Assignment 1: Coretta Leases A Workshop In Which She Weaves

Assignment1 Coretta Leases A Workshop In Which She Weaves Rugs From

Assignment1 Coretta Leases A Workshop In Which She Weaves Rugs From

Coretta leases a workshop, in which she weaves rugs from marsh grass she gathers (for free) from a nearby riverside. She finds that if she works alone for a day, she can weave two rugs. If she hires one additional worker, together they can weave 5 rugs in a day, each specializing in some of the tasks. Adding a third worker also allows for more specialization, bringing production up to 8 rugs. Adding a fourth worker adds only 2 rugs to production, because there are no more gains from specialization to be had. Hiring a fifth worker adds only one additional rug to daily production, because now the workshop space is getting crowded.

a. What is the output in this example? What is the fixed input? What is the variable input?

The output in this scenario is measured by the number of rugs produced per day, which varies based on the number of workers employed. When she works alone, she produces 2 rugs; with additional workers, production increases as follows: 5 rugs with two workers, 8 rugs with three workers, 10 rugs with four workers, and 11 rugs with five workers. The fixed input in this situation is the workshop space and equipment, which remains constant regardless of the number of rugs produced. The variable input is the labor—specifically, the number of workers hired each day—since this can change and directly affects the output.

b. Find marginal product and complete column (2) and (3): from 0 to 1 worker: _____ from 1 to 2 workers: _____ from 2 to 3 workers: _____ from 3 to 4 workers: _____ from 4 to 5 workers: _____

Marginal product (MP) is calculated by the change in total output divided by the change in labor input. Assuming she starts with zero workers and no rugs, the calculations are:

  • From 0 to 1 worker: 2 rugs - 0 rugs = 2 rugs
  • From 1 to 2 workers: 5 rugs - 2 rugs = 3 rugs
  • From 2 to 3 workers: 8 rugs - 5 rugs = 3 rugs
  • From 3 to 4 workers: 10 rugs - 8 rugs = 2 rugs
  • From 4 to 5 workers: 11 rugs - 10 rugs = 1 rug

Thus, the marginal products are:

  • 0 to 1 worker: 2 rugs
  • 1 to 2 workers: 3 rugs
  • 2 to 3 workers: 3 rugs
  • 3 to 4 workers: 2 rugs
  • 4 to 5 workers: 1 rug

c. Suppose that rent for the workshop space costs Coretta $70 per day (whether she produces anything or not). Her cost for labor (including the opportunity cost of “hiring” herself!) is $60 per worker per day. Complete columns (4) through (6) in the table above, showing her average fixed, variable, and total costs of producing various numbers of rugs.

The fixed cost (FC) per day is $70, regardless of output. The variable cost (VC) depends on the number of workers: $60 per worker per day, multiplied by the number of workers. The total cost (TC) is fixed cost plus variable cost.

Calculations:

  • Number of workers: 1
  • VC = 1 x $60 = $60
  • TC = $70 + $60 = $130
  • Average Fixed Cost (AFC) = $70 / 1 = $70
  • Average Variable Cost (AVC) = $60 / 2 rugs = $30
  • Average Total Cost (ATC) = $130 / 2 = $65
  • Number of workers: 2
  • VC = 2 x $60 = $120
  • TC = $70 + $120 = $190
  • AFC = $70 / 2 = $35
  • AVC = $120 / 5 rugs = $24
  • ATC = $190 / 5 = $38
  • Number of workers: 3
  • VC = 3 x $60 = $180
  • TC = $70 + $180 = $250
  • AFC = $70 / 3 ≈ $23.33
  • AVC = $180 / 8 rugs = $22.5
  • ATC = $250 / 8 ≈ $31.25
  • Number of workers: 4
  • VC = 4 x $60 = $240
  • TC = $70 + $240 = $310
  • AFC = $70 / 4 = $17.5
  • AVC = $240 / 10 rugs = $24
  • ATC = $310 / 10 = $31
  • Number of workers: 5
  • VC = 5 x $60 = $300
  • TC = $70 + $300 = $370
  • AFC = $70 / 5 = $14
  • AVC = $300 / 11 rugs ≈ $27.27
  • ATC = $370 / 11 ≈ $33.64

d. Find marginal cost and complete column (7).

Marginal cost (MC) is calculated as the change in total cost divided by the change in output. Using the total costs from above:

  • From 2 to 5 rugs: (TC at 5 rugs - TC at 2 rugs) / (5 - 2) = ($370 - $190) / 3 ≈ $60
  • Similarly, for the other intervals, marginal cost estimates are:
  • Between 2 and 5 rugs (production increase from 2 to 3 workers): approximately $60 per rug
  • Between 5 and 8 rugs (3 to 4 workers): ($310 - $250) / (8 - 5) ≈ $20 per rug
  • Between 8 and 10 rugs (4 to 5 workers): ($370 - $310) / (11 - 10) = $60 per rug

Second Part: Heritage Corporation's AVC and SMC Analysis

Heritage Corporation estimates that its total variable costs follow a cubic specification, with the regression analysis providing estimates for average variable costs (AVC) at different output levels. The computer output indicates the intercept and coefficients for the polynomial model, allowing analysis of the shape and significance of the AVC curve and deriving the marginal cost function.

a. Do the estimated coefficients have the required signs to yield a-shaped AVC curve? Discuss the significance using the p-values.

The cubic form of AVC typically includes coefficients that produce a U-shaped or an inverted U-shaped curve depending on their signs. The intercept is positive at 175. The coefficients for the polynomial terms, labeled as Q-like variables, are critical. If the quadratic term (Q²) coefficient is negative and the cubic term (Q³) coefficient is positive, the AVC curve can exhibit a U-shape. Significance is assessed via p-values: if p-values are below 0.05, the coefficients are statistically significant. According to the output, the intercept and Q coefficients are highly significant (p

b. Heritage Corporation’s marginal cost function is SMC = ___________________________________.

Considering the cubic relationship, marginal cost (SMC) is derived as the derivative of total variable costs concerning output. Mathematically, SMC is the derivative of AVC multiplied by output or directly obtained from the polynomial form of total variable costs. Given the coefficients, the SMC function can be expressed as: SMC = a + 2bQ + 3cQ², where a, b, and c are the estimated parameters from the regression, representing the marginal cost's polynomial form.

c. At what level of output does AVC reach a minimum? What is the value of AVC at its minimum? Qm in = ___________ AVC min = _______________________

The minimum point of AVC occurs where its derivative with respect to Q equals zero. Calculating the critical points involves solving the first derivative of the AVC function set equal to zero, typically leading to a quadratic or cubic equation. Once Qm is found, plug into AVC formula to get the minimum AVC value.

d. Compute AVC and SMC when Heritage produces 8 units. AVCQ= 8 = _______ SMCQ =8 = _______

Using the estimated AVC and SMC functions, substitute Q = 8 to compute AVC and SMC values at that production level, informing decisions about efficiency and costs at this output level.

References

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