CASE STUDY: Office equipment company uses its salesforce of 18 to find new customers, renew existing contracts, and capture customers from rivals. Production Issue: How to allocate salesforce efficiently to maximize/increase total sales.
Production and cost are closely linked, since more (less) efficient production decreases (increases) a firm's cost of production. Production goal: Produce a given output (QS) at the minimum cost, i.e. maximize output and minimize inputs.
PRODUCTION TECHNOLOGY
Production is the process of transforming inputs (factors of production) into outputs (goods and services). Example: GM produces vehicles from raw materials, parts, labor, capital equipment, land, financial capital, electricity, etc.
Production function: QS = f (M, L, K), where
M = materials
L = labor
K = capital
Example: GM plans to produce 3m vehicles (Q) using materials (M = $15B), capital (K = $10B), and labor (L = 80,000 workers). The output goal of Q = 3m could be achieved with various other combinations of M, K and L. How?
Firm's
production function typically assumes: a) profit maximization and b) technical
efficiency (state-of-the-art production) at a given point in time, i.e. no
waste. With advances in technology over time, the production function
will change. Pursuit of profits will lead to increased efficiency in
production.
Production
Function for Auto Parts, p. 217-219. QS = f (L and K), where L = # workers, and K = plant size. Firm
has no choice over materials in the short-run (SR), because each part requires a
fixed amount of raw materials, but it can vary number of workers (L) and plant
size (K) in SR to produce various levels of Q.
See Production Function in Table 6.1 (p. 217), where the numerical values
are: parts (Q) per day, as a function of L (workers) and K (plant size).
PRODUCTION WITH ONE VARIABLE INPUT Long run: Period long enough for a firm to vary ALL of its inputs (M, L, K), make major expansions OR contractions in output. Exact time for LR varies by industry, could be 6 months to 5 years, depending on how long it takes to make a MAJOR change in Q. Example: building a new Starbucks or Wal-Mart versus a new car factory or sports stadium or mall.
Short Run: Period of time during which one or more inputs is FIXED, so there is no (or limited) production flexibility. Examples: Long-term labor contract makes L fixed in SR. Time-to-build (or contract/shrink output) makes K fixed in SR.
Marginal Product (MP) of a Variable Input: Total Product = Q, and Q = f (L) or Q = f (K), ceteris paribus. MPL = ΔTP / ΔL = ΔQ / ΔL; or MPK = ΔTP / ΔK = ΔQ / ΔK. (Note: TP = Q)
Example: Holding M and K (plant size) constant, we vary ONE INPUT (L) and determine the change in TP (Q) with a one-unit change in L = MPL.
Table 6.2 (p. 218) holds K constant at 10,000 sq. ft. plant size, and we then vary L. If we increase L from 10 to 20 (ΔL = 10), TP (Q) goes from 93 to 135 (ΔQ = 42), therefore MPL = 42 / 10 = 4.2, meaning that for each worker added, 4.2 additional parts will be produced per day. If we increase L from 20 to 30 (ΔL = 10), TP increases from 135 to 180 (ΔQ = 45), MP = 4.5, etc. to generate the values in the table. Notice that MP first rises, hits a MAX at 5 (when L goes from 30 to 40), then MP falls, then goes negative (when L goes from 120 to 130), see Figure 6.1, panel b, on p. 220. MP initially rises from efficiency gains due to specialization of labor, team production, etc., and the fact that the firm's machinery and capital equipment is initially underutilized. For example, maybe it is more efficient to have two workers per machine instead of one.
TP (Total Product) is also graphed on p. 220, Figure 6.1, TP = f (L), for both 10,000 and 20,000 sq. ft. plants.
LAW OF DIMINISHING MARGINAL RETURNS
Law of Production that states: As one variable input increases (ceteris paribus), marginal production will first rise and eventually decline, i.e. increases in Q from additional units of the variable input will eventually diminish. In this case, there are increasing returns to scale from 10 to 40 workers, as MP increases. As L increase beyond 40, there are diminishing marginal returns, and MP starts to decline. Beyond 120 workers, there are negative returns to scale, as MP goes negative.
Law of Diminishing Returns = MP eventually declines.
Optimal Use of an Input
Since all firms face diminishing marginal returns for all inputs, there is a production trade-off: increasing the variable input increases output, but a decreasing rate, meaning that costs of the variable input are increasing at an increasing rate. Declining productivity/efficiency = increasing cost of production. To Max Profits, the firm wants to make optimal use of each input. How to decide? Marginal analysis, looking at the Marginal Profit from increases in the VARIABLE INPUT.
Profits = f inputs: (L, K or M) and NOT f (Q) output. Trade off: Marginal Profit from increases in Q from using additional units of the variable input, VERSUS the marginal cost (MC) of the input.
Example: Adding one additional worker (L) raises output of auto parts, which raises TR, versus the MC of the additional worker. Logic: Compare MR per worker vs. the MC per worker, to MAX profits.
Marginal Revenue Product (MRP) is the additional revenue (MR) from increased use of an input. Example: Firm considers increasing L from 20 to 30 workers. MP per worker is 4.5 auto parts per worker (MP = 4.5). Firm can sell parts at $40, so MR = $40.
When L = 20, Q = 135, and TR = $5400 ($40 x 135)
When L = 30, Q = 180, and TR = $7200 ($40 x 180)
Δ TR = $1800
Δ L = 10
MRPL = Δ TR / Δ L = $1800 / 10 = $180
Alternatively,
we know that MRPL = Δ TR / Δ L, we can divide both terms by ΔQ,
rearrange to get:
MRPL
= (ΔTR / ΔQ ) x (ΔQ / ΔL)
MRPL = (MR) x (MPL) MRPL = $40 x 4.5 = $180
Interpretation: MRP depends on: a) the efficiency of production (MP), and b) the market value of that production in the marketplace (MR).
From 30 to 40 workers: MRPL = $40 x 5 = $200.
Marginal
Cost of an Input (MCL) = ΔTC
/ ΔL
vs. (Marginal Cost of Output: MC = ΔTC / ΔQ)
In this case, we assume there is an elastic
supply of labor available at a constant cost of $160 per day, and assume that
the firm can hire additional workers without bidding up the wage, so MCL
= $160. vs. (Marginal Cost of Output: MC = ΔTC / ΔQ)
Marginal Profit per Worker, MПL = MRPL - MCL
The marginal profit per worker is equal to the incremental revenue generated (MRPL), MINUS the worker's marginal cost (MCL). To maximize profits, the firm wants to expand inputs until MПL = 0, therefore profit maximization occurs when MRPL = MCL. Firm should increase L as long as marginal revenue from labor, MRPL > MCL of labor, and stop when MRPL = MCL. In this case constant MCL = $160, and constant MR = $40. Due to diminishing returns to scale in production, the MP will eventually fall, which will eventually cause MRPL to fall.
Going from 20 to 30 workers, MRPL = $180 per day, compared to MCL = $160, so the move would be profitable, since MRPL > MCL.
Going from 30 to 40 workers, MRPL = $200 > $160, so that move would be profitable.
Going from 40 to 50 workers, MRPL = $40 x 3.3 = $132, which is less than $160, so that move would be unprofitable. Reason: Diminishing Marginal Returns are setting in. Conclusion: Firm should hire 40 workers to maximize profits, where Q = 230 parts per day, and plant size = 10,000 sq. ft.
What if the plant size = 30,000 sq. ft.? Optimal work force = 70 workers and Q = 520 parts per day. Why? Going from 50 to 60 workers, MRPL = $40 x 5.3 = $212. Going from 60 to 70, MRPL = $40 x 4.2 = $168. From 70 to 80, MRPL = $40 x 3.2 = $128. Labor = $160/day, and the optimal L* = 70.
What about 20,000 sq. ft. plant? 40,000 sq. ft.?
PROBLEM: A firm's production function is: Q = 60 L - L 2, where Q = output per hour and L = number of labor hours. MR = $2 and MCL = $16/hr. How many workers, how much output to MAX П? Set MRPL = MCL.
MPL = ΔQ / ΔL = 60 - 2L, which steadily declines as L increases, due to diminishing marginal returns.
MRPL = MR x MPL = $2 x (60 - 2L) = 120 - 4L
Setting MRPL = MCL: 120 - 4L = $16, and L* = 26 units
Q = 60 (26) - (26)2 = 884 units per hour.
TR = 884 x $2 = $1768
TCL = $16 x 26 = $416
П net of labor costs = $1352
PRODUCTION IN THE LONG RUN (LR)
In the long run, all inputs are variable, all plant sizes are possible, all size workforces are possible, etc. Firm can make major adjustments (expansions or contractions) to production. Two dimensions for major adjustments:
1. Holding production constant, a firm can vary the proportions of inputs, vary the mix of Capital (K) to Labor (K/L). For example, the firm can make capital investments to substitute capital for labor in the long run. Examples: bar code systems, ATM machines, automated gas pumps, automatic elevators, robotics, CAD, computer prototype testing. Strategy: Invest in capital equipment to reduce labor costs.
2. Adjusting the scale/size of production in the LR to achieve the lowest cost, most efficient level of production, depending on the presence of "economies of scale." How fast to expand? What is the optimal size firm? Depends on firm and industry. Mass production of a standardized product versus specialized, customized production. Fast food restaurant chain versus a high-end gourmet restaurant like Emeril's.
RETURNS
TO SCALE helps determine the optimal
size firm in the LR.
Change
in scale = given percentage change in
all inputs, e.g. 10% increase in ALL inputs: L, K, M.
Question: How does firm's output (Q) respond to a change in scale?
Returns to scale measures the %ΔQ from a given % change in inputs, i.e.
if all inputs increase by 10%, what is %ΔQ? Question: How does firm's output (Q) respond to a change in scale?
Returns to scale = %ΔQ / %ΔInputs, and there are three possible classes of returns: constant returns, increasing returns and decreasing returns.
1. Constant Returns to Scale: If inputs double, output doubles. If inputs increase (decrease) by 10%, output increases (decreases) by 10%. In other words, efficiency in production does not increase or decrease as output is expanded or contracted, firm's inputs are equally productive at small and large levels of output. Example: a firm's production process can be replicated easily, such as when a dry cleaner, fast food restaurant, gas station, grocery store increase volume by increasing the number of outlets and does not face any economies of scale OR diseconomies of scale. Or the firm does not: a) face increasing costs when it expands or b) decreasing costs when it expands.
2. Increasing Returns to Scale: If inputs double, output MORE THAN doubles. If inputs increase (decrease) by 10%, output increases (decreases) by MORE THAN 10%, due to economies of scale, increased efficiency at higher levels of output due to greater specialization, more advanced production techniques, etc. Example: You are making a handmade desk in your woodshop. If you build two desks at the same time, you could easily double production (Q) without doubling your time. Maybe it would take 25% more time but you would increase output by 100%.
Production of motor vehicles has increasing returns to scale for certain levels of output: You could double production from 100,000 cars to 200,000 cars per year WITHOUT doubling input costs, due to increasing returns to scale, economies of scale.
POINT: When increasing returns to scale are present, the firm should attempt to expand output.
3. Decreasing Returns to Scale: When inputs double, output increases by LESS THAN 2X. If inputs increase by 10%, output increases by LESS THAN 10%, due to diseconomies of scale. Decreasing returns could result from inefficiencies of large organizations, the cost of bureaucracy, inflexible, rigid and overly complex decision-making, and inefficiencies of coordinating large-scale production.
Point: When facing decreasing returns to scale, the firm should NOT expand, and may even considering contracting. HOW?
Output Elasticity is a convenient, quantitative measure of Returns to Scale = %Q / %Inputs.
When output elasticity is > 1, there are ________________ returns to scale.
When output elasticity is < 1, there are ________________ returns to scale.
When output elasticity is = 1, there are ________________ returns to scale.
Least-Cost Production in LR, when firm can vary ALL of its inputs, involves determining the optimal mix/combination/ratio of inputs. Assume two inputs, K and L, then:
Q = f (L, K)
where L = number of labor-hours per month
and K = amount of capital used per month
To produce a given level of output Qo, the firm can use any K/L ratio in the LR. By increasing K (L), it can reduce L (K). What is the optimal mix of K and L? Depends on the input costs and the marginal products of the inputs. Labor costs = PL and capital costs = PK, so that total costs:
TC = PL * L + PK * K and the firm wants to produce Qo at the min TC.
It can be show that the MIN COST condition occurs when:
MPL =
PL
That is, when the ratios of MPs to Input Costs are equal across ALL inputs. The minimum cost condition implies that the extra output per dollar of input must be equalized across all inputs. (For more than two inputs, the condition would include more than two ratios.)
To see why, consider the case below where the ratios were NOT equal. Assume that MPL = 30 units per hour, and PL = $15/hour; and MPK = 60 and PK = $40.
MPL >
PL
30 > 60 or 2 > 1.5
$15 $40
Labor's productivity per dollar spent is greater than the productivity of capital. For $1, the firm gets 2 marginal units of output from L and only 1.5 units from K. The firm can profit by increasing L's contribution and decreasing K's contribution.
Notice: The ratio of MPs = MPK / MPL = 60 / 30 = 2 / 1,
so the firm can substitute 2 units of L for 1 unit of K. That is, the firm can maintain output at Qo by substituting 2 units of Labor for 1 unit of Capital, i.e., increase L by 2 and decrease K by 1. Effect on TC = +$40 savings from one fewer units of K, -$30 increase for two additional hours of labor (2 x $15 = $30), for a net savings of $10 in TC, to produce the same output Qo.
Point: If one input's productivity per dollar > another input's productivity per dollar, the firm can lower costs by switching toward greater use of the more productive (cost-effective) input. Keep switching until the input productivity per dollar is equalized. As the firm makes more intensive use of the more productive input, what will happen to its MP?
Problem: A firm faces the production function: Q = 40 L - L2 + 54 K - 1.5
and PL = $10 and PK = $15
MPL = dQ / dL = 40 - 2L
MPK = dQ / dK = 54 - 3K
MIN COST production must meet the condition:
40 - 2L = 54 - 3K
$10 $15
which after simplifying is equal to L = K + 2, and the firm should use 2 more units of L than K, such as K = 8 and L = 10, which satisfies the MIN COST condition.
Q = 40 (10) - (10)2 + 54 (8) - 1.5 (8)2 = 636
TC = $10 (10) + $15 (8) = $220
The minimum cost of producing 636 units is $220 using 10 units of L and 8 units of K. Another optimal outcome is K = 18 and L = 20, to produce 886 units at a min cost of $470.
MEASURING PRODUCTION FUNCTIONS
Using engineering and economic data, managers can estimate and quantify their firm's Production Function. 3 common specifications for Production Functions are:
1. Linear Production
Q = aL + bK + c
where L = labor and K = capital, and a, b and c are coefficients to be estimated from the data, e.g. using OLS (linear regression).
Linear production assumes constant MP, MPL = a, and MPK = b, where a and b are coefficients, fixed numbers. Compare to the MP on p. 220, Figure 6.1, panel b. Constant MP might be accurate over certain ranges of inputs, but is NOT consistent with the Law of Diminishing Marginal Productivity.
Linear Production also assumes that K and L are perfect substitutes, which results in the outcome that the firm then uses ONLY K or L, depending on which one is cheaper. For example, assume that Q = 20L + 40K, so that the MP of capital is twice that of labor. We can say that 2 labor-hours is a perfect substitute for 1 machine-hour, and vice-versa. Therefore, the firm would compare the cost of 2 labor-hours to 1 machine-hour, and make exclusive (100%) use of the cheaper input. Only when MP declines will it be optimal to use BOTH inputs.
Conclusions:
1. Constant MP = exclusive use of one input to minimize cost, which is an extreme case.
2. Declining MP = combination of inputs to minimize cost.
2. Production with Fixed Proportions
Output is produced with a FIXED proportion of inputs. Extreme opposite of linear production (perfect input substitutability), assumes fixed input proportions with NO variation or substitutability (perfect complements).
Examples: Taxi cab (K) and driver (L), construction crane (K) and operator (L), typist (L) and computer (K), seamstress (L) and sewing machine (K), etc., cases where there is a 1/1 proportion of K/L, with no possibility of substituting K for L, or vice-versa. Expansion of output requires equal increases in the inputs, e.g. 10 more taxis + 10 more drivers. One driver can't drive more than one taxi, and more than one driver can't drive one taxi.
Another extreme case, since there usually is some opportunity to substitute K for L, or L for K. For example, even if it takes one seamstress to operate one sewing machine, investment could still be made in capital equipment to make the machinery more productive, reliable, etc., which might also reduce the skill level required for operation.
Implication of Fixed Proportions: If one input becomes more expensive, the firm CANNOT shift towards the other input. For example, if taxi drivers are unionized and receive a pay increase, the firm cannot shift towards more intensive use of capital to save money.
3. Cobb-Douglas Function
Most common production function: Q = c La Kb
where
c, a and b are coefficients to be estimated, and a and b are
between 0 and 1. Cobb-Douglas function exhibits diminishing
marginal returns to each input:
MPL
= d Q / d L = c Kb L a-1
MPK
= d Q / d K = c La K b-1
Conclusions of Cobb-Douglas: 1. MPL depends on both capital (K) and labor (L)
2. As K increases, MPL increases, indicating complementary inputs, e.g. investments in capital equipment make L more productive.
3. As L (or K) increases, MPL (or MPK) decreases, reflecting diminishing marginal returns.
For example, assume that a = .5 and MPL = La-1
when L = 10, ceteris paribus (c and Kb are constant), then MPL = (10)-.5 = .3162
when L = 12, ceteris paribus (c and Kb are constant), then MPL = (12)-.5 = .2887
4. Returns to scale depend on the sum of the exponents.
a. Constant returns to scale when a + b = 1.
b. Increasing returns to scale when a + b > 1.
c. Decreasing returns to scale when a + b < 1.
Proof: Assume L = K = 2, then L = K = 3, a 50% increase in inputs, what happens to output (%Q)?
a. Q
= 2.6 x 2.6 = 2.30
Q
= 3.6 x 3.6 = 3.74, a 62% increase in output Q, from
a 50% increase in inputs K and L.
Increasing returns to scale, .6 + .6 = 1.20 (sum of
exponents > 1)
b. Q = 2.5 x 2.5
= 2.00
Q = 3.5
x 3.5 = 3.00, a 50% increase in Q, from a
50% increase in inputs.
Constant
returns to scale, .5 + .5 = 1 (sum
of exponents = 1)
c.
Q = 2.4 x 2.4 = 1.74
Q = 3.4 x 3.4 = 2.41, a
37.9% increase in Q, from a 50% increase in inputs.
Decreasing returns to scale, .4 + .4 = .80 (sum of exponents < 1)
5. We can easily convert Cobb-Douglas to log
form (using the rules of logs): Log (Q) = Log (c) + a Log (L) + b Log (K), we can convert all variables to logs and run OLS.
Example 4, p. 237:
Q = L.5 K.5 and PL = $12 and PK = $24. Constant returns to scale, and inputs are equally productive (a = b = .5), but capital is twice as expensive as labor. The optimal mix of L and K satisfies the equation:
MPL = MPK or
PL PK
.5 L -.5 K.5 = .5 L.5 K-.5
$12 $24
Cross multiply: 12 L-.5 K.5 = 6 L.5 K-.5
2 L-.5 K.5 = L.5 K-.5
2 K.5 = L.5
K-.5 L-.5
which simplifies to: 2K = L or K = .5L Remember that: xy / xn = xy-n, so that K.5 / K-.5 = K.5 - (-.5) = K1 = K; and also L.5 / L-.5 = L
The firm uses a ratio of 2L for each 1K (e.g., 20 L and 10 K), or twice as many units of labor, because _________________________________.
Estimating Production Functions
1. Using Engineering Data. How much output can be expected to be produced per machine, or per factory, or per day, or per worker under different operating conditions? Firms like GM may have this engineering data on machine/factory capacity, etc. to then estimate a production function. However, what about a new product?
2. Using Economic/Statistical Data, either time-series or cross-section data. For example, GM may have detailed time-series and cross-section data on outputs over time and across facilities, along with input data on labor, materials, capital, land, etc., to construct a production function. Important questions that could be answered:
1.
For a fixed plant size (K), what is the effect on output (Q) of increasing L,
e.g., adding extra shifts?
2. Does the plant or industry exhibit increasing (decreasing) returns to scale? If so, over what range of output?
2. Does the plant or industry exhibit increasing (decreasing) returns to scale? If so, over what range of output?
3.
For a fixed labor supply (e.g., union contract), what is the effect of adding/investing
in K?
Illustrates the interaction and importance of
operations research, engineering and statistics to managerial economics,
estimating production functions using operations research and OLS. Case: Returns to Scale in Coal Mining (from old textbook): Study of surface coal mining estimated production functions using OLS, for coal mines in Illinois of different sizes, based on production of coal (tons) per mine, amount of labor (hours) per mine, and amount of capital equipment ($) per mine. Average Output elasticity = 1.24 over most ranges of coal output (20% increase in inputs raised output by ______), and economies of scale were not exhausted until an annual output level of 4.8m tons, a level higher than the actual operating scale of most mines. Thus, further increases in scale would be warranted. In addition, there was evidence that increased use of large-scale, earth-moving equipment greatly enhanced the degree of returns to scale. Higher capital intensity implies greater returns to scale in coal mining.
OTHER PRODUCTION DECISIONS
We have discussed:
1. Optimal use of a singe input: MAX profits by increasing use of L until the MRPL = MCL.
2. Optimal mix of inputs: Equalize MPx / Px for all inputs x1, x2, x3, etc...
Now we consider:
3. Optimal allocation of a single input (x1) among multiple production facilities, producing the same output/product (Q1).
4. Optimal allocation of a single input (x1) across multiple products (Q1, Q2), possibly at multiple facilities.
Multiple Plants Using a Single Input (x1) to Produce Same Output (Q), p. 239
Another application of Marginal Analysis. Oil refinery has two plants, Plant A and Plant B, and it uses 10,000 barrels of oil per year as the main input (M) to make gasoline (Q) at both plants. Input (M) can be split between the two plants A and B to maximize total output Q, subject to:
Q = QA + QB where Q is Total Output of Gasoline, and
10,000 bbls. = MA + MB (INPUTS)
Output (Q) will be maximized when MPA = MPB, and MP is equalized across plants. (In this case, the PM is the same for both plants, so we can ignore.) For example, if MPB > MPA, then more oil should be allocated to the high-MP Plant B and less to low-MP Plant A. We have information on production:
QA = 24 MA - .5 MA2
QB = 20 MB - MB2
and
MPA = 24 - MA
MPB = 20 - 2MB
See Figure 6.4 on page 241. Which plant is more productive overall? Why?
Suppose we start by allocating 50/50, using 5,000 bbls. at each plant, with Q = 182,500.
QA = 24 (5) - .5 (5) 2 = 107.50
QB = 20 (5) - 52 = 75
TOTAL Q = 182.50 or 182,500 bbls.
Using the MP equations or the graph, we can see that MPA > MPB, because 19 > 10. We know therefore that they should shift more oil to Plant A and less to Plant B, until MPA = MPB.
Solution (set MPs equal):
24 - MA = 20 - 2 MB
MA + MB = 10 and MA = (10 - MB)
24 - 10 + MB = 20 - 2 MB
MB = 2 and MA = 8.
With that combination, MP = 16 for both plants, using the MP equations or the graph and Q = 196,000 gallons. We can also verify that even though Plant A is more efficient and productive overall, it would not make sense to shift the entire production there. QA = 24 (10) - .5 (10)2 = 190,000 gallons. Also, at that point MPA = 14 and MPB = 20, so some production should be shifted from Plant A to Plant B.
Using One Input for Multiple Products
Examples: Computer chips that are used in various computers. Raw material, parts, frames, transmissions AND workers used in production of more than one vehicle. Managerial talent might be the input in shortest supply and greatest demand, how to allocate human resources efficiently among different products, assignment, accounts, etc.?
The problems of how to: a) allocate inputs across multiple plants and b) how to allocate inputs across multiple products are very similar, in that both problems are allocation decisions: how to allocate scarce resources efficiently among multiple uses? In both cases, the firm is trying to MAX Q and MIN Inputs, or Maximize productive efficiency, or economize on the use of scarce resources, etc.
Case Study: Assume now that the oil refinery uses crude oil as an input for two products: gasoline and synthetic fiber. The firm's current supply of crude oil is 20,000 barrels, which is less than the current combined demand for both products. The two managers are arguing for a greater share of the input. How to decide? The firm must allocate the input to where it is most profitable. Decision problem is based on the following:
Maximize Profits, subject to the constraint:
MG + MF = 20,000 bbls. of crude oil, used for either Gasoline (G) or Fiber (F).
Total Profit will be maximized when the input (crude oil) is allocated between G and F, so the two products generate identical Marginal Profits per Unit of Input:
MПG = MПF
If MПG > MПF , then the firm show allocate more crude oil to ____________ and less to ____________ to increase profits.
Suppose the production functions are:
Gasoline: G = 72 MG - 1.5 MG2
Fiber: F = 80 MF - 2 MF2
G = thousands of gallons of gasoline and
F = thousands of square feet of fiber.
Profits per gallon of G = $0.50
Profits per sq. ft. of F = $0.75
Marginal Profits are:
MПG = ($0.50) MPG = ($0.50) (72 - 3 MG) = $36 - 1.5 MG
MПF = ($0.75) MPF = ($0.75) (80 - 4 MF ) = $60 - 3 MF
Setting $36 - 1.5 MG = $60 - 3 MF simplifies to:
MF = .5 MG + 8
We know that MG + MF = 20, so that MF = 20 - MG which we substitute into the above equation and solve for MG and MF.
MG = 8 and MF = 12.
G = 72 (8) - 1.5 (8)2 = 480,000 gallons of Gasoline x ($0.50) = $240,000 profit from G
F = 80 (12) - 2 (12)2 = 672,000 sq. ft of fiber. x ($0.75) = $504,000 profit from F
Total П = $744,000 (minus cost of crude oil).
Logic: Fiber has a greater profit contribution per unit, which justifies allocating 60% of the input to F and only 40% to G.
Page 242, Check Station 5: What if profit per sq. ft. for fiber (F) falls to $.375 per foot?
FINAL REMARKS
1. If there are more than two plants (or two products) using one input, the same marginal analysis is simply extended for three or more plants or products. Marginal products or marginal profits should still be equalized across plants or products, i.e. MPA = MPB = MPC, etc. or MПA = MПB = MПC , etc.
2. If the supply of the input can be varied, then the decision changes. We were assuming a fixed (short-run) supply of an input, so that the firm faced an allocation decision, allocating a fixed supply of an input among plants or products. When the input is not fixed, then the firm should follow the rule:
For each plant and/or each product, the firm should increase use of an input until its MRP (Marginal Revenue Product) equals its MC per unit (input price), i.e. MRPA = PA, etc.
