Economy

Bangladesh University of Business & Technology

Assignment on Production & Cost

Submitted by:Rabiul Islam

ID-001MBM-5th Intake

Production functionIn micro-economics, A production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs. Almost of all macroeconomic theories, like macroeconomic theory, real business cycle theory, neoclassical growth theory (classical and new) presuppose (aggregate) production function. Heckscher-Ohlin-Samuelson theory in international trade theory also presupposes production function. In this sense, production function is one of the key concepts of necoclassical macroeconomic theories. It is also important to know that there is a subversive criticism on the very concept of production function.

Concept of production functions

In micro-economics, A production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs. A meta-production function (sometimes metaproduction function) compares the practice of the existing entities converting inputs X into output y to determine the most efficient practice production function of the existing entities, whether the most efficient feasible practice production or the most efficient actual practice production.[1] In either case, the maximum output of a technologically-determined production process is a mathematical function of input factors of production. Put another way, given the set of all technically feasible combinations of output and inputs, only the combinations encompassing a maximum output for a specified set of inputs would constitute the production function. Alternatively, a production function can be defined as the specification of the minimum input requirements needed to produce designated quantities of output, given available technology. It is usually presumed that unique production functions can be constructed for every production technology.

By assuming that the maximum output technologically possible from a given set of inputs is achieved, economists using a production function in analysis are abstracting away from the engineering and managerial problems inherently associated with a particular production process. The engineering and managerial problems of technical efficiency are assumed to be solved, so that analysis can focus on the problems of allocative efficiency. The firm is assumed to be making allocative choices concerning how much of each input factor to use, given the price of the factor and the technological determinants represented by the production function. A decision frame, in which one or more inputs are held constant, may be used; for example, capital may be assumed to be fixed or constant in the short run, and only labour variable, while in the long run, both capital and labour factors are variable, but the production function itself remains fixed, while in the very long run, the firm may face even a choice of technologies, represented by various, possible production functions.

The relationship of output to inputs is non-monetary, that is, a production function relates physical inputs to physical outputs, and prices and costs are not considered. But, the production

function is not a full model of the production process: it deliberately abstracts away from essential and inherent aspects of physical production processes, including error, entropy or waste. Moreover, production functions do not ordinarily model the business processes, either, ignoring the role of management, of sunk cost investments and the relation of fixed overhead to variable costs. (For a primer on the fundamental elements of microeconomic production theory, see production theory basics).

The primary purpose of the production function is to address allocative efficiency in the use of factor inputs in production and the resulting distribution of income to those factors. Under certain assumptions, the production function can be used to derive a marginal product for each factor, which implies an ideal division of the income generated from output into an income due to each input factor of production.

Production function as an equation

There are several ways of specifying the production function.

In a general mathematical form, a production function can be expressed as:

Q = f(X1,X2,X3,...,Xn)where:Q = quantity of outputX1,X2,X3,...,Xn = factor inputs (such as capital, labour, land or raw materials). This general form does not encompass joint production, that is a production process, which has multiple co-products or outputs.

One way of specifying a production function is simply as a table of discrete outputs and input combinations, and not as a formula or equation at all. Using an equation usually implies continual variation of output with minute variation in inputs, which is simply not realistic. Fixed ratios of factors, as in the case of laborers and their tools, might imply that only discrete input combinations, and therefore, discrete maximum outputs, are of practical interest.

One formulation is as a linear function:

Q = a + bX1 + cX2 + dX3,...where a,b,c, and d are parameters that are determined empirically.

Another is as a Cobb-Douglas production function (multiplicative):

Other forms include the constant elasticity of substitution production function (CES) which is a generalized form of the Cobb-Douglas function, and the quadratic production function which is a specific type of additive function. The best form of the equation to use and the values of the parameters (a,b,c, and d) vary from company to company and industry to industry. In a short run

production function at least one of the X's (inputs) is fixed. In the long run all factor inputs are variable at the discretion of management.

Production function as a graph

Any of these equations can be plotted on a graph. A typical (quadratic) production function is shown in the following diagram. All points above the production function are unobtainable with current technology, all points below are technically feasible, and all points on the function show the maximum quantity of output obtainable at the specified levels of inputs. From the origin, through points A, B, and C, the production function is rising, indicating that as additional units of inputs are used, the quantity of outputs also increases. Beyond point C, the employment of additional units of inputs produces no additional outputs, in fact, total output starts to decline. The variable inputs are being used too intensively (or to put it another way, the fixed inputs are under utilized). With too much variable input use relative to the available fixed inputs, the company is experiencing negative returns to variable inputs, and diminishing total returns. In the diagram this is illustrated by the negative marginal physical product curve (MPP) beyond point Z, and the declining production function beyond point C.

Quadratic Production Function

From the origin to point A, the firm is experiencing increasing returns to variable inputs. As additional inputs are employed, output increases at an increasing rate. Both marginal physical product (MPP) and average physical product (APP) is rising. The inflection point A, defines the

point of diminishing marginal returns, as can be seen from the declining MPP curve beyond point X. From point A to point C, the firm is experiencing positive but decreasing returns to variable inputs. As additional inputs are employed, output increases but at a decreasing rate. Point B is the point of diminishing average returns, as shown by the declining slope of the average physical product curve (APP) beyond point Y. Point B is just tangent to the steepest ray from the origin hence the average physical product is at a maximum. Beyond point B, mathematical necessity requires that the marginal curve must be below the average curve (See production theory basics for an explanation.).

Stages of production

To simplify the interpretation of a production function, it is common to divide its range into 3 stages. In Stage 1 (from the origin to point B) the variable input is being used with increasing efficiency, reaching a maximum at point B (since the average physical product is at its maximum at that point). The average physical product of fixed inputs will also be rising in this stage (not shown in the diagram). Because the efficiency of both fixed and variable inputs is improving throughout stage 1, a firm will always try to operate beyond this stage. In stage 1, fixed inputs are underutilized.

In Stage 2, output increases at a decreasing rate, and the average and marginal physical product is declining. However the average product of fixed inputs (not shown) is still rising. In this stage, the employment of additional variable inputs increase the efficiency of fixed inputs but decrease the efficiency of variable inputs. The optimum input/output combination will be in stage 2. Maximum production efficiency must fall somewhere in this stage. Note that this does not define the profit maximizing point. It takes no account of prices or demand. If demand for a product is low, the profit maximizing output could be in stage 1 even though the point of optimum efficiency is in stage 2.

In Stage 3, too much variable input is being used relative to the available fixed inputs: variable inputs are over utilized. Both the efficiency of variable inputs and the efficiency of fixed inputs decline through out this stage. At the boundary between stage 2 and stage 3, fixed input is being utilized most efficiently and short-run output is maximum.

Shifting a production function

As noted above, it is possible for the profit maximizing output level to occur in any of the three stages. If profit maximization occurs in either stage 1 or stage 3, the firm will be operating at a technically inefficient point on its production function. In the short run it can try to alter demand by changing the price of the output or adjusting the level of promotional expenditure. In the long run the firm has more options available to it, most notably, adjusting its production processes so they better match the characteristics of demand. This usually involves changing the scale of operations by adjusting the level of fixed inputs. If fixed inputs are lumpy, adjustments to the scale of operations may be more significant than what is required to merely balance production capacity with demand. For example, you may only need to increase production by a million units

per year to keep up with demand, but the production equipment upgrades that are available may involve increasing production by 2 million units per year.

Shifting a Production Function

If a firm is operating (inefficiently) at a profit maximizing level in stage one, it might, in the long run, choose to reduce its scale of operations (by selling capital equipment). By reducing the amount of fixed capital inputs, the production function will shift down and to the left. The beginning of stage 2 shifts from B1 to B2. The (unchanged) profit maximizing output level will now be in stage 2 and the firm will be operating more efficiently.

If a firm is operating (inefficiently) at a profit maximizing level in stage three, it might, in the long run, choose to increase its scale of operations (by investing in new capital equipment). By increasing the amount of fixed capital inputs, the production function will shift up and to the right.

Homogeneous and homothetic production functions

There are two special classes of production functions that are frequently mentioned in textbooks but are seldom seen in reality. The production function Q = f(X1,X2) is said to be homogeneous of degree n, if given any positive constant k, f(kX1,kX2) = knf(X1,X2). When n > 1, the function exhibits increasing returns, and decreasing returns when n < 1. When it is homogeneous of degree 1, it exhibits constant returns. Homothetic functions are functions whose marginal technical rate of substitution (slope of the isoquant) is homogeneous of degree zero. Due to this, along rays coming from the origin, the slope of the isoquants will be the same. Homothetic functions are of form F(h(X1,X2)) where F(y) is a monotonically increasing function (the derivative of F(y) is positive (dF / dy > 0)), and function h(X1,X2) is a homogeneous function of any degree.

Aggregate production functions

In macroeconomics, production functions for whole nations are sometimes constructed. In theory they are the summation of all the production functions of individual producers, however this is an impractical way of constructing them. There are also methodological problems associated with aggregate production functions.

Criticisms of production functions

There are two major criticism against the standard form of the production function. On the history of production functions, see Mishra [2007][2].

On the concept of capital

During the 1950s, 60s, and 70s there was a lively debate about the theoretical soundness of production functions. (See the Capital controversy.) Although most of the criticism was directed primarily at aggregate production functions, microeconomic production functions were also put under scrutiny. The debate began in 1953 when Joan Robinson criticized the way the factor input, capital, was measured and how the notion of factor proportions had distracted economists.

According to the argument, it is impossible to conceive of an abstract quantity of capital which is independent of the rates of interest and wages. The problem is that this independence is a precondition of constructing an iso-product curve. Further, the slope of the iso-product curve helps determine relative factor prices, but the curve cannot be constructed (and its slope measured) unless the prices are known beforehand.

Often natural resources are omitted from production functions. When Solow and Stiglitz sought to make the production function more realistic by adding in natural resources, they did it in a manner that economist Georgescu-Roegen criticized as a "conjuring trick" that failed to address the laws of thermodynamics. Neither Solow nor Stiglitz addressed his criticism, despite an invitation to do so in the September 1997 issue of the journal Ecological Economics.[3] For more recent retrospectives, see Cohen and Harcourt [2003][4].

On the functional form

The production functions are usually written as above

Q = f(X1, X2, ... , Xn),

where X1, X2, ... , Xn are arbitrarily chosen inputs. But, this is far from what the firms are doing. Any firm has a well defined design of a product. What is necessary in this specific production, the inputs are precisely determined. For example, to produce a passenger car, it is necessary to have a steering wheel, 5 pneumatic wheels (5th wheel for a reserve), a radiator, a combustion

engine, four sheets, and so on. Car needs more than 10 thousand of parts. Even if you have various parts in different proportions which are necessary to make a car, you will not arrive to knock down a complete car.

More realistic description of input-output relations is

s (a1, a2, ..., an) => s

, where s is the scale of the production (of any product) and a1, a2, ..., an are cefficients of various inputs such as parts and materials which are needed in the production of the product. Usual expression of production function thus disfigures the basic relationships: which determines which. In the real world, it is the output (the amount of products) which determines the inputs (the quantities of parts and materials).

Total, average, and marginal product

Total Product Curve

The total product (or total physical product) of a variable factor of production identifies what outputs are possible using various levels of the variable input. This can be displayed in either a chart that lists the output level corresponding to various levels of input, or a graph that summarizes the data into a “total product curve”. The diagram shows a typical total product curve. In this example, output increases as more inputs are employed up until point A. The maximum output possible with this production process is Qm. (If there are other inputs used in the process, they are assumed to be fixed.)

The average physical product is the total production divided by the number of units of variable input employed. It is the output of each unit of input. If there are 10 employees working on a

production process that manufactures 50 units per day, then the average product of variable labour input is 5 units per day.

Average and Marginal Physical Product Curves

The average product typically varies as more of the input is employed, so this relationship can also be expressed as a chart or as a graph. A typical average physical product curve is shown (APP). It can be obtained by drawing a vector from the origin to various points on the total product curve and plotting the slopes of these vectors.

The marginal physical product of a variable input is the change in total output due to a one unit change in the variable input (called the discrete marginal product) or alternatively the rate of change in total output due to an infinitesimally small change in the variable input (called the continuous marginal product). The discrete marginal product of capital is the additional output resulting from the use of an additional unit of capital (assuming all other factors are fixed). The continuous marginal product of a variable input can be calculated as the derivative of quantity produced with respect to variable input employed. The marginal physical product curve is shown (MPP). It can be obtained from the slope of the total product curve.

Because the marginal product drives changes in the average product, we know that when the average physical product is falling, the marginal physical product must be less than the average. Likewise, when the average physical product is rising, it must be due to a marginal physical product greater than the average. For this reason, the marginal physical product curve must intersect the maximum point on the average physical product curve.

Notes: MPP keeps increasing until it reaches its maximum. Up until this point every additional unit has been adding more value to the total product than the previous one. From this point onwards, every additional unit adds less to the total product compared to the previous one – MPP

is decreasing. But the average product is still increasing till MPP touches APP. At this point, an additional unit is adding the same value as the average product. From this point onwards, APP starts to reduce because every additional unit is adding less to APP than the average product. But the total product is still increasing because every additional unit is still contributing positively. Therefore, during this period, both, the average as well as marginal products, are decreasing, but the total product is still increasing. Finally we reach a point when MPP crosses the x-axis. At this point every additional unit starts to diminish the product of previous units, possibly by getting into their way. Therefore the total product starts to decrease at this point. This is point A on the total product curve. (Courtesy: Dr. Shehzad Inayat Ali).

Marginal productIn economics, the marginal product or marginal physical product is the extra output produced by one more unit of an input (for instance, the difference in output when a firm's labour is increased from five to six units). Assuming that no other inputs to production change, the marginal product of a given input X can be expressed as

where ΔX is the change in a firm's production inputs and ΔY is the change in quantity of production output.

In neoclassical economics, this is the mathematical derivative of the production function.... Note that the "product" Y is typically defined ignoring external costs and benefits. In the "law" of diminishing marginal returns, the marginal product of one input is assumed to fall as long as some other input to production does not change.

In the neoclassical theory of competitive markets, the marginal product of labor equals the real wage. In aggregate models of perfect competition, in which a single good is produced and that good is used both in consumption and as a capital good, the marginal product of capital equals its rate of return. As was shown in the Cambridge capital controversy, this proposition about the marginal product of capital cannot generally be sustained in multicommodity models in which capital and consumption goods are distinguished.

Marginal product is the slope of the total product curve and is given by:

MP = Total product Quantity of labor units

CostThe economic cost of a decision depends on both the cost of the alternative chosen and the benefit that the best alternative would have provided if chosen. Economic cost differs from accounting cost because it includes opportunity cost.

As an example, consider the economic cost of attending college. The accounting cost of attending college includes tuition, room and board, books, food, and other incidental expenditures while there. The opportunity cost of college also includes the salary or wage that otherwise could be earning during the period. So for the two to four years an individual spends in school, the opportunity cost includes the money that one could have been making at the best possible job. The economic cost of college is the accounting cost plus the opportunity cost.

Thus, if attending college has a direct cost of $20,000 dollars a year for four years, and the lost wages from not working during that period equals $25,000 dollars a year, then the total economic cost of going to college would be $180,000 dollars ($20,000 x 4 years + $25,000 x 4 years).

Components of Economic Costs

Total cost (TC): Total Cost equal fixed cost plus variable costs. TC = FC + VC. o Variable cost (VC): Variable costs are the costs paid to the variable input. Inputs

include labor, capital, materials, power and land and buildings. Variable inputs are inputs whose use vary with output. Conventionally the variable input is assumed to be labor.

Total variable cost (TVC) or (VC) total variable costs is the same as variable costs.

o Fixed cost (FC) fixed costs are the costs of the fixed assets those that do not vary with production.

Total fixed cost (TFC) or (FC) Average cost (AC) average cost are total costs divided by output. AC = FC/q + VC/q

o Average fixed cost (AFC) = fixed costs divided by output. AFC = FC/q. The average fixed cost function continuously declines as production increaes.

o Average variable cost (AVC) = variable costs divided by output. AVC = VC/q. The average variable cost curve is typically U-shaped. It lies below the average cost curve and generally has the same shape - the vertical distance between the average cost curve and average variable cost curve equals average fixed costs. The curve normally starts to the right of the y axis because with zero production

Marginal cost (MC) Cost curves

Total cost

One can decompose Total Costs as Fixed Costs plus Variable Costs. In the Cost-Volume-Profit Analysis model, Total Costs are linear in volume.

In economics, and cost accounting, total cost (TC) describes the total economic cost of production and is made up of variable costs, which vary according to the quantity of a good produced and include inputs such as labor and raw materials, plus fixed costs, which are independent of the quantity of a good produced and include inputs (capital) that cannot be varied in the short term, such as buildings and machinery. Total cost in economics includes the total opportunity cost of each factor of production in addition to fixed and variable costs.

The rate at which total cost changes as the amount produced changes is called marginal cost. This is also known as the marginal unit variable cost.

If one assumes that the unit variable cost is constant, as in cost-volume-profit analysis developed and used in cost accounting by the accountants, then total cost is linear in volume, and given by: total cost = fixed costs + unit variable cost * amount.

Variable cost

Decomposing Total Costs as Fixed Costs plus Variable Costs.

Variable costs are expenses that change in proportion to the activity of a business.[1] In other words, variable cost is the sum of marginal costs. It can also be considered normal costs. Along with fixed costs, variable costs make up the two components of total cost. Direct Costs, however, are costs that can easily be associated with a particular cost object.[2] Not all variable costs are direct costs, however; for example, variable manufacturing overhead costs are variable costs that are not a direct costs, but indirect costs.Variable costs are sometimes called unit-level costs as they vary with the number of units produced.

Direct labor and overhead are often called conversion cost,[3] while direct material and direct labor are often referred to as prime cost.[3]

Explanation

For example, a manufacturing firm pays for raw materials. When activity is decreased, less raw material is used, and so the spending for raw materials falls. When activity is increased, more raw material is used and spending therefore rises. Note that the changes in expenses happen with little or no need for managerial intervention.

A company will pay for line rental and maintenance fees each period regardless of how much power gets used. And some electrical equipment (air conditioning or lighting) may be kept running even in periods of low activity. These expenses can be regarded as fixed. But beyond this, the company will use electricity to run plant and machinery as required. The busier the company, the more the plant will be run, and so the more electricity gets used. This extra spending can therefore be regarded as variable.

In retail the cost of goods is almost entirely a variable cost; this is not true of manufacturing where many fixed costs, such as depreciation, are included in the cost of goods.

Although taxation usually varies with profit, which in turn varies with sales volume, it is not normally considered a variable cost.

In most of the concerns, salary is paid on monthly rates. Though there may exist a labour work norm based on which the direct cost (labour) can be absorbed in to cost of the product, salary cannot be termed as variable in this case.

Labour is in some cases considered as a variable cost.

Fixed cost

Decomposing Total Costs as Fixed Costs plus Variable Costs.

In economics, fixed costs are business expenses that are not dependent on the activities of the business [1] They tend to be time-related, such as salaries or rents being paid per month. This is in contrast to variable costs, which are volume-related (and are paid per quantity).

In management accounting, fixed costs are defined as expenses that do not change in proportion to the activity of a business, within the relevant period. For example, a retailer must pay rent and utility bills irrespective of sales.

Along with variable costs, fixed costs make up one of the two components of total cost. In the most simple production function, total cost is equal to fixed costs plus variable costs.

Areas of confusion

Fixed costs should not be confused with sunk costs. From a pure economics perspective, fixed costs may not be fixed in the sense of invariate; they may change (and probably will over time), but are fixed in relation to the quantity of production for the relevant period. For example, a company may have unexpected and unpredictable expenses unrelated to production. On the other hand, production output may vary sharply without changing the fixed costs.

Strictly speaking, there is not absolute fixed cost in long-run if the relevant range is long enough. Investments in facilities, equipment, and the basic organization that can't be significantly reduced even for short periods of time without making fundamental changes are referred to as committed fixed costs. Discretionary fixed costs usually arise from annual decisions by management to spend on certain fixed cost items.

In business planning and management accounting, usage of the terms fixed costs, variable costs and others will often differ from usage in economics, and may depend on the intended use. Some cost accounting practices such as activity-based costing will allocate fixed costs to business activities, in effect treating them as variable costs. This can simplify decision-making, but can be confusing and controversial.[2] [3]

In accounting terminology, fixed costs will broadly include almost all costs (expenses) which are not included in cost of goods sold, and variable costs are those captured in costs of goods sold. The implicit assumption required to make the equivalence between the accounting and economics terminology is that the accounting period is equal to the period in which fixed costs do not vary in relation to production. In practice, this equivalenceies does not always hold, and depending on the period under consideration by management, some overhead expenses (e.g. sales, general and administrative expenses) can be adjusted by management, and the specific allocation of each expense to each category will be decided under cost accounting.

Average costIn economics, average cost is equal to total cost divided by the number of goods produced (the output quantity, Q). It is also equal to the sum of average variable costs (total variable costs divided by Q) plus average fixed costs (total fixed costs divided by Q). Average costs may be dependent on the time period considered (increasing production may be expensive or impossible in the short term, for example). Average costs affect the supply curve and are a fundamental component of supply and demand.

Overview

Average cost is distinct from the price, and depends on the interaction with demand through elasticity of demand and elasticity of supply. In cases of perfect competition, price may be lower than average cost due to marginal cost pricing.

Average cost will vary in relation to the quantity produced unless fixed costs are zero and variable costs constant. A cost curve can be plotted, with cost on the y-axis and quantity on the x-axis. Marginal costs are often shown on these graphs, with marginal cost representing the cost of the last unit produced at each point; marginal costs are the first derivative of total or variable costs.

A typical average cost curve will have a U-shape, because fixed costs are all incurred before any production takes place and marginal costs are typically increasing, because of diminishing marginal productivity. In this "typical" case, for low levels of production there are economies of scale: marginal costs are below average costs, so average costs are decreasing as quantity increases. An increasing marginal cost curve will intersect a U-shaped average cost curve at its minimum, after which point the average cost curve begins to slope upward. This is indicative of diseconomies of scale. For further increases in production beyond this minimum, marginal cost is above average costs, so average costs are increasing as quantity increases. An example of this typical case would be a factory designed to produce a specific quantity of widgets per period: below a certain production level, average cost is higher due to under-utilised equipment, while above that level, production bottlenecks increase the average cost.

Relationship to marginal cost

When average cost is declining as output increases, marginal cost is less than average cost. When average cost is rising, marginal cost is greater than average cost. When average cost is neither rising nor falling (at a minimum or maximum), marginal cost equals average cost.

Other special cases for average cost and marginal cost appear frequently:

Constant marginal cost/high fixed costs: each additional unit of production is produced at constant additional expense per unit. The average cost curve slopes down continuously, approaching marginal cost. An example may be hydroelectric generation, which has no fuel expense, limited maintenance expenses and a high up-front fixed cost (ignoring irregular maintenance costs or useful lifespan). Industries where fixed marginal costs obtain, such as electrical transmission networks, may meet the conditions for a natural monopoly, because once capacity is built, the marginal cost to the incumbent of serving an additional customer is always lower than the average cost for a potential competitor. The high fixed capital costs are a barrier to entry.

Minimum efficient scale / maximum efficient scale: marginal or average costs may be non-linear, or have discontinuities. Average cost curves may therefore only be shown over a limited scale of production for a given technology. For example, a nuclear plant would be extremely inefficient (very high average cost) for production in small quantities; similarly, its maximum output for any given time period may essentially be fixed, and production above that level may be technically impossible, dangerous or extremely costly. The long run elasticity of supply will be higher, as new plants could be built and brought on-line.

Low or zero fixed costs / constant marginal cost: since there is no economy of scale, average cost will be close to or equal to marginal cost. Examples may include buying and selling of commodities (trading) etc...

Relationship between AC, AFC, AVC and MC

1. The Average Fixed Cost curve starts from a height and goes on declining continuously as production increases.

2. The Average Variable Cost curve, Average Cost curve and the Marginal Cost curve start from a height, reach the minimum points, then rise sharply and continuously.

3. Marginal Cost curve is the determining curve, while the rest are determined curves.

4. The movement in the Marginal Cost curve determines the movement and direction of the other curves.

5. The Average Fixed Cost curve nears the Average Cost curve initially and then moves away from it. The Average Variable Cost Curve is never parallel or intersects the Average Cost curve due to the existence of the Average Fixed Cost in all units of production.

6. The Marginal Cost curve always passes through the minimum points of the Average Variable Cost and Average Cost curves, though the Average Variable Cost curve attains the minimum point prior to that of the Average Cost curve.

Marginal costIn economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. That is, it is the cost of producing one more unit of a good.[1] Mathematically, the marginal cost (MC) function is expressed as the first derivative of the total cost (TC) function with respect to quantity (Q). Note that the marginal cost may change with volume, and so at each level of production, the marginal cost is the cost of the next unit produced.

A typical Marginal Cost Curve

In general terms, marginal cost at each level of production includes any additional costs required to produce the next unit. If producing additional vehicles requires, for example, building a new factory, the marginal cost of those extra vehicles includes the cost of the new factory. In practice, the analysis is segregated into short and long-run cases, and over the longest run, all costs are marginal. At each level of production and time period being considered, marginal costs include all costs which vary with the level of production, and other costs are considered fixed costs.

A number of other factors can affect marginal cost and its applicability to real world problems. Some of these may be considered market failures. These may include information asymmetries, the presence of negative or positive externalities, transaction costs, price discrimination and others.

Cost functions and relationship to average cost

In the simplest case, the total cost function and its derivative are expressed as follows, where Q represents the production quantity, VC represents variable costs, FC represents fixed costs and TC represents total costs.

Since (by definition) fixed costs do not vary with production quantity, it drops out of the equation when it is differentiated. The important conclusion is that marginal cost is not related to fixed costs. This can be compared with average total cost or ATC, which is the total cost divided by the number of units produced and does include fixed costs.

For discrete calculation without calculus, marginal cost equals the change in total (or variable) cost that comes with each additional unit produced. For instance, suppose the total cost of making 1 shoe is $30 and the total cost of making 2 shoes is $40. The marginal cost of producing the second shoe is $40 - $30 = $10.

Cost curveIn economics, a cost curve is a graph of the costs of production as a function of total quantity produced. In a free market economy, productively efficient firms use these curves to find the optimal point of production, where they make the most profits. There are a few different types of cost curves, each relevant to a different area of economics.

The short-run average total cost curve (SATC or SAC)

Typical short run average cost curve

The average total cost curve is constructed to capture the relation between cost per unit and the level of output, ceteris paribus. A productively efficient firm organizes its factors of production in such a way that the average cost of production is at lowest point and intersects Marginal Cost. In the short run, when at least one factor of production is fixed, this occurs at the optimum capacity where it has enjoyed all the possible benefits of specialization and no further opportunities for decreasing costs exist. This is usually not U-shaped, it is a checkmark-shaped curve. This is at the minimum point in the diagram on the right.Example: Q=2K.5L.5 STC=Pk(K)+Pw(Q2/4K) SATC or SAC= (Pk(K)/Q)+Pw(Q/4K)Short run average cost equals average fixed costs plus average variable costs. Average fixed cost continuously falls as production increases. The shape of the average variable cost curve is directly determined by diminishing marginal returns to the variable input (conventionally labor). Average variable cost equals w/APL or the wage rate divided by the average product of labor.

The long-run average cost curve (LRAC)

Typical long run average cost curve

Essentially, the long-run average cost curve depicts what the minimum per-unit cost of producing a certain number of units would be if all productive inputs could be varied. Given that LRAC is an average quantity, one must not confuse it with the long-run marginal cost curve, which is the cost of one more unit. The LRAC curve is created as an envelope of an infinite number of short-run average total cost curves. The typical LRAC curve is U-shaped, reflecting

economies of scale when negatively-sloped and diseconomies of scale when positively sloped. Contrary to Viner, the envelope is not created by the minimum point of each short-run average cost curve. This mistake is recognized as Viner's Error.

In a long-run perfectly competitive environment, the equilibrium level of output corresponds to the minimum efficient scale, marked as Q2 in the diagram. This is due to the zero-profit requirement of a perfectly competitive equilibrium. This result, which implies production is at a level corresponding to the lowest possible average cost, does not imply that other production levels are not efficient. All points along the LRAC are productively efficient, by definition, but are not equilibrium points in a long-run perfectly competitive environment.

In some industries, the LRAC is always declining (economies of scale exist indefinitely). This means that the largest firm tends to have a cost advantage, and the industry tends naturally to become a monopoly, and hence is called a natural monopoly. Natural monopolies tend to exist in industries with high capital costs in relation to variable costs, such as water supply and electricity supply.

The average cost is the total cost divided by the number of units produced.

The marginal cost curve (MC)

Typical marginal cost curve

A marginal cost that graphically represents the relation between marginal cost incurred by a firm in the short-run product of a good or service and the quantity of output produced. This curve is constructed to capture the relation between marginal cost and the level of output, holding other variables, like technology and resource prices, constant. The marginal cost curve is U-shaped. Marginal cost is relatively high at small quantities of output, then as production increases, declines, reaches a minimum value, then rises. The marginal cost is shown in relation to marginal revenue, the incremental amount of sales that an additional product or service will bring to the firm. This shape of the marginal cost curve is directly attributable to increasing, then decreasing marginal returns (and the law of diminishing marginal returns - Diminishing returns). Marginal cost equal w/MPL. For most production processes the marginal product of labor initially rises, reaches a maximum value and then continuously falls as production increases. Thus marginal cost initially falls, reaches a minimum value and then increases.The marignal cost curve

intersects first the average variable cost curve then the average total cost curve at their minimum points. When the marginal cost curve is above an average cost curve the average curve is rising. When the marginal costs curve is below an average curve the average curve is falling. This relations holds regardless of whether the marginal curve is rising or falling.

The long run marginal cost curve is the minimum cost incurred per unit change in output when all factors of production are variable. The long run marginal cost curve is shaped by economies of scale rather than the law of diminishing marginal returns. The long run marginal cost curve tends to be flatter than its short run counterpart due to increased input flexibility. The long run marginal cost curve intersects the long run average cost curve at its minimum point. When long run marginal costs is below long run average costs long run average costs is falling.[ When long run marginal costs is above long run average costs. avearge costs is falling. Long run marginal costs equals short run marginal costs at the least cost level of production (long run).

Combining cost curves

Cost curves in perfect competition compared to marginal revenue

Cost curves can be combined to provide information about firms. In this diagram for example, firms are assumed to be in a perfectly competitive market. The marginal cost curve will cut the average cost curve at its lowest point. In a perfectly competitive market a firm's profit maximising price would be at or above the price at which the average cost curve cuts the marginal cost curve. If the marginal revenue is above the average total cost price the firm is deriving an economic profit.

Cost curves and production functions

Assuming that factor prices are constant, the production function determines all cost functions. The variable cost curve is the inverted short run production function or total product curve and its behavior and properties are determined by the production function. Because the production function determines the variable cost function it necessarily determines the shape and properties of marginal cost curve and the average cost functions.

Cost functions

Total Cost = Fixed Costs (FC) + Variable Costs (VC)

C = 420 + 60Q + Q2

FC = 420VC = 60Q + Q2

Marginal Costs (MC) = ∂C/∂Q

MC = 60 +2Q MC equals slope of the total cost function and the variable cost function.

VC' = 60 +2Q = MCC' = 60 + 2Q = MC

Average Total Cost (ATC) = Total Cost/Q

ATC = (420 + 60Q + Q2)/QATC = 420/Q + 60 + Q

Average Fixed Cost (AFC) = FC/Q

AFC = 420/Q Average Variable Costs = VC/Q

AVC = (60Q + Q2)/QAVC = 60 + Q

MC curve determines the shape of the ATC and AVC functions.

If MC curve is above average cost or average variable cost curves, then curves are rising.

If MC is below average cost curve or average variable cost curve, then curves are falling.If MC equals average cost, then average cost is at its minimum value.If MC equals average variable cost, then average variable cost is at its minimum value.

Relationship of functions – Example: MC = ATC at minimum ATC

To find minimum ATC take the first derivative of the ATC function and set it equal to zero and solve for Q.

ATC = 420/Q + 60 + QATC = 420Q-1 + 60 + QATC’ = -420Q-2 +1ATC’ = (-420/Q2) + 1ATC’ = 0(-420/Q2) + 1 = 0-420/Q2 = -1-420 = -Q2Q = √420Q = 20.494MC = 60 +2QMC = 60 +2(20.494)MC = 60 + 40.988MC = 100.988ATC = 420/Q + 60 + QATC = 420/(20.494) + 60 + 20.494ATC = 20.494 + 60 + 20.494ATC = 100.988

Relationship between short run and long run cost functions

1. The SRTC can be tangent to the LRTC at only one point. The SRTC cannot intersect the curve. The SRTC can lie wholly “above” the curve with no tangency point.2. The SRTC curve is tangent to LRTC at long run cost minimizing level of production. At the point of tangency LRTC = SRTC. At all other levels of production SRTC will exceed LRTC.3. Average cost functions are the total cost function divided by the level of output. Therefore the LRATC is also tangent to the SRATC at cost minimizing level of output. At the point of tangency LRATC = SRATC. At all other levels of production SRATC > LRATC4. The slope of the total cost curves equals marginal cost. Therefore when LRTC is tangent to SRTC, SRMC = LRMC.5. At the long run cost minimizing level of output LRTC = SRTC; SRATC = LRATC and SRMC = LRMC.6. The long run cost minimizing level of output may be different from minimum SRATC.7. If constant returns to scale then (min) SRATC = LRATC = SRMC = LRMC.8. If increasing returns to scale (min) SRATC will occur at higher level of production than long run cost minimizing level of production. While LRTC = SRTC; SRATC = LRATC and SRMC = LRMC. SRMC does not equal LRMC and LRMC does not equal LRAC.9. With decreasing returns (min) SRATC will be tangent to the LRAC curve at a lower level of production lower than long run cost minimizing level of production. While LRTC = SRTC; SRATC = LRATC and SRMC = LRMC. SRMC does not equal LRMC and LRMC does not equal LRAC

10. A firm that is experiencing increasing (decreasing) returns to scale and is producing at min SRAC can always reduce average cost by expanding (reducing) the use of the fixed input.11. LRATC will always equal or be less than SRATC.

U-shaped curves

Both the SRAC and LRAC curves are typically expressed as U-shaped. However, the shape of the curves are not due to the same factors. For the short run curve the initial downward slope is largely due to declining average fixed costs while the upward slope is due to diminishing marginal returns to the variable input. With the long run curve the shape is due to economies of scale. At low levels of production long run production functions generally exhibit increasing returns to scale which means that the long run average costs is falling. The upward slope of the long run average cost function is due to decreasing returns to scale which set in at relatively high levels of production.