Download Unit 5 -- Cost Functions and Utils 25

Managerial Economics
Cost Analysis
In a world of scarcity, firms have to pay for their inputs, i.e., land, labor, and capital.
However too much production can be as dangerous as too little; history shows that overexpansion can often drive a fast growing company into bankruptcy by pushing up costs
far faster than revenues.
Explicit Costs and Implicit Costs
Economists classify these types of costs: explicit (=accounting) costs, and implicit costs.
Explicit costs are out of pocket, obvious kinds of costs, e.g. expenses on books, tuition,
gas, etc.. Implicit costs are not really expenses you incur, but involve income or values
you are giving up by not doing something that you could have chosen to do. For
instance, if you decide to go to school full time instead of working a $20,000 job, you are
giving up earning $20,000. This is your implicit cost. Sample problem: Suppose you are
running a small business and incur the following expenses: labor = $80,000; raw materials
= $30,000; finance charges on a loan = $3,000. You are not paying explicit rent, because
you own the building you are operating in. If you would rent it out, however, you could
be earning $12,000. You also estimate your own time to be worth $25,000. What are your
expenses?
Answer: Explicit costs = $80,000 + $30,000 + $3,000 = $113,000.
Implicit costs = $12,000 + $25,000 = $37,000.
Total economic costs = explicit + implicit costs = $150,000
Accounting versus Economic Profits
To calculate accounting and economic profits we need to know the company’s total
revenue. Let’s suppose it is $140,000. Then, accounting profits are: total revenue minus
explicit costs or: $140,000 - $113,000 = $27,000. Economic profits are: total revenue minus
total economic costs or; $140,000 - $150,000 = -$10,000 (i.e. a loss of $10,000).
The above firm reaps a positive accounting profit; but the negative economic profit
indicates that from an economic point of view, the owner should discontinue the
operation.
Total and Per Unit Costs
Seven cost functions which we will discuss are:
1. Total Variable Cost (TVC) – The cost of all variable resources
Examples: Cost of labor, materials, office supplies
2. Total Fixed Cost (TFC) – The cost of all fixed inputs
Examples: Cost of the building, large pieces of machinery, certain taxes
3. Total Cost (TC) – This the sum of TVC and TFC.
4. Average Variable Cost (AVC) – This is variable cost per product.
5. Average Fixed Cost (AFC) – This is fixed cost per product.
6. Average Total Cost (ATC) – This is total cost per product.
7. Marginal Cost (MC) – This is the cost of producing an additional unit of the
product.
Cost Calculations
Using the above abbreviations and Q for the quantity of output:
ATC = TC/Q
AFC = TFC/Q
AVC = TVC/Q
MC = change in TC/change in Q
Example: Let’s suppose you are making 50 bottles of wine each week. You know that
your fixed costs add up to $300, and your variable costs amount to $900. You also know
that if you were to make an extra 5 bottles, your total cost would rise by $60. What is your
total cost; average total cost; average total cost; average variable cost; average fixed
cost; and marginal cost?
Answer: Total cost = $300 + $900 = $1200
ATC = $1200/50 = $24; AVC = $900/50 = $18; AFC = $300/50 = $6; MC = $60/5 = $12
A table with cost data might show the following:
Cost Curves
The curves above show typical shapes of a firm’s total cost, total variable cost and total
fixed cost curves. Total fixed cost is constant at $50 for all levels of production. Total cost
and total variable cost increase with higher levels of output. Note that total fixed cost
and total variable cost always add to total cost.
Marginal and average cost curves
There is an important link between MC and ATC cost curves: when MC is bellow ATC then
ATC declines and when MC is above ATC, ATC rises. For a U- shaped ATC curve, the point
where MC equal ATC, is also the point where ATC hits its minimum level. In brief:
if MC< ATC,
ATC falls;
if MC = ATC
ATC is minimum and no change in ATC;
if MC> ATC
ATC rises.
The Figure drawn above shows this relationship. Graphically, a rising MC curve will
intersect ATC (and also AVC) where ATC (and AVC) is minimum; before that MC is bellow
ATC (and AVC) and after that MC is above ATC (and AVC). (Note that AC and ATC are
the same thing.)
The Long Run Average Cost Curve
The long-run average cost curve is derived from a number of short-run average cost
curves. For each fixed plant size (short run), you look at the lowest costs for that size plant.
These bottom portions of the different short run cost curves make up the long run
average cost curve.
A firm’s long run average cost curve is the “envelope” of many short run average cost
curves. All inputs are variable and the firm has the choice of building or changing to a
variety of plant or facility sizes. A small operation (SRAC1) which wants to produce 300
units will have average costs of $26. A larger one, which produces 700 units, can produce
each product for $17 (economies of scale). When the firm gets too large (SRAC6),
average costs rise to $20 (diseconomies of scale).
Increasing, Decreasing, and Constant Returns to Scale
Note that increasing returns to scale is closely associated with the concept economies of
scale (the downward sloping part of the long run average total cost curve.) Decreasing
returns to scale relates to diseconomies of scale (the upward part of the curve).
Increasing returns to scale occurs when a firm increases its inputs, and a more than
proportionate increase in production results. For example, one year a firm employs 200
workers and 50 machines and produces 1000 products. A year later it increases the
number of workers to 40 0 and the machines to 100 (inputs doubled) and the output rises
to a level of 2500 (more than doubled).
Increasing returns to scale is often accompanied by decreasing long run average costs
(economies of scale). A firm which gets bigger may experience this because of
increased specialization, more efficient use of large pieces of machinery (for example,
use of assembly lines), volume discounts, etc.
Decreasing returns to scale happens when the firm’s output rises by less than the
percentage increases in inputs. In the last example, had the firm’s output risen to 1500,
we would experience decreasing returns to scale.
Decreasing returns to scale can be associated with rising long run average costs
(diseconomies of scale). An organization may become too big, thus creating too many
layers of management, too many departments, and too much red tape. This lead to a
lack of communications, inefficiency, and delays in decision making.
Constant returns to scale occurs when the firm’s output rises proportionate to the
increase in inputs. What would output have to be for this to take place?
Appendix
Finding Minimum Average Variable Cost
Given the total cost function
TC = 1,000 + 10Q – 0.9Q2 + 0.04Q3
Find the rate of output that results in minimum average variable cost.
Solution : Marginal cost is the first derivative of the total cost function
dTC
dQ
MC
10 1.8Q 0.12 Q 2
Now, find the variable cost function (VC) by subtracting the fixed cost component
($1,000) from the total cost function. That is,
VC = 10Q – 0.9Q2 + 0.04Q3
Then find average variable cost (AVC) by dividing VC by output (Q). That is,
TVC 10Q 0.9Q 2
AVC
Q
Or, AVC = 10 –Q0.9Q + 0.04Q2
0.04Q 3
Because the minimum point of AVC occurs at its intersection with marginal cost, equate
the AVC and MC functions and solve for Q. That is,
AVC = MC
Or, 10 – 0.9Q + 0.04Q2 = 10 – 1.8Q + 0.12Q2
Rearranging terms yields a quadratic equation
– 0.08 Q2 + 0.9Q = 0
or, Q(– 0.08Q + 0.9) =0
Which has the roots
Q1 = 0 and Q2 = 11.25
Disregarding the root associated with a zero output rate, it is seen that minimum AVC is
achieved at an output rate of 11.25 units.
Alternatively, the minimum point of AVC could be found by setting the first derivative of
AVC equal to zero and solving for Q. That is,
d ( AVC )
dQ
0.08Q = 0.9
Q = 11.25
0.9 0.08Q
0
Managerial Economics
Production, cost theory and Decisions of the firm
The production theory and the cost analysis described in previous lectures are among
the fundamental building blocks of microeconomics. A thorough understanding of
production and cost is necessary for an appreciation of how economic scarcity gets
translated into prices in the marketplace. This appendix develops these concepts further
and introduces the concept of profit maximization.
A NUMERICAL PRODUCTION FUNCTION
Production theory and cost analysis have their roots in the concept of a production
function, which shows the maximum amount of output that can be produced with
various combinations of inputs. Table 1 starts with a numerical example of a constant
returns-to-scale production function, showing the amount of inputs along the axes and
the amount of output at the grid points of the table.
Along the left-hand side are listed the varying amounts of land, going from 1 unit to 6
units. Along the bottom are listed amounts of labour, which also go from 1 to 6. Output
corresponding to each land row and labour column is listed inside the table.
If we are interested in knowing exactly how much output there will be when 3 units of
land and 2 units of labour are available, we count up 3 units of land and then go over 2
units of labour. The answer is seen to be 346 units of product. (Can you identify some
other input combinations that will produce q = 346?). Similarly, we find that 3 units of land
and 6 of labour produce 600 units of q. Remember that the production function shows
the maximum output available given engineering skills and technical knowledge
available at a particular time.
TABLE IA-1. A Tabular Picture of a Production Function Relating Amount of Output to
Varying Combinations of Labour and Land Inputs.
When you have 3 land units and 2 labour units available, the engineer tells you the
maximum obtainable output is 346 units. Note the different ways to produce 346. Do the
same for 490. (The production function shown in the table is a special case of the
Cobb-Douglas production function, one given by the formula Q =
THE LAW OF DIMINISHING MARGINAL PRODUCT
Table –1 can nicely illustrate the law of diminishing return. First, recall that the marginal
product of labour is the extra production resulting from 1 additional unit of labour when
land and other inputs are held constant. At any point in Table –1, we can find the
marginal product of labour by subtracting the output from the number on its right in the
same row. Thus, when there are 2 units of land and 4 units of labour, the marginal product
of an additional labour would be 48, or 448 minus 400 in the second row.
By the "marginal product of land" we mean, of Course, the extra product resulting from 1
additional unit of land when labour is held constant. It is calculated by comparing
adjacent items in a given column. Thus, when there are 2 units of land and 4 units of
labour, the marginal product of land is shown in the fourth column as 490 – 400, or 90.
We can easily, find the marginal product of each of our two factors by comparing
adjacent entries in the vertical columns or horizontal rows of Table –1.
Having defined the concept of marginal product of an input, we now can easily define
the law of diminishing returns: The law of diminishing returns states that as we increase
one input and hold other inputs constant, the marginal product of the varying input will
at least after some point decline.
To illustrate this, hold land constant in Table – 1 by sticking to a given row – say, the row
corresponding to land equal to 2 units. Now let labour increase from 1 to 2 units, from 2 to
3 units, and so forth. What happens to q at each step?
As labour goes from 1 to 2 units, the level of output increases from 200 to 282 units, or by
82 units. But the next dose of labour adds only 64 units, or 346 – 282, Diminishing returns
have set in. Still further additions of a single unit of labour give its, respectively, only, 54
extra units output, 48 units, and finally 42 units. You can easily verify that the law holds for
other rows and that the law holds when land is varied and labour held constant.
We can use this example to verify our intuitive justification of the law of diminishing returns
– the assertion that the law holds because the fixed factor decreases relative to the
variable factor. According to this explanation, each unit of the variable factor has less
and less of the fixed factor to work with. So it is natural that extra product should drop off.
If this explanation is to hold true, output should increase proportionately when both
factors are increased together. In the long run, when the firm can change all inputs, it
enjoys constant returns to scale. For example, when the firm uses 1 unit of both land and
labor, it can produce 141 units of output. Now, if the firm doubles the number of inputs
(both land & labor), it can produce double the amount it was producing. So, the firm with
2 units of land and 2 units of labor can produce 282 units of output that is just double of
141. When labour increases from 1 to 2 and land simultaneously increases from 1 to 2, we
should get the same increase in product as when both increase simultaneously from 2 to
3. This can be verified in Table –1. In the first move we go from 141to 282, and in the
second move the product increases front 282 to 423, an equal jump of 141 units.
LEAST-COST FACTOR COMBINATION FOR A GIVEN OUTPUT
The numerical production function shows us the different ways to produce a given level
of output. But which of the many possibilities should the firm use? If desired level of output
is q = 346, there are no less than four different combinations of land and labour, shown as
A, B, C, and D in Table – 2.
As far as the engineer is concerned, each of these combinations is equally good at
producing an output of 346 units. But the manager, interested in minimizing costs, wants
to find the combination that costs the least.
Let us suppose that the price of labour is $2 and the price of land $3. The total costs when
input prices are at this level are shown in the third column of Table – 2. For combination A,
the total labour and land cost will be $20, equal to [(1 X $2) + (6 X $3)]. Costs at B, C and
D will be, respectively, $13, $12 and $15. At the assumed input prices (i.e. $2 for labour per
unit and $3 for land per unit), C is the least costly way to produce the given output.
If either of the input prices changes, the equilibrium proportion of the inputs will also
change so as to use less of the input that has gone up most in price. As soon as input
prices are known, the least-cost method of production can be found by calculating the
costs of different input combinations.
TABLE – 2 Inputs and Costs of producing a Given Level of Output
Now, assume that the firm has chosen 346 units of output to produce. Then it can use any
of the four choices of input combinations
shown as A, B, C and D. As the firm moves
down the list, production becomes more labour-intensive and less land-intensive. (Fill in
the missing numbers.)
The firm’s choice among the different techniques will depend on input prices. When price
of labour is $2 and Price of land is $3, verify that the cost-minimizing combination is C.
Show that lowering the price of land from $3 to $1 leads the firm to choose a more land
intensive combination at B.
Equal-Product Curves
The commonsense numerical analysis of the way in which a firm will combine inputs to
minimize costs can be made more vivid by the use of diagrams. We will use the
diagrammatic approach by putting together two new curves, the equal-product curve
and the equal-cost line.
Let’s turn Table –1 into a continuous curve by drawing a smooth curve through all the
points that yield q = 346. This smooth curve, shown in Figure – 1, indicates all the different
combinations of labour and land that yield an output of 346 units. This is called an equalproduct curve or isoquant. You should be able to draw on Figure –1 the corresponding
equal product curve for output equal to 490 by getting the data from Table –1. Indeed,
an infinite number of such equal-product contour lines could be drawn in.
Equal-Cost Lines
Given the price of labour and land, the firm can evaluate the total cost for points A, B, C
and D or for any other point on the equal-product curve. The firm will minimize its costs
when it selects that point on its equal-product curve that has the lowest total cost.
An easy technique for finding the least-cost method of production is to construct equalcost lines. This is done in figure–2, where the family of parallel straight lines represents a
number of equal-cost curves when the price of labour is $2 and the price of land $3.
Every point on a given equal-cost line represents the same total cost. The lines are straight
because factor prices are constant, and they all have a negative slope equal to the
ratio of labour price to land price, $2/$3, and hence are parallel.
To find the total cost for any point, we simply read off the number appended to the
equal-cost line going through that point. The lines are all straight and parallel because
the firm is assumed to be able to buy all it wishes of either input at constant prices. The lies
are somewhat flatter that 450 because the price of labour PL is somewhat less than the
price of land PA. More precisely, we can always say that the arithmetic value of the slope
of each equal-cost line must equal the ratio of the price of labour to that of land – in this
case.
Equal-Product and Equal-Cost Contours: Least-Cost Tangency
Combining the equal-product and equal-lines we can determine the optimal, or costminimizing, position of the firm. Recall that the optimal input combination comes at that
point where the given output of q = 346 can be produced at least cost. To find such a
point, simply superimpose that single equal-product curve where q = 346, upon the family
of equal-cost lines, as shown in Figure – 3. The firm will always keep moving along the
convex curve of Figure – 3 as long as it is able to cross over to lower cost lines. Its
equilibrium will therefore be at C, where the isoquant touches (but not crosses) the lowest
isocost line. This is a point of tangency, where the slope of the equal-product curve just
matches the slope of an equal-cost line and the curves are just kissing.
We already know that the slope of the equal-cost curves is PL/PA. But what is the slope of
the straight line tangent to the curve at the point in question. For the equal-product
curve, this slope is a “substitution ratio” between the two factors. It depends upon the
relative marginal products of the two factors of production, namely, MP L/MPL.