Takesh Luckho Production and Cost Theory. Producers Theory Last time, we looked at the main economic theories that are used to explain the rationale behind.

Takesh Luckho

Production and Cost Theory

Producer’s Theory• Last time, we looked at the main economic theories that

are used to explain the rationale behind consumption decisions made by the rational consumers of a good– We saw how a consumer maximises his satisfaction

and look at some comparative statics on the subjects. • Now, let us focus on the reasons and theories that may

dictate firms/companies in making production decisions. In doing so, we will try to tackle the following questions:– How much will a firm produce of a good?– What combination of inputs will it used in the

production process?

• To answer these questions, we will look at the follow theories:– Product theory: which deals with the relationship between

a firm’s factor inputs and its final output. – Cost theory: which deals with the relationship between a

firm’s output decisions and the costs of production. • But first of all, it is very Important to differentiate between the

long-run and the short-run:– Recall from Basic Economics that there are two types of

cost involved in production: • Fixed Cost (fixed amount at all levels of output) and • Variable Cost (varies with the level of output)

Producer’s Theory

• On one hand, the Short run is defined as a situation where one of the factors being used in the production process is limited/available in fixed amount. In other, the short run is defined as a situation where a least a fixed cost exists.

• On the other hand, the Long run is defined as a situation where all the factors being used in the production process varies with the level of output. In other words, the long run is characterised by the presence of only variable costs.

• Examples of the short-run and long run. Let say that a firm want to produce textile using labour and capital.– In the short-run, labour is the variable cost as the hiring of

labour will depend on the level of output the firm want to produce. Capital is the fixed cost as firm has to invest on a fixed technology/machineries to launch production irrespective of the amount of output it want to produce.

– In the long run, capital also becomes a variable cost. The firm can decide on the level of extra investment it want to make on machineries or technology in other to achieve the targeted level of output.

Producer’s Theory

Product Theory: The Production Function• A Production function is a mathematical equation that

specifies the amount of output that a firm/business can produce at any given combinations of input.

• Alternatively, a production function can be defined as function that is showing the minimum level of input that is required to produce a given level of output, at a given technology level,.– In Mathematical terms the production function may be written

as: Q = f (combination of inputs), where the inputs are

• Land – all natural resources • Labour – all physical and mental human effort involved in

production• Capital – buildings, machinery and equipment used for

production

• Hence, Q = f( Land, Labour, Capital)

The Production Function in the Short-Run: Total Product, Average Product and Marginal Product

• Assuming that a country has only two types of input (labour and capital) and combination of which are used to produce some unit of good x.

• Given that the level of capital is fixed in the short-run, a simple production function can be drawn as:

Qx =Qx(z), where z represent units of labour used in production

• Total product is defined as the amount of output that a firm can produce by making use of a certain amount of input.– For example, if a firm can make 100 pieces of good x by

using some level of labour and a fixed level of capital, the then total product of labour is given as 100.

– In mathematical terms, TP = Q (z),

• Average product is defined as the quantity of output that is produced per unit of the factor being used. – For example if in the situation above, the firms make

use of 10 unit of labour to produce 100 piece of good x. Then the Average Product of labour is set as 10.

– In mathematical terms, AP = (TP/ labour unit used) = Q (z)/z

– The Average Product of Labour also used as a good measure of the productivity level of the firm. In other words, it shows how much output is actually being generated per labour-employed by the firm.

The Production Function in the Short-Run: Total Product, Average Product and Marginal Product

• Marginal Product is defined as the additional level of output that can be produced by using an extra unit of the input factor, assuming ceteris paribus conditions.– Recalling the example above, if by making use of an

extra unit of labour, the firm can now increase its production level of good x from 100 to 105, then the marginal product of labour is said to be 5.

– In Mathematical terms

MP = ∆TP/∆input

= ∆Q(z)/∆z

The Production Function in the Short-Run: Total Product, Average Product and Marginal Product

Graphical Representation

Some Quick Observations• Marginal Product curve has an inverted U-Shape. It rises

until a maximum point and then falls.• Average Product curve follows the same pattern as MP.• MR rises faster and falls faster than the AP, cutting the

AP curve at its higher point (maximum point)• Recalling that the MP is also known as the gradient of

Total Product, the shape of the TP curve depends on the behaviour of the MP curve– When MP increase, TP also rises at an increasing

rate– When MP is maximum – Gradient along the TP curve

is maximised. Sometimes known as a “Rest Point”.– When MP falls, TP rises at a decreasing rate– TP is maximised at a point when MP = 0

• According to the law of diminishing returns (to factors), an increase in the amount of the variable input being used by the firm will lead to a continuous fall in the marginal product of that factor.

• In other words, using more and more unit of labour (assuming that capital is fixed) will lead to smaller increase in the output level of the firm.– This implies that if one factor remains constant, the

excessive use of the other factor will at some point have a negative impact on the output level of the firm.

• The law of diminishing returns helps to explain the shape of the marginal product curve.

Law of Diminishing Marginal Returns to factors

Law of Diminishing Marginal Returns to factors

• The rising portion of the MP curve shows increase returns to factors

• The falling portion of the MP curve shows diminishing return to factors

• When MP is maximised – this shows constant return to factors; and

• When MP becomes negative – this show negative returns to factors.

Long-Run Production Function• In the long run, both capital and labour are variable.

Hence the simple production function becomes:Qx =Qx(L,K),

• As we are dealing with a combination of two variable factors to produce one more unit of x, we cannot talk about the concept of return to factors. return to scale happens on an individual factor but not on a combination of factors

• In the Long-Run, we talk about the concept of return to scale– Increasing returns to scale, where the output responds

by a greater percentage than the change in the input quantities.

– Constant returns to scale, where the output responds by the same percentage as the change in the input quantities

– Decreasing returns to scale, where the output responds by a smaller percentage than the change in the input quantities.

• There are many reasons for increasing returns to scale. They would include: – Specialisation and division of labour – with a larger

labour force as your disposal, you can gain from the advantages of increasing specialisation of tasks. This may save time as less ‘tooling up’ and ‘tooling down’ between tasks is required.

– Increased Technical Efficiency – Technical efficiency may occur only after a certain size of production is attained

• The reasons for decreasing returns to scale include: – Management problems. When a firm gets too large, its

management may face problems with communication, co-ordination and control of the firm

– Low labour morale. Employees may become less committed to their work, as managers appear distant and unaware of individual problems and views.

– Bottlenecks. The more complex a firm becomes with size, the more likely it is that the obstructions in the output flow will occur for both human and technical reasons.

Cobb-Douglas Production Function• Cobb Douglas function – a common production

technology that is used in every textbooks to represent this relationship between output and inputs.– Long-run Cobb Douglas production function -- Q(L,K) = aLßKØ ,

• Q is the total output• a is a constant• L is labour and K is capital• ß and Ø are output elasticities

– In the short-run, given that capital is fixed, the Cobb Douglas function becomes: Q(L,k) = akLß , k is a fixed amount of capital

• Output elasticity measures the responsiveness of output to a change in levels of either labor or capital used in production, ceteris paribus. In the long-run these output elasticities can be used to determine whether the production function is exhibits CRS, IRS or DRS.– If ß + Ø = 1, then the production function exhibits CRS– ß and Ø >1, then the production function exhibits IRS– ß and Ø < 1, then the production function exhibits DRS

Cost Theory• Till now, we have seen how the various factors of

production enters a production process and thus impact of a firm’s final output level.

• Let’s use these information to show how costs vary with the amount a firm produces.

• The first thing that we must do is to be clear on meaning of the word ‘cost’.– Total Cost in Accounting refers to the book value of the costs

that is incurred before a sale is made. It is also known as the total cost of production (or total cost of sales).

– Total Cost in Economics is a much broader term. It included not only the book value of the product being sold but also the opportunity cost of buying this product.

– In other words, Total cost in economics may be given as the money cost of the product plus its opportunity cost.

• Hence total cost = explicit cost (money cost) + implicit cost (opportunity cost)

Explicit Cost and Implicit Cost

• A simple example: Paul builds a cabinet. He spends 2 hours building the cabinet. He could have been working instead and normally makes $25/hour at his job. Since he was building a cabinet he wasn't paid for this time. The materials to make the cabinet cost him $20.– His Explicit Costs are: $20 in materials – His Implicit Costs are: $25/hr x 2 hrs= $50 of

foregone pay – His Total Costs are: $20 in materials + $50 of

foregone pay = $70 Total Costs

Types of Cost• In the short run, as some factors are fixed in

supply, their total costs are also expected to be fixed in the sense that they do not vary with output.– On one hand, a Fixed Cost is defined as the portion of

total cost that do not vary with the level of output/production

– E.g of a fixed cost is the Rent on land you have to pay the same whether the firm produces or not.

• On the other hand, a Variable Cost may be defined as the portion of total cost that do vary with the level of production of the firm– E.g of a variable cost is raw material The more you

produce, the more of raw materials you will use.• Hence, Total Cost = Fixed Cost + Variable Cost

Q0

TFC(Total Fixed Cost)

Costs

TVC(Total Variable Cost)

TC(Total Cost)

“TOTAL” COST CURVES

Graphical Representation

• Average cost (AC) may be defined as the cost per unit of production: AC can be calculated as follows:

Average Cost = Total Cost/Output = TC/Q• Also need to differentiate between Average

Fixed Cost (AFC) and Average Variable Cost (AVC)– AFC is the average per-unit cost of using the firm’s fixed

inputs. AFC = Fixed Cost/Q– AVC is the average per-unit cost of using the firm’s

variable inputs. AVC = Variable Cost/Q

• Marginal Cost (MC) is the cost of producing one extra unit of the product. MC = Change in total cost = TC/Q or = TC1 – TC0. – MC is the gradient along the TC curve.

Types of Cost

Q0

AFC(Average Fixed Cost)

Costs

As more output is produced, the Average Fixed Cost decreases.

Graphical Representation

AVC(Average Variable Cost)

Minimum AVC

The Average Variable Cost is U shaped. First it decreases, reaches a minimum and then increases.

q1

Q

0

Costs

AVC(Average Variable Cost)

MC (Marginal Cost)

AC(Average Cost)

Graphical Representation

Observation• Properties of Average and Marginal Cost Curves

– They are U-Shape curves – Initially falls and eventually rises

– Marginal Cost is lower than the Average Cost curve. It decreases faster and rises faster, intercepting the AC curve at the lowest point.

– MC always touches the AC curve at its minimum value.

– The Marginal Cost Curve also falls faster, rise faster and touches the AVC at its minimum.

– Being the gradient, MC gives the shape of the TC initially rises at a diminishing rate up to the point where MC is minimal. After that point it starts to rise at an increasing rate.

Connection between Product Functions and Cost Curves

• Average Product and Marginal Product curves are mirror reflection of the Average Cost and Marginal Cost Curve.

• Graphs or some Simple Mathematical manipulation can be used to prove this.– Average Product (of Labour) = Total Product/Labour =

Q/L– Average Cost = Average Variable Cost + Fixed Cost

= L/Q + k– After simple Mathematics, AC = 1/AP + k

• Hence, the Short-Run Law of Diminishing Returns can be thus mirrored on Short-Run Cost Curves.– The Shape of the Marginal Cost curve may be thus

explained by the law of diminishing returns.

Connection between Product Functions and Cost Curves

Negative ReturnI.R D.R

The Long Run Cost Function

• In long run, all inputs are variable.– Total Cost = Total Variable costs

• How to draw a long-run total cost curve?– The LRAC curve is an envelope of SRAC

curves, and outlines the lowest per-unit costs the firm will incur over a range of output.

LAC

SAC1

Q0

COST

SAC2

Long-run Average Cost Curve

Minimum efficient scale is the lowest output level for which LRAC is minimized

Economies of Scale• The concept of increasing returns to scale is closely

linked to that of economies of scale. • A firm experiences economies of scale if costs per unit of

output fall as the scale of production increases. • Clearly, if a firm is getting increasing returns to scale

from its factors of production, then as it produces more it will be using smaller and smaller amounts of factors per unit of output.

• In other words, assuming ceteris paribus conditions, a firm deriving economics of scale will be producing at a lower cost per unit

• When the cost of production increases as the scale of production increase, the firm is said to be deriving diseconomies of scale

• There are two types of economies of scale– Internal Economies of Scale and External Economies of Scale

Graphical Representation

Internal Economies of Scales • Internal Economies of Scales is achieved when a

firm/business experiences a fall in its per-unit cost following an expansion in its production scale. Internal Economies of Scales can happened due to the following reasons:– Bulk-buying economies – obtaining discounts and lower prices

for the raw materials due to bulk purchase.– Technical economies – lower per-unit cost due to the use of

more advance technologies. – Financial economies – big firms can raise money at lower

interest rates.– Marketing economies – an increase in production level will

spread the marketing cost (which are usually fixed cost) over a wider range and thus lead to a fall in per-unit cost.

– Managerial economies – average administrative cost will decrease with a rise in output level, hence leading to a fall in average unit cost.

External Economies of Scales • External Economies of Scale occurs when a

firm/business benefits from lower unit cost of production as results of a growth in the industry size. External EOS can happened due to the following factors:– Economies of Concentration – Concentration of firms

and suppliers in the same region may lead to a fall in the cost delivering raw materials for the whole industry. Concentration of firms also reduces the cost of information seeking.

– Infrastructure – Concentration of firms in a particular region may lead to an improvement in the Transport and Communication infrastructures of that region, at a lower unit cost for individual firms.

– Skilled Labour and Training- As an industry gets bigger, training and education become more focused on the industry needs.

Profit Maximisation• Profit is the return for entrepreneurial risk-

taking. • In economics the definition of profit differs from

that of accounting– Profit = Normal profit + Supernormal profit

• Where Normal profit is the minimum acceptable return to the entrepreneur. It is the opportunity cost of his enterprise. It is sometime known as breakeven profit

• Supernormal profit is the level of profit over an above normal profit

• How to know where profit is maximised?– Two approaches to get the profit maximising output

level• Total Revenue and Total Curve Approach • Mathematical Approach (Using MR and MC)

Total Cost and Total Revenue Approach

Mathematical Approach• Simpler way of getting the profit maximising

output. Assume that we have a Profit function as – Π = Total Revenue – Total Cost – Π = TR(Q) – TC (Q)– dΠ/dQ = dTR(Q)/dQ – dTC(Q)/dQ– At Maximum profit dΠ/dQ = 0– Hence dTR(Q)/dQ – dTC(Q)/dQ = 0

dTR(Q)/dQ = dTC(Q)/dQ MR = MC

• Hence, the Profit-maximising rule of a firm says that maximum profit is achieved at a point where marginal revenue levels equals marginal cost levels.

Profit Maximisation Rule: MR =MC

Profit Maximisation in the Short-Run• In the long-run, the optimal output (profit

max output) of the firm will be at a point where MR=MC

• However, in the sort run, this relation is necessary but no sufficient.– This is because of the presence of a fixed

cost.– The profit maximising output will depend on

the extent the firm is covering its variable costs

• Assume that the firm is making a loss, Should it close down or continue to stay in business?

• Using some maths to answer this question:– Assuming that the firm is maximising its profit at a point

where MR=MC (let say Q*) and at this point the firm is incurring a loss.

• Π(Q*) = TR(Q*) – TC (Q*) < 0

• What are its choices?– Stay in business and make loss

Π(Q*) = TR(Q*) – VC(Q*) – FC < 0 , – Close down and incur the fixed cost (as even if the firm

close down it must continue to pay its fixed cost) Π(0) = TR(0) – VC(0) – FC = - FC,

• The firm should stay into business as long as – Π(Q*) ≥ Π(0)TR(Q*) – VC(Q*) – FC ≥ - FCTR(Q*) ≥ VC(Q*) P.Q* ≥ VC(Q*)

• Dividing through out by Q* P > AVC (Q*)

• Despite making a loss, in the short-run, a firm can continue doing business as long as it covers its Average variable cost (i.e P>AVC)

• But in the long-run, if a firm is making a loss, it must automatically close down.

• Quick recall– Short Run Total Cost :

• STC(Q) = SVC (Q) + FC, assuming Labour is variable, capital is fixed and the price of labour and capital is given by w (wages) and r (interest) respectively

• STC(Q) = wL + rǨ where Ǩ is fixed

– Long Run Total Cost :• LTC(Q) = SVC (Q) where both Labour and capital are variable

costs• LTC(Q) = wL + rK