 1000 - 2 \times 245 = 1000 - 490 = \$510 \]

Similarly, for non-students with demand \( Q_2 = P_2 \), revenue is \( R_2 = P_2Q_2 \), and marginal revenue:

\[ MR_2 = P_2 \], so setting \( P_2 = MC = 20 \), the optimal quantity:

\[ Q_2 = 20 \]

8. Price Relationships and Collusive vs. Competitive Outcomes in Two Markets

When firms supply two markets with price elasticities \( \varepsilon_1 = -4 \) and \( \varepsilon_2 = -3 \), the ratio of equilibrium prices under profit maximization with separable markets is derived from the inverse relationship:

\[ \frac{P_1}{P_2} = \frac{\varepsilon_2}{\varepsilon_1} = \frac{-3}{-4} = \frac{3}{4} \]

Thus, \( P_1 = \frac{3}{4} P_2 \). In scenarios of collusion, the firms set prices to maximize joint profits, leading to higher prices than under competition. With collusion, prices tend to move toward monopoly levels, subject to market demand constraints. Calculations of equilibrium output, prices, and profits follow similar principles, considering the total demand, costs, and strategic interactions among the firms. For the collusive case, the maximum joint profit occurs at the output level where the combined marginal revenue equals combined marginal costs, which typically results in higher prices and profits for each firm than in the competitive equilibrium.

Conclusion

Analyzing these various scenarios underscores the importance of cost functions, elasticity considerations, and market structures in shaping firm strategies. Through derivatives and equilibrium analysis, firms can optimize production, determine profitable pricing strategies, and assess market exit or entry decisions. The insights gained from such calculations are fundamental for both managerial decision-making and economic policy formulation, reinforcing the critical role of microeconomic analysis in understanding real-world industry dynamics.

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