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Chapter 7 Handouts Production Analysis

Production: transforming inputs (land, labor, capital, materials) into outputs (goods and services).

Short Run Production: the time period in which at least one input is fixed (and at least one variable)

• Typically assume K (capital) is fixed

o Ex: size of plant, or production capacity

o K is usually expressed in terms of machine hours

• Typically assume L (labor) is variable.

o Ex: can easily adjust number of labor hours to meet production needs (can hire more people or have existing staff working extra hours to increase labor input; or can lay off people or having existing staff work less hour to decrease labor input)

o L is typically expressed in terms of labor hours

• The short run will be different for each industry or firm depending on several factors:

o Their current production level relative to capacity,

o Their financial ability to increase capital, and

o Length of long-term contracts.

Production Function: Q=f(K, L, ….)

• Q= quantity of output produced

• Function of amount of K, L, and other inputs such as materials

• The bar over K indicates the level of capital is fixed

• It shows the maximum amount that can be produced given a fixed amount of inputs

3 measures of productivity:

1. Total Product (TP): shows the maximum produced given various levels of K, L.

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• Measures the total productivity of a production process or the number of units produced in a given time frame.

2. Marginal Product (MP): measures the change in total product when the level a variable input is changed.

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• As we increase labor by one unit we can see the contribution to total output. MP tells us when we add a worker how many extra units of output will be produced by this worker.

• MP is an individual measure of productivity (each worker’s or each labor hour’s contribution in terms of the amount of output produced)

3. Average Product (AP): measures the average productivity level of our variable input, labor.

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• Instead of using an individual measure of productivity we look at the entire labor force or a subset of the labor force and measure its collective, average, productivity.

• AP is most commonly used to compare productivity levels across different production sites or across countries.

• AP is also used then as a measure of standard of living since productivity and wages are directly linked.

Example: Daily Plant Output of Copper Wire

|Capital (Machine |Labor (person |TP or Output (in pounds)|MP (pounds per person-per hour)|AP (pounds per person-per hour) |

|Hours) |hours) | | | |

|40 |0 |0 |----- |-------- |

|40 |10 |1000 | | |

|40 |20 |2500 | | |

|40 |30 |4500 | | |

|40 |40 |6000 | | |

|40 |50 |7000 | | |

|40 |60 |7500 | | |

|40 |70 |7500 | | |

|40 |80 |6000 | | |

A. Fill in the table above

B. Graph TP on upper graph; MP, AP on lower graph to see typical relationship between the three different measures

• Note that there are diminishing returns to labor or “diminishing marginal returns” after the 30th worker MP falls.

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Mathematical determination of short run production:

TP=Q=50L + 6L2 – 0.5L3

1. Derive the MPL equation:

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o What is MPL of the 3rd worker?

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2. Derive the APL equation:

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o What is the AP of the first 3 workers?

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3. What is TP with 5 workers?

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4. At what labor unit is production maximized? What is the value of TP at its maximum?

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• Rearrange to use quadratic formula -1.5L2 +12L +50

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where a is coefficient of highest power, b is coefficient of next highest power, c is coefficient of next highest power

o Quadratic Formula: [pic]

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o get two roots: -3.02; 11.02

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o since at L=11.02 is where TP will be maximized; plug into TP function to get the value of TP

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▪ TP=551 + 728.64 -669.14 =610.5 units is the maximum production possible. Occurs at L=11.02 units.

Optimal Level of a Single Input

• How does a firm determine how much labor to hire in order to maximize profit and not simply maximize production?

• By analyzing the revenue generating capability of the input: labor compared to the cost of the input

The economic productivity of an input is called its Marginal Revenue Product (MRP): measures the additional revenue generated by the last unit employed. Basically, how much does the next unit of labor contribute to revenue; or how much does the next unit of capital contribute to revenue?

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▪ Ex: if the next unit of labor has MP=2 (it can produce 2 units of output) and this output can be sold for $5 each (MR=$5) then:

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▪ The next unit of labor will generate $10 of economic value for the firm.

o Firm’s will maximize profit by operating where MR=MC (or MR-MC=0 or Mπ=0 as discussed in Chapter 2).

o Remember, If there is not quantity for which MR=MC, then firms should produce the largest Q where MR > MC (because the next unit adds more to revenue than to cost so it improves profitability). Firms never want to produce where MR < MC because the next unit adds more to cost than revenue and dampens profits.

o We use the same type of analysis for determining the quantity of labor to hire. As long as the MRP (revenue generated by the extra labor) is greater than the MC (marginal cost of labor) then it will improve the overall profitability of the firm.

Assume we have the total product of labor at various labor units and we know that the price of labor (PL) or wage, (w)=$12 per unit. Assume the final product will be sold for P=$5 each.

|Labor Units |Total Labor Cost |Marginal Cost of Labor|Total Product of Labor|Marginal Product of |Marginal Revenue Product |

|(QL) | | |(TP) |Labor MPL |MRPL |

|1 | | |3 | | |

|2 | | |7 | | |

|3 | | |10 | | |

|4 | | |12 | | |

|5 | | |13 | | |

o Fill in the values for Total Labor Cost (PL * QL or w*QL)

o Fill in the values for MC Labor (ΔTC/ΔQL)

o Fill in the values for MPL (ΔTP/ΔQL)

o Fill in the values for MRPL (MPL x MR)

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o How many units of labor should be hired?

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o Start with unit 1:

o Go to unit 2:

o Go to unit 3:

o Go to unit 4:

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o Notice that the MC is the price of labor PL or wage (w). This gives managers another easy rule to remember in hiring:

o If MRP > PL or wage (w) you should continue to hire workers.

o Ideally, you should hire until MRP = PL, and

o You should never hire a worker if MRP < PL.

o Note: we are assuming there are no other additional variable costs associated with production. If we are talking about preparation of tax returns then in addition to the labor cost there would be computer maintenance cost or cost of paper, etc. In this case you would subtract any additional costs per unit and marginal revenue product (MRP) is called Net marginal Revenue Product (NMRP).

Example using Net Marginal Revenue Product (NMRP) Top Gun Marketing, Inc., offers overhead banner fly-by promotion services using their Cessna aircraft and banner creation facilities. The Padres Island firm specializes in restaurant promotion via fly-bys at outdoor events and other high traffic centers, where each 10 minute increment of advertising costs $300. Over the past year, the following relation between fly-by advertising and incremental restaurant guests per month has been observed:

Sales (units) = 5,200 + 50A - 0.5A2

o Here A represents a 10-minute fly-by advertisement, and sales are measured in numbers of restaurant guests.

o Therefore, to think of advertising as production in this case the marginal productivity of advertising is the derivative or:

As a manager you must recommend an appropriate level of advertising. In doing so you must determine the profit-maximizing level of employment for the input, advertising, in this "production" system. After consultation with the restaurant, he determined that the value of output is $10 per guest, the net marginal revenue earned by the client (price minus all marginal costs except fly-by advertising).

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| |Using the rule for optimal resource employment, determine the profit-maximizing number of 10-minute ads. |

ANSWER:

| |The marginal product of advertising was found above as: ________________ |

| |The rule for determining the optimal amount of a resource to employ is: |

| |MRPA = PA |

| |The optimal advertising level is found where: |

Optimal Levels of Multiple Inputs

To maximize profits when multiple inputs must be hired (skilled labor, unskilled labor, capital, etc) the firm must determine the cost-minimizing input proportions at the optimal level of output. Hence, they must determine the optimal quantity to produce and then determine how best to produce that output level given the productivity various inputs and the cost of those inputs.

• The same general principle applies as the single input model:

o Hire as long as MRP of an input ≥ P of that same input.

o However, you must consider how to allocate cost across different inputs with different productivity levels (e.g. do you hire workers or buy machines and automate a production process or use some combination—the productivity of each input (L, K) is different and the cost is different (PL or wage and PK or “rental rate”, r)

• Need the ratio of marginal productivity (MP) to cost (price of the input) to be equal across the different inputs:

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o Interpreting these numbers and understanding why they must be equal for optimization: If MPL = 10 and PL or wage= $20 and MPK = 50 and Pk or rental rate=$200 then

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▪ for each unit of capital hired 50 units of output will be produced

▪ Capital, K, is clearly more productive.

▪ However, each unit of labor costs $20 whereas each unit of capital, r, costs $200 so labor is clearly less expense.

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▪ Clearly, the firm should increase its employment of labor and decrease its employment of capital. As this happens, the marginal productivity of labor will decrease (diminishing returns) and marginal productivity of capital will increase (less capital; more productive) until the ratios become equalized.

o Once the input proportions (MP/P) are equalized you must double-check to make sure the marginal revenue product (MRP) ≥ P for each input—otherwise the firm has the cost-minimizing input levels but they’re not generating the proper amount of output to maximize profits.

Numerical Example of Optimal Input Level. Laboratory Testing, Inc., provides routine drug tests for employers in the Los Angeles metropolitan area. Tests are supervised by skilled technicians using equipment produced by two leading competitors in the medical equipment industry. Records for the current year show an average of 24 tests per hour performed on the A-1, and 51 tests per hour on a new machine, the Caltec. The A-1 is leased for $16,000 per month, and the Caltec is leased at a rate of $34,000 per month. On average, each machine is operated 25 eight-hour days per month. Labor and all other costs are fixed.

|A. |Does company usage reflect an optimal mix of testing equipment? |

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| |The rule for an optimal combination of A-1 (A) and Caltec (C) equipment is: |

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| |Of course, marginal products and equipment prices must both be in the same relevant time frame, either hours or |

| |months. |

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| |On a per hour basis, the relevant question is: |

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| |Yes, each machine generates .3 units of output per dollar spent. |

|B. |At a price of $5 per test should the company lease more machines? |

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| |The rule for optimal input employment is: |

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| |MRP = MP * MRQ = Input Price or cost | |

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| |Or, in per month terms: | |

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| |In both cases, we see that each machine returns more than its marginal cost (price), and expansion would be profitable. |

Recommended Problems from Chapter 7 are:

ST 7.1

P 7.6

P7.8 (for part B you don’t have to understand the derivation of the optimal rule just what the rule is and have a basic understanding of the intuition of the rule—not the mathematical derivation of the rule).

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