Production+Processes,+and+Costs

Production, Process, and Costs #21 Chris Naffziger, Jason Fitzpatrick, Dwight Todd

Production Process from a managerial perspective involves evaluating many different factors of production and aligning them in such a way that will allow for optimal output while minimizing costs. A useful tool in this process is called the production function; a function that defines the maximum amount of output that can be produced with a given set of inputs. Mathematically, the production function is denoted as Q=F(K,L), where Q is quantity, K is units of capital, and L is units of labor.(157) Production decisions can be evaluated in the short run (where capital is a fixed input) and in the long run (where all factors of production can be modified). Three important factors used to evaluate the production process are total product, average product, and marginal product. Total product is the maximum level of output that can be produced with a given amount of inputs. Average product is a measure of the output produced per unit of input. (Average Product of Labor=Quantity/Labor; Average Product of Capital=Quantity/Capital) Marginal Product is the change in output attributable to the last unit of an input. (Marginal Product of Labor=Change in Quantity/Change in Labor; Marginal Product of Capital=Change in Quantity/Change in Capital) (159)

As the inputs of prouduction vary, the marginal product will change as well. This can be explained by stating that as the usage of an input increases, marginal product initially increases (increasing marginal returns), then begins to decline (decreasing marginal returns), and eventually becomes negative (negative marginal returns). (161) What this means is that managers must not only make sure that the firm is producing on the production function, but they must also make sure that the right amount of inputs are being utilized, thus avoiding negative marginal returns. To help determine the amount of inputs, managers can use the concept of the value marginal product (VMP), which is the value of the output produced by the last unit of an input (VMPlabor=P X MPlabor; VMPcapital=P X MPcapital; where P is the price of the unit sold). To maximize profits, a manager should use inputs at levels at which the marginal benefit equals the marginal cost. (162,163)


 * Algebraic Forms of Production Functions** (165-168)


 * Linear Production Function** Q=F(K,L) = aK + bL; where a and b are constants, inputs are perfect substitutes.
 * Marginal Product for Linear Function**; MPcapital=a; MPlabor=b.

Both capital and labor must be used in the fixed proportion.
 * Leontief Production Function** (fixed-proportions production function) Q=F(K,L) = min{bK,cL}; where b and c are constants.

inputs is not linear. Inputs need not be used in fixed proportions. Assumes some degree of substitutability, but not perfect.
 * Cobb-Douglas Production Function** Q=F(K,L) = K^a,L^b; where a and b are constants, the relationship between output and
 * Marginal Product for Cobb-Douglas Function**; MPcapital=aK^a-1, L^b; MPlabor=bK^a, L^b-1.


 * Isoquants**

To look at the choice of combinations of inputs to produce output, a helpful tool is the isoquant curve. An isoquant defines the combinations of capital and labor that yield the producer the same output. The reason that isoquants are typically drawn with a convex shape is that inputs such as capital and labor are not perfectly substitutable. The rate at which labor and capital can substitute for each other is called the marginal rate of technical substitution (MRTS) MRTS(kl)=MPlabor/MPcapital. (169) The following graph shows an example of an isoquant. ([|www.econ.iastate.edu])



The Law of Diminishing Rate of Technical Substitution states that as a producer uses less of an input, increasingly more of the other input must be employed to produce the same level of output. Whenever an isoquant exhibits a diminishing marginal rate of technical substitution, the corresponding isoquants are convex from the origin. (171)


 * Isocosts**

We've just discussed how isoquants convey the different combinations of inputs that produce the same output. Similarly, an Isocost line shows a combination of inputs that will cost the firm the same amount of money. The formula for an Isocost line is showed as wL + rK=Cost; where w is the price of labor and l is the price of labor. (172) It is important to note here that isocosts with lower costs will lie below ones with higher costs, as well as the fact that when input prices are constant the isocost lines will be parallel. The following graph shows two different isocost lines as well as where the isoquant curve intersects each of them. ([|www.econ.iastate.edu])




 * Cost Minimization**

Managers are interested in producing at the lowest cost in order to maximize the net margin. This can be accomplished by producing where the MRTSkl=price of labor/price of capital. (175) A principle that surfaces when looking at cost minimization is the Optimal Input Substitution rule. This states that to minimize the cost of producing a given level of output, the firm should use less of an input and more of other inputs when that input's price rises. (176)


 * Short Run Costs**

The sum of fixed and variable costs is the firm's short-run cost function, a function that summarizes the minimum possible cost of producing each level of output when variable factors are being used in the cost minimizing way. Fixed costs (FC) are costs that do not change with changes in output, while variable costs (VC(Q)) change with changes in output. (178,9) Total cost is also defined as the sum of fixed and variable costs.

The following are helpful equations to help analyze costs: (180-184)

Average Fixed Cost=AFC=FC/Q

Average Variable Cost=AVC=VC(Q)/Q

Average Total Cost=ATC=C(Q)/Q

Marginal Cost=MC=change in C/change in Q

C(Q)=VC(Q) + FC

ATC=AVC + AFC

The following graph shows the relationship between MC, ATC, and AVC:([|http://livingeconomics.org])


 * Fixed and Sunk Costs**

A Sunk Cost is a cost that is lost forever once it has been paid, the portion of the fixed cost that cannot be recouped. Since they are lost forever once they have been paid, they are irrelevant to decision making. The Decision Maker should ignore sunk costs to maximize profits or minimize losses. (184)

As the time goes by, all costs become variable. In the long-run, the firm can now adjust different inputs and other factors which will ultimately alter the ATC curve. There will be different ATC curves as well for different quantities produced. The Long-Run Average Cost Curve is a curve that defines the minimum average cost of producing alternate levels of output, allowing for optimal selection of both fixed and variable factors of production. The long-run average cost curve is the lower envelope of all the short-run cost curves.(187) The following graph depicts the Long-Run Average Cost Curve visually. ([|www.fao.org])
 * Long-Run Costs**


 * Economies of Scale**

Economies of Scale exist when long-run average costs decline as output is increased. Conversely, diseconomies of scale exist when long-run average costs rise as output is increased. A third possibility, Constant Returns to Scale, occur when long-run average costs remain constant as output is increased.(188) The following graph shows Economies and Diseconomies of Scale. ([|http://img.tfd.com])




 * Multiple-Output Cost Functions**

Up until this point, we have discussed costs as a function of producing only one product. However, most firms produce more than one product. In this case, we use the Multiproduct Cost Function. This is a function that defines the cost of producing given levels of two or more types of outputs assuming all inputs are used efficiently. The function is given by C(Q1,Q2); where Q1 is the number of units produced of product 1 and Q2 is the number of units produced of product 1. (189)


 * Economies of Scope and Cost Complementarity**

With a firm producing more than one good, the cost of production depend partially on how much of the other good(s) will be produced. Two situations can arise from this, Economies of Scope and Cost Complementarity. Economies of Scope exist when the total cost of producing two goods together is less than the total cost of producing each good separately. This is shown as C(Q1, 0) +C(0,Q2)>C(Q1,Q2). Cost Complementarities exist in a multiproduct cost function when the marginal cost of producing one output is reduced when the output of another product is increased. (189-190)

Questions

1. A production manager is trying to figure out his Average Product of Labor for his plant. He can either have 15, 20, or 25 workers on the clock at any time. With 15 workers on, 2500 units can be produced. With 20 workers on, 2900 units can be produced. With 25 workers on, 3300 units can be produced. Which number of workers has the highest Average Product of Labor, 15, 20, or 25?

Answer: 15 workers (Average Product of Labor=166.7)

2. Why do marginal returns eventually become negative? A. Workers become tired as time goes on B. Turnover can be high at certain times C. Too many workers can get in each other's way

Answer: C (You can reach a point when their are actually too many workers to be effective and efficient.)

3. A firm manager is thinking about hiring one more worker, which he has found will increase the output by 250 units. Currently, each unit sells for $13. What is the value of Marginal Product of Labor? A. $3500 B. $3250 C. $3000 D. $325

4. A company has a linear production function, where their machines make the product three times faster than the workers can.

5.

6. A manager has found that his firms Marginal Rate of Technical Substitution is 2/3. She is currently paying her production employees $10 an hour, and renting her capital for $15 an hour. Is this firm producing at their lowest cost possible?

Answer: Yes, because the MRTS=price of labor/price of capital

7. Woodbridge Brewing Company has found out that barley (an input in beer production) prices are going to increase next year. Will their isocost line recede or grow outward?

Answer: The isocost line will recede, allowing them to produce less unless they increase their spending on barley.

8. Looking at Question #7, will Woodbridge Brewing Company be able to produce at the same Isoquant curve point that they have been once the barley prices go up if they can't increase their spending on barley?

Answer: No. Unless they are willing to pay more for barley they will not be able to produce on the same Isoquant curve.

9. A manager needs to lower costs. He has to find a variable cost that he can hopefully lower. Which of the following is a variable cost? A. 5 year lease signed last year on the production facility B. Employee Benefits program that is up for renegotiation in 3 years C. Sales Reps vehicle leasing program that is renewed every 6 months

Answer: C. (A and B are both fixed costs)

10. You pay a one time non-refundable fee of $10,000 to be able to sell your product at the local YMCA's vending machines for a 1 year period starting in 2 years. One week after signing the contract, the YMCA is forced to close and has no plans for re-opening. Should this cost affect your future financial decision making and why?

Answer: No. This is a Sunk Cost, which should not affect your decision making

Baye, Micheal R. __Managerial Economics and Business Strategy__. 2006, pp. 157-190