EC611--(Ch 06) Production Theory & Estimation

EC611--Managerial Economics

Production Theory and Estimation

Dr. Savvas C Savvides, European University Cyprus

Managerial Economics DR. SAVVAS C SAVVIDES 2

The Organization of Production

Inputs (the factors of production and material things that go into the production of goods and services):

Labor, Capital, Land, Raw materials

Fixed Inputs (inputs that don’t vary with the level of output):

Plant, machinery, bank loan, permanent staff

Variable Inputs (inputs that vary with the level of output):

hourly labour, raw materials


Choosing outputCOSTS REVENUES

Technology & costs of

hiring factors of production

TC curves(short & long run)

AC(short &long run)

MC

Demandcurve

AR

MR

CHECK: produce in SR?close down in LR?

Choose output level


The Production FunctionThe amount of output produced depends upon the inputs used in the production process

The production function specifies the maximum amount of output which can be produced with specific level of inputs, given the level of existing technological know-how (table, equation, or graph)

In general form, a production function may be expressed as:

Q = f ( X1 , X2 , X3 , … , Xk )

where the X’s are the various inputs used


Short-run vs. Long-run

The short run is the period in which a firm can make only partial adjustment of inputs

e.g. the firm may be able to vary the amount of labour, but cannot change capital.

The long run is the period in which a firm can adjust all inputs to changed conditions.


Total Product

Marginal Product

Average Product APL = TP / L (6L2 – 0.2L3)/L = 6L – 0.2L2

Output Elasticity EL = MPL / APL (12L– 0.2L2)/(6L – 0.2L2)

TP = Q = f(L)

MPL =∆TP

∆LThe marginal product of labour is the partial derivative of output with respect to the variable factor (in this case labour), but holding constant the inputs of other factors.

Example: What is the MPL of Q = 6L2 – 0.2L3

Answer: We take first derivative of the equation w.r.t. L

MPL = dQ / dL = 12L – 0.6L2

Let’s assume labour is the only variable factor (with capital fixed)

Prod. Function:One Variable Input

dTPdL=


14545675151.5406040010

-325-7562525040080020

1.6435.257.6281.681.7528.850.4172.861.8520.838.483.241.9311.221.622.42

--000

EL= (12L– 0.2L2)

/(6L – 0.2L2)

APL= 6L – 0.2L2

MPL =12L – 0.6L2

Q =6L2 – 0.2L3L




-500

0

500

1000

Labor

Out

put

Labor 0 2 4 6 8 10 15 20 25 26

Output 0 22.4 83.2 172.8 281.6 400 675 800 625 540.8

MPL 0 21.6 38.4 50.4 57.6 60 45 0 -75 -93.6

APL 0 11.2 20.8 28.8 35.2 40 45 40 25 20.8

1 2 3 4 5 6 7 8 9 10


The Law of Diminishing Returns

Holding all factors constant except one, the law of diminishing returns says that:

beyond some value of the variable input,further increases in the variable input lead to steadily decreasing marginal product of that input.

e.g. trying to increase labour input without also increasing capital will bring diminishing returns.


-200-100

0100200300400500600700800900

1 2 3 4 5 6 7 8 9 10

Stage IStage II

Stage III

The Three Stages of Production

0 1510862 4 20 25


The Three Stages of Production

Stage I

Stage IIStage III


Marginal Revenue ProductThe Marginal Revenue (or Value) Product of Labour is the extra (additional) revenue (benefit) that the firm receives from production by an extra (additional) worker.

MRPL = ∆TR /∆L = d TR / dL

Recall that TR = f(Q) and Q = f ( L)

Then, by the “Function-in-a-function (Chain) Rule” of differentiation, we have

dTR/dL = (dTR/dQ) * (dQ/dL)

(We get the same result if we multiply and divide by dQ).

= MR * MPL

In perfect competition, where firms are price-takers, MR = P, then

MRPL = MPL * P


Optimal Use of the Variable Input

Marginal Resource (Factor) Cost MRCL = ∆TC

∆L

Optimal Use of Labor MRPL = MRCL

As long as the incremental market value of the extra output produced (that is, the marginal revenue brought in) is greater than the cost of hiring the extra labour (the cost of wages), then the firm should go ahead and employ the extra labour. Remember that MRPL = MPL * P (the price of the product).

Profits are maximized where the marginal revenue brought in (theMRPL) is exactly equal to the cost of labour to be employed, or where: MRPL = w The simple case can be illustrated with a constant wage rate (horizontal wage line w).

dTCdL=


60045045675156006006040010

6000080020

60057657.6281.6860050450.4172.8660038438.483.2460021621.622.426000000

WMRPL= MPL * P

MPL =12L – 0.6L2

Q =6L2 – 0.2L3L

Assume that the selling price of the product (equal to MR in perfect competition) is £10 and the wage rate is £600

Marginal Revenue Product


With diminishing marginal productivity, the firm maximizes profit when the marginal cost of employing an extra worker equals the MRPL...

The marginal revenue product of labour is the revenue obtained by selling the output produced by an extra worker

W0

MRPL

Employment

Wag

e, M

RP

L

…this occurs at E where w = MRPL. Employment is L*.

Below L*, extra employment adds more to revenue than to labour costs. Above L*, the reverse is so.This decision is consistent with the rule MR = SMC for maximizing profit.L*

£600

10

MRPL = MPL * MR

Optimal Use of the Variable Input


Example:A producer of pocket calculators has a fixed amount of plant and

equipment (capital), but can vary the number of workers. The production function is given by the following relationship:

Q = 98L – 3L2

Being a small producer, the firm can produce and sell all its output at £20 each ( MR=20). It can also hire as many workers at £40 per day (MC=40).

Question: How many workers should the firm hire per day?

Answer: The optimization Rule is: MRPL = MCL

MPL = dQ/dL = 98 – 6L MRPL = 20(98 – 6L) since MR = 20To max Profits: MRPL = MCL

20(98 – 6L) = 40 since MC =40L = 16

Thus, in order to max Profits, this firm should hire 16 workers per day.

Optimal Use of One Input--Example


Representative Prod. FunctionAssume two inputs, labour and capital:

Q = f(L, K)

654321

Capi tal

12141412831

Labour

23456

Output Quantity (Q)

710121210

33363633233640403628

2830302818

40424036282940363124


Prod. Function:Two InputsQ = f(L, K)

654321

Capi tal

12141412831

Labour

23456

Output Quantity (Q)

710121210

33363633233640403628

2830302818

40424036282940363124


Isoquants

Prod. with Two Inputs: Isoquants

Isoquants show combinations of two inputs that can produce the same level of output.


Prod. with Two Inputs: IsoquantsFirms will only use combinations of two inputs that are in the economic (or “feasible”) region of production, which is defined by the portion of each isoquant that is negatively sloped.

Ridge Lines


ExampleFor Isoquant 12, moving from point N to point R: ∆K= -2.5 ∆L=1

MRTS = -(-2.5/1) = 2.5

Marginal Rate of Technical Substitution

Recall that all points on an isoquant refer to the same level of output. Therefore, moving down an isoquant, the gain in Q from using more L must be equal to the loss in Q from using less K.

That is, (∆L)(MPL) = - (∆K) (MPK)

MRTS = MPL/MPK = -∆K/∆L (slope of isoquant)


Perfect Substitutes Perfect Complements

Substitute & Complement Inputs


Optimal Combination of Inputs

Isocost lines represent all combinations of two inputs that a firm can purchase with the same total cost. It is basically the budget lineof the firm.

C wL rK= +

C wK Lr r

= −

C Total Cost=

( )w Wage Rateof Labor L=

( )r Cost of Capital K=


Isocost Lines

With w = £1, r = £2 and a budget of £50 we have Isocost 1.

K = (50 / 2) – 1 / 2 ( L ) = 25 – 0.5 L Slope = - 0.5 Intercept = 25

If the budget increases to £60, then the isocost line shifts to the right.

The slope w / r (or relative price ratio of the two inputs) does not change. Only the intercept (C / r ) changes.



Changes in Relative Prices of Inputs

If wages drop to w = £0.75, r = £2 (and original budget of £50), the isocost line rotates outward to the right on the horizontal (labour) axis.

K = (50 / 2) – 0.75 / 2 ( L ) = 25 – 0.375 L

Slope = - 0.375 (vs. - 0.5 before) Intercept = 25 (same as before)

If w = 1.25 slope is 0.625. Isocost rotates inwards. (same intercept)



The optimum input combination is reached where the firm is able to attain the highest possible isoquant with the available budget (represented by the isocost).

MRTS = MPL/MPK = -∆K/∆L (slope of isoquant) MRTS = w/r (slope of isocost)

MPL/MPK = w/r ( MPL/ w ) = ( MPK / r)



Example:

Assume that the MPL= 40 units of output and MPK=120 units. Assume also that w= £20 and r = £30.

(a) Why is this firm not max. Q or min. Costs?

(b) How can the firm max. Q or min. Costs?

Answer:

(a) Because ( MPL/ w ) = 40/20 = 2

whereas ( MPK / r) = 120/30 = 4

(b) By hiring fewer workers and using more capital. This way, MPL increases and MPK decreases. This process will continue until ( MPL/ w ) = ( MPK / r)



LR Production:Returns to ScaleReturns to Scale refers to the proportionate change in output resulting from a certain change in inputs.This is a LR concept.

Production Function Q = f(L, K)

λQ = f(hL, hK)

If λ > h, we have increasing returns to scale.

If λ = h, we have constant returns to scale.

If λ < h, we have decreasing returns to scale.

Returns to scale can also and readily be measured using the output elasticity:

EL = 1 (CRTS) EL > 1 (IRTS) EL < 1 (DRTS)


Constant Returns to

Scale

Increasing Returns to

Scale

Decreasing Returns to

Scale

Long Run: Returns to Scale

λ = h100% = 100%

λ > h200% > 100%

λ < h50% < 100%


Economies and Diseconomies of ScaleReasons for Economies of Scale

Specialization: As a firm’s scale of operations increases, there are more opportunities for developing specialization in the use of inputs as well. This reduces unit costs

Dimensional Factors / Indivisibilities: As the scale of operations increase, firms are able to economize on certain inputs which do not have to be employed in the same proportion as the scale of operations has increased. This is true especially of certain fixed costs (large capacity machinery, a large telephone electronic switchboard, etc) or head office overhead expenses (marketing, accounting managerial staff).

Reasons for Diseconomies of ScaleOne of the major reasons firms may experience increasing unit costs as their operations increase is that the complexities of management structures means that they are less efficient (more bureaucratic) with many layers of supervision and authority. Besides the costs of these management structures, decision-making is inefficient. Think of the complex managerial structure of a multinational company.


Empirical Production FunctionsCobb-Douglas Production Function

Q = AKaLb

Such production functions can be estimated using natural logarithms to “linearize” the function in order to use regression techniques:

ln Q = ln A + a ln K + b ln L

The values of a and b as estimated from the above equation are the respective output elasticities of Labor and Capital.

If a + b = 1 constant returns to scale If a + b > 1 increasing returns to scale If a + b < 1 decreasing returns to scale


Innovations and Global Competitiveness

Innovations are perhaps the single most important determinant of a firm’s LR competitivenessProduct Innovation—introduction of new or improved productsProcess Innovation—Introduction of new or improved production process (process re-engineering)

given same resources there’s a shift outward of the isoquantBoth product and process innovation are, and should be, continuous

and may be in the form of small “doses” rather than through “big bang” breakthroughs.

Keen competition at home and abroad usually stimulates innovations.

There are frequently high risks in the introduction of innovations.


Innovations and Global Competitiveness

Product Cycle Model—innovating companies eventually lose market share (locally and internationally) due to cheaper imitators. The lead time by innovators in exploiting the benefits of their innovations is becoming shorter.Just-In-Time Production System—process innovation often provides a much longer time for exploiting the benefits and much bigger returns (ROCE) because of long-lasting economies (higher efficiency, lower inventories, etc)Competitive Benchmarking

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EC611--(Ch 06) Production Theory & Estimation