Consider The Following Benefit And Cost Functions

Consider The Following Benefit And Cost Functions Bx 600x 12x2a

Consider the following benefit and cost functions: B(X) = 600X - 12X^2 and C(X) = 20X^2. Use this information to answer the following questions. a. What are the MB and the MC? b. What level of X maximizes the net benefit? c. What is the net benefit (NB)?

Consider the quarterly demand and supply for the Petram Company: Qd = 1000 + 0.5M + 0.25A – 100P and Qs = -750 + 100P, where Q is quantity per quarter, P is price, M is income, and A is advertising expenditure. Given A = 1000 and M = 20,000, answer the following questions. a. What is the equilibrium price and quantity? b. What is the inverse demand?

Consider the benefit and cost of Mr. T's Coffee Shop: B(Q) = 50 + 18Q – 2Q^2 and C(Q) = 40 + 6Q. Answer these questions: a. What is the marginal benefit, MB(Q)? b. What is the marginal cost, MC(Q)? c. At what level of output is the net benefit maximized?

Given the market demand and supply: Qd = 50 – P and Qs = 2.5 + 1.5P. Answer these questions: a. What is the equilibrium price and quantity? b. If the government sets a price floor of $25, what is the surplus or shortage? If the government buys the surplus, what would be the cost to the government?

The C & A Lawnmower Firm operates in a highly competitive industry with MB(Q) = 100, and total cost C(Q) = 100,000 + 20Q + 0.1Q^2. Answer these questions: a. What is the net benefit-maximizing level of output? b. What is the total benefit function? c. What is the maximum net benefit?

Sample Paper For Above instruction

Economic Analysis of Benefit and Cost Functions, Market Equilibria, and Decision-Making

The primary goal of this paper is to analyze various economic functions and market scenarios provided in the prompts, utilizing fundamental microeconomic principles including marginal analysis, equilibrium determination, and optimization strategies. We will systematically address each problem, illustrating the application of these principles through detailed calculations and theoretical explanations.

A. Analysis of Benefit and Cost Functions

1. Marginal Benefit (MB) and Marginal Cost (MC)

The benefit function is given as B(X) = 600X - 12X^2, and the cost function as C(X) = 20X^2. The marginal benefit is derived by differentiating the benefit function with respect to X: MB = dB/dX = 600 - 24X. Similarly, the marginal cost is obtained by differentiating the cost function: MC = dC/dX = 40X.

Thus, the MB function is MB = 600 - 24X, and the MC function is MC = 40X.

2. Maximizing Net Benefit

The net benefit (NB) is calculated as NB = B(X) - C(X). To find the level of X that maximizes NB, we differentiate NB with respect to X and set it equal to zero:

d(NB)/dX = MB - MC = (600 - 24X) - 40X = 0

Solving for X:

600 - 24X - 40X = 0

600 = 64X

X = 600 / 64 ≈ 9.375

The maximum net benefit occurs at approximately X = 9.375 units.

3. Computing the Net Benefit at Optimal Level

Calculate NB at X ≈ 9.375:

B(9.375) = 600(9.375) - 12(9.375)^2 ≈ 5625 - 12(87.89) ≈ 5625 - 1054.7 ≈ 4570.3

C(9.375) = 20(9.375)^2 ≈ 20(87.89) ≈ 1757.8

Therefore, NB ≈ 4570.3 - 1757.8 ≈ 2812.5

B. Market Equilibrium Analysis

1. Equilibrium Price and Quantity for Petram Company

The demand function: Qd = 1000 + 0.5M + 0.25A – 100P

With A=1000 and M=20,000:

Qd = 1000 + 0.5(20,000) + 0.25(1000) – 100P = 1000 + 10,000 + 250 – 100P = 11,250 – 100P

The supply function: Qs = -750 + 100P

At equilibrium: Qd = Qs

11,250 – 100P = -750 + 100P

Adding 100P to both sides and adding 750: 11,250 + 750 = 200P

12,000 = 200P

P = 12,000 / 200 = 60

Substituting P = 60 back into the supply function:

Qs = -750 + 100(60) = -750 + 6000 = 5250

Equilibrium price is $60 and quantity is 5,250 units.

2. Inverse Demand Function

The inverse demand function derived from Qd = 11,250 – 100P:

P = (11,250 – Qd) / 100

C. Marginal Benefit, Marginal Cost, and Optimal Output for Mr. T's Coffee Shop

1. Marginal Benefit MB(Q)

Given B(Q) = 50 + 18Q – 2Q^2, differentiate to find MB:

MB(Q) = dB/dQ = 18 – 4Q

2. Marginal Cost MC(Q)

Given C(Q) = 40 + 6Q, differentiate:

MC(Q) = dC/dQ = 6

3. Maximizing Net Benefit

Net benefit NB(Q) = B(Q) – C(Q):

To find the maximizer, set MB = MC:

18 – 4Q = 6

> 18 – 6 = 4Q

> 12 = 4Q

> Q = 3 units

At Q = 3, net benefit is maximized.

Calculate NB at Q=3:

B(3) = 50 + 18(3) – 2(3)^2 = 50 + 54 – 18 = 86

C(3) = 40 + 6(3) = 40 + 18 = 58

Net Benefit ≈ 86 – 58 = 28

D. Market Equilibrium and Policy Impact

1. Equilibrium Price and Quantity

Demand: Qd = 50 – P

Supply: Qs = 2.5 + 1.5P

At equilibrium, Qd = Qs:

50 – P = 2.5 + 1.5P

> 50 – 2.5 = P + 1.5P

> 47.5 = 2.5P

> P = 47.5 / 2.5 = 19

Substituting P back into Qd or Qs:

Qd = 50 – 19 = 31

Equilibrium price is $19, and quantity is 31 units.

2. Price Floor Effect

Set price floor at $25:

Qd at P=25: Qd = 50 – 25 = 25

Qs at P=25: Qs = 2.5 + 1.5(25) = 2.5 + 37.5 = 40

Surplus = Qs – Qd = 40 – 25 = 15 units

Cost of surplus to government if purchased: 15 units × $25 = $375

E. Profit Maximization in Competitive Industry

1. Optimal Output and Maximum Net Benefit

The marginal benefit MB(Q) = 100, implying a constant marginal benefit per unit, which equals price.

Total cost: C(Q) = 100,000 + 20Q + 0.1Q^2

Maximize net benefit: NB(Q) = MB × Q – C(Q)

Net benefit is maximized where MB = MC, and MC = derivative of C(Q): 0.2Q + 20.

Set MB = MC:

100 = 0.2Q + 20

> 100 – 20 = 0.2Q

> 80 = 0.2Q

> Q = 80 / 0.2 = 400 units

Calculate total benefit at Q=400:

TB = MB × Q = 100 × 400 = 40,000

Total cost at Q=400:

C(400) = 100,000 + 20(400) + 0.1(400)^2 = 100,000 + 8,000 + 0.1(160,000) = 100,000 + 8,000 + 16,000 = 124,000

Maximum net benefit: NB = TB – TC = 40,000 – 124,000 = -84,000 (indicating a loss, so in real setting, the maximum profit point must be where MB = MC within feasible range).

2. Total Benefit Function

Total benefit function: TB(Q) = 100Q

3. Maximum Net Benefit

Net benefit at Q=400 units (as above): approximately -$84,000, indicating that the firm’s operation at this level results in a loss, highlighting the importance of careful capacity and cost management.

Conclusion

This comprehensive analysis demonstrates core microeconomic principles including the calculation of marginal benefits and costs, identifying optimal production levels, market equilibrium, and evaluating policy impacts. Proper application of these concepts enables firms and policymakers to make informed decisions to maximize benefits and minimize costs.

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