National University Of Singapore EC2101 Microeconomic Analys

National University Of Singapore Ec2101 Microeconomic Analysis I Dep

Answer the following questions. Briefly explain your answer. a) You obtained the following short-run cost information of a firm. When the firm produces 2 units of output, its total cost is $500,040. When the firm produces 10 units of output, its average fixed cost is $50,000. What is the average fixed cost when the firm produces 2 units of output? What is the average fixed cost when the firm produces 5 units of output? b) A profit-maximizing firm in a perfectly competitive market currently produces at an output level where its short-run average total cost curve is upward sloping. What does this imply about the firm’s economic profit?

Question 2 A firm produces a product with labor and capital. The price of labor is w=2, and the price of capital is r=1. a) Suppose the production function is q= LK . What is the equation of the long-run total cost curve? Suppose in the short run, capital is fixed at K=4, what is the equation of the short-run total cost curve? b) Suppose the production function is q= L+K . What is the equation of the long-run total cost curve? Suppose in the short run, capital is fixed at K=4, what is the equation of the short-run total cost curve? c) Suppose the production function is q= min(2L,K) . What is the equation of the long-run total cost curve? Suppose in the short run, capital is fixed at K=4, what is the equation of the short-run total cost curve?

Question 3 A firm has production function q=2LK +K . The price of labor is w=1 and the price of capital is r=1. (L and K do not have to be integers.) a) Suppose in the short run, capital is fixed at K=5. What is the equation of the short-run total cost curve? b) In the long run, what is the cost-minimizing input combination when the firm produces q=18? What is the level of total cost at this input choice? c) Suppose now w=8 and r=1. What is the long-run cost-minimizing input combination when the firm produces q=3? What is the level of total cost at this input choice? Suppose in the short run, capital is fixed at K=1. What is the level of total cost in the short run if the firm produces q=3?

Question 4 In a perfectly competitive industry, every firm has identical cost structure. The short-run total cost curve of an individual firm is STC(q)=40+10q+0.1q2. Currently there are 10 firms in the market. The market demand curve is D(P)=P. a) What is the equation of an individual firm’s short-run supply curve? b) What is the short-run equilibrium price and quantity in this market? c) At the equilibrium, what is the total producer surplus in the market? What is the profit of an individual firm?

Paper For Above instruction

In this analysis, we explore fundamental concepts of microeconomic theory through specific questions related to production costs, firm behavior under perfect competition, and cost minimization strategies. These aspects are crucial for understanding firm decision-making processes in the short and long run, as well as market outcomes such as equilibrium prices, quantities, and producer surplus.

Fixed and Average Fixed Costs

Starting with the initial question, the firm’s total cost at producing 2 units is $500,040. To find the average fixed cost (AFC) at this level, we need to determine the fixed component of total costs. When producing 10 units, the average fixed cost is given as $50,000. Total fixed costs (TFC) can be found by multiplying AFC by the quantity. Therefore, TFC at 10 units equals $50,000 * 10 = $500,000. Given total cost at 2 units is $500,040, fixed costs are approximately $500,000, implying variable costs are minor or negligible at this scale, leading to the AFC when producing 2 units being approximately $500,000 / 2 = $250,000.

Similarly, when producing 5 units, AFC equals total fixed costs divided by 5: $500,000 / 5 = $100,000. These calculations suggest that fixed costs remain constant regardless of output, while variable costs change with production levels, consistent with common cost structures in microeconomics.

Implication of Upward Sloping ATC Curve

When a firm’s short-run average total cost (ATC) curve is upward sloping at the current production level, it indicates increasing costs per additional unit of output. Typically, this suggests the firm is experiencing diminishing returns to scale in the short run, and its economic profit is likely to be zero or negative. Since profits are maximized when marginal cost equals marginal revenue and, in perfect competition, when price equals minimum average total cost, an upward sloping ATC above the market price signifies the firm may be incurring losses or operating at breakeven points.

Cost Functions with Different Production Technologies

Next, analyzing different production functions reveals how costs fluctuate with technological constraints:

  • Production function q=LK: The long-run total cost (LTC) function minimizes cost for a given output q. Since q=LK, for LTC, the choice of L and K should satisfy the cost minimization condition, leading to LTC = 2√(w r) q or LTC = 2√(2 1) q = 2 √2 q ≈ 2.828 q. In the short run, with fixed capital K=4, the cost is C = wL + rK, so L = q/K = q/4, giving C = 2(q/4) + 1*4 = 0.5q + 4.
  • Production function q= L+K: The LTC in this case is linear, as LTC = wL + rK, with L + K = q. Choosing prices, LTC = 2L + 1K, and minimizing costs for a given q results in LTC = (w + r)q = (2 + 1)q = 3q. In the short run, fixing K=4, the cost becomes C = 2L + 4, with L = q - 4, so C = 2(q - 4) + 4 = 2q - 8 + 4 = 2q - 4.
  • Production function q= min(2L,K): The LTC is determined by cost-efficiently producing q units, with K ≥ 2L. To produce q units, set 2L = K and total cost minimization yields LTC = 2wL + rK. Since K=2L, costs are C= 22L + 12L = (4 + 2)L = 6L, and L= q/2, so LTC = 6*(q/2) = 3q. In the short run, with fixed K=4, the maximum q produced is limited by K, i.e., q= min(2L, 4). Adjustments lead to the same cost relationships valid within those constraints.

Production Function q=2LK +K

In the case where the production function is q=2LK + K, the optimal input combination can be obtained by setting marginal products proportional to input prices. When K is fixed at K=5, the short-run cost is C = wL + rK = L + 5. To produce q=18 units in the long run, the cost-minimizing approach involves purchasing K and L optimally. Solving for L and K, the total cost at this point is minimized at L = √(q / (2K)), leading to a total cost approximately aligned with C= L + K, evaluated at the optimal point, which yields the minimal value for specific q.

When the input prices change (w=8, r=1), the new optimal input combination shifts toward minimizing costs with the higher wage, resulting in different values for L and K. Fixing K=1 in the short run for producing q=3 results in a total cost of C= L + 1, where L is derived from the production function constraints.

Market Equilibrium and Producer Surplus

The firm's short-run total cost function, STC(q)=40+10q+0.1q2, enables us to derive individual supply curves by equating marginal cost to price. The marginal cost (MC) is obtained by differentiating the total cost: MC=10+0.2q. Setting P=MC provides the supply function, i.e., supply at different prices depending on the equilibrium conditions.

In a market with 10 identical firms and a demand D(P)=P, equilibrium occurs where the sum of individual supplies equals market demand. Solving for the equilibrium price and quantity involves setting the aggregate supply equal to demand, which yields specific market clearing prices and quantities. The total producer surplus can then be calculated as the difference between total revenue and total variable costs at equilibrium, providing insight into market efficiency. The individual firm’s profit equals total revenue minus total costs at the equilibrium output, capturing profitability in the industry.

Conclusion

Understanding these core principles of cost structures, firm decision-making, and market equilibrium enables economists and business analysts to predict firm behavior and market outcomes under various scenarios. The interplay of short-run constraints and long-run optimization shapes industry dynamics, costs, and profitability, serving as a foundation for more advanced economic analysis and policy considerations.

References

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