ECP 3703 Managerial Economics Homework 3 Fall 2013 Professor

Ecp 3703 Managerial Economics Homework 3 Fall 2013 Professor T Le

Ecp 3703 Managerial Economics Homework 3 Fall 2013 Professor T Le

This assignment involves multiple questions related to production functions, cost analysis, and input substitution in managerial economics. It covers evaluating a firm's labor hiring recommendation using a given production function, optimizing labor input choices with a constrained budget, analyzing the impact of input prices on production isoquants and isocost lines, and deriving costs for a quadratic total cost function to determine the minimum average total cost.

Paper For Above instruction

Managerial economics provides vital tools for firms to optimize production, minimize costs, and make informed decisions in response to input price changes. This paper discusses the practical application of these tools by analyzing several scenarios involving production functions, cost minimization, and input substitution, illustrating core principles with detailed explanations and recommendations.

Evaluation of Labor Hiring Recommendation at Elwyn Company

The Elwyn Company’s production function is given by Q = 300S + 200U - 0.2S^2 - 0.3U^2, where S and U are hours of skilled and unskilled labor, respectively. The wages are $10 per hour for skilled labor and $5 per hour for unskilled labor. The engineer recommends hiring 400 hours of skilled labor and 100 hours of unskilled labor. To evaluate this recommendation, we must understand how these choices impact output and costs.

First, calculating the output at the recommended levels:

Q = 300(400) + 200(100) - 0.2(400)^2 - 0.3(100)^2 = 120,000 + 20,000 - 0.2(160,000) - 0.3(10,000) = 140,000 - 32,000 - 3,000 = 105,000 units.

Next, assessing the marginal contributions of each input, which derive from the partial derivatives of the production function. The marginal product of skilled labor (S) is:

MP_S = ∂Q/∂S = 300 - 0.4S = 300 - 0.4(400) = 300 - 160 = 140 units per hour.

Similarly, the marginal product of unskilled labor (U) is:

MP_U = ∂Q/∂U = 200 - 0.6U = 200 - 0.6(100) = 200 - 60 = 140 units per hour.

Note that both inputs provide equal marginal products of 140 units per hour, implying each contributes equally to output at these levels. Cost-wise, total expenditure is:

Cost = 10(400) + 5(100) = $4,000 + $500 = $4,500.

Given the marginal products and costs, this combination appears efficient in terms of marginal productivity per dollar spent, as MP per dollar for skilled labor is 14 units, and for unskilled labor also 14 units. However, the firm should evaluate whether this output level maximizes profit relative to costs, considering diminishing marginal returns which are evident from the decreasing marginal products due to the quadratic terms.

Optimal Labor Choice with a $5000 Budget

If the firm has a total of $5,000 to spend on labor, the problem involves choosing S and U to maximize output or profit, subject to the budget constraint:

10S + 5U = 5000, which simplifies to S + 0.5U = 500.

To maximize output, the firm should allocate labor toward inputs with higher marginal products per dollar. The marginal products per dollar are:

For skilled labor: MP_S / wage_S = (300 - 0.4S) / 10

For unskilled labor: MP_U / wage_U = (200 - 0.6U) / 5

At the optimal point, the firm should equalize the marginal product per dollar across inputs:

(300 - 0.4S) / 10 = (200 - 0.6U) / 5

Cross-multiplied:

5(300 - 0.4S) = 10(200 - 0.6U)

1500 - 2S = 2000 - 6U

Rearranged:

6U - 2S = 500

Using the budget constraint S + 0.5U = 500, express S in terms of U:

S = 500 - 0.5U

Substitute into the previous equation:

6U - 2(500 - 0.5U) = 500

6U - 1000 + U = 500

7U = 1500

U = approximately 214.29 hours

S = 500 - 0.5(214.29) ≈ 500 - 107.14 = 392.86 hours

Thus, to maximize output within the budget, the firm should hire approximately 393 hours of skilled labor and 214 hours of unskilled labor. This allocation balances marginal productivity per dollar and budget constraints for optimal production.

Impact of Price increase of Inputs on Isoquants and Isocosts

When the price of one input increases, while the other remains constant, the firm's cost structure changes. Isoquants, representing combinations of inputs yielding the same output, are unaffected by input prices and thus remain unchanged. However, the isocost line, which indicates combinations of inputs at a given total cost, shifts due to input price changes.

An increase in the price of one input causes the isocost line to pivot inward on the axis corresponding to the more expensive input, indicating higher costs for the same input combinations. This change often leads the firm to substitute away from the more expensive input toward the cheaper one, depending on input substitutability, as described by the elasticity of substitution.

Consequently, the firm's optimal input combination shifts along the new tangency point of the isoquant and the isocost line, favoring the input whose price has not increased. This substitution behavior aligns with the cost minimization condition that the marginal rate of technical substitution (MRTS) equals the input price ratio:

\( MRTS_{LK} = \frac{MP_L}{MP_K} = \frac{w_K}{w_L} \)

Cost Analysis with Quadratic Total Cost Function

The total cost function is given as:

C = 32 + 2Q^2

a. Average total cost (ATC):

ATC = C / Q = (32 + 2Q^2) / Q = 32 / Q + 2Q

b. Average variable cost (AVC):

The variable cost is the portion that depends on output, which is 2Q^2; thus, AVC = (2Q^2) / Q = 2Q.

c. Marginal cost (MC):

MC is the derivative of total cost with respect to Q:

MC = dC/dQ = 4Q

d. Average fixed cost (AFC):

The fixed cost is 32; thus, AFC = 32 / Q.

e. The total average cost (ATC) is minimized where its derivative with respect to Q equals zero:

d(ATC)/dQ = -32 / Q^2 + 2 = 0

Solving for Q:

-32 / Q^2 + 2 = 0

2 = 32 / Q^2

Q^2 = 32 / 2 = 16

Q = 4 units.

At Q = 4, total average cost reaches its minimum, which can be confirmed by plugging back into the ATC formula:

ATC(4) = 32 / 4 + 2(4) = 8 + 8 = 16.

Conclusion

The analysis demonstrates how different economic tools help managers make optimal decisions regarding input utilization and cost management. Carefully evaluating marginal products relative to input costs guides efficient labor hiring. Understanding input price changes' impacts on isoquants and isocost lines informs substitution decisions. Deriving cost functions is vital for identifying optimal production levels that minimize costs, thus maximizing profitability. These principles form the backbone of effective production and cost management in managerial economics.

References

  • Varian, H. R. (2014). Intermediate Microeconomics: A Modern Approach. W. W. Norton & Company.
  • Pindyck, R. S., & Rubinfeld, D. L. (2013). Microeconomics. Pearson.
  • Salvatore, D. (2012). Microeconomics: Theory and Applications. Oxford University Press.
  • Case, K. E., Fair, R. C., & Oster, S. M. (2017). Principles of Economics. Pearson.
  • Perloff, J. M. (2013). Microeconomics. Pearson.
  • McConnell, C. R., Brue, S. L., & Flynn, S. M. (2014). Microeconomics. McGraw-Hill Education.
  • Hirschey, M. (2018). Managerial Economics. Cengage Learning.
  • Frank, R., & Bernanke, B. (2015). Principles of Economics. McGraw-Hill Education.
  • Krugman, P., & Wells, R. (2012). Microeconomics. Worth Publishers.
  • Rosen, H. S. (2012). Microeconomic Analysis. McGraw-Hill Education.

Related Assignments