Unit 5.

Production or Manufacturing of output, revenues, costs, and profits

as functions of input level

GM Labor IssuesGM (General Motors) officials are about to begin labor contract renewal negotiations with the UAW (United Auto Workers Union). GM officials are concerned about lagging worker productivity in their plants vis-à-vis the competition. For example, Ford workers produced an average of 33.2 vehicles per year and were paid an average wage of $43 per hour, while the corresponding figures for GM were 27.9 cars at $45 per hour. Based on this information, how much higher are GM’s labor costs per car than Ford’s? What GM labor output would result in GM’s labor costs per car being equal to Ford’s with current wages?

Managing Office Space

Global Technologies is currently leasing the office space used by staff at corporate headquarters in Seattle. Other than a change in the price or cost of office space, what economic factors might explain management’s decision to increase leased office space by 10%? Suppose in the geographic area of Global’s Corporate headquarters (i.e. Seattle) management has reason to believe that other companies may be cutting back on their needs for office space. Based on this information, what advice do you have for Global Technologies management regarding the future purchase of office space in Seattle?

How Many Workers to Hire?

A concession stand owner/operator hires workers (= L) at a daily wage (= W) of $40 to sell pop (= q) at $2.00 (= P = price) per bottle. If q = 200L1/2 and total fixed costs = $100, how many workers should be hired? For simplicity, assume labor costs are the only variable costs.

Who’s To Blame?

The salaries of professional athletes in major sports often are above $1 million per year. Who or what is the main economic explanation as to why this happens? Is the main cause the athletes themselves, the owners, television networks, sports clothing companies, and/or the fans?

Production-Related Qs of Interest to Firm Managers (Examples)

How does output Q change as input Q changes?Can output Q be increased w/o increasing input Q?To what extent can one input be substituted for another in the production process?What input Q would minimize costs? Maximize profits?How should inputs be acquired?

Revenues & Costs(Input side relationships)

1. Graphical

2. Mathematical

$ concepts = f(Q of input)

$

Q of input

Mgmt Concerns______________

Demand Economics: P x Q = RevenuesD factorsConsumer utility - CostElasticities of D ============

etc.= Profits

Supply Economics:S factorsProduction processesInput productivity

etc.

Firm Decisions that Impact TR

1. Products to produce

2. Product prices

3. Quantity of products

4. How to market/promote products

Firm Decisions that Impact TC

1. Quantity of products produced

2. What inputs to use to produce products

3. What quantity and combinations of inputs to use

4. How to acquire inputs

5. How to make inputs more productive

Production

= the process by which inputs are combined and transformed into outputs

Production Analysis

Production FunctionQ = F(K,L)The maximum amount of output that can

be produced with K units of capital and L units of labor.

Short-Run vs. Long-Run DecisionsFixed vs. Variable Inputs

Types (or lengths) of production periods

1. Short run (SR) => period of time for which a firm is stuck with a fixed or given quantity of at least one input

2. Long run (LR) => period of time for which a firm can vary or alter the quantities of all inputs

Assume q = f( ,L) = a short-run production function where:

Q = physical units of output

= physical units of fixed capital

L = physical units of variable labor

r = per unit cost (rental rate) of capital

w = per unit cost (wage rate) of labor

K

K

Linear Production FunctionTP

L

TP = aL

APMP

a AP = MP

L

SR Production ConceptsConcept/Definition Math Calculation

q=linear fn of L q=nonlinear fn of L1. TP = total product = q=aL = q=aL2-bL3 (e.g.)

= total physical units of output = total quantity of output (=q)

2. AP = average product = TP/L = TP/L = output per unit of input = q/L = q/L

= output of ‘average’ input = aL/L = (aL2-bL3)/L = slope of line from origin

to TP curve = a = aL = bL2

3. MP = marginal product = TP/L = TP/L = additional output per unit = TP/L = TP/L of additional input = a = 2aL – 3bL2

= slope of TP curve = output of last input unit

NOTE: when MP > AP, AP is increasingwhen MP < AP, AP is decreasingwhen MP = AP is either at a maximum or constant

Nonlinear Production Function

Input Productivity Increase

Total Product

Cobb-Douglas Production FunctionExample: Q = F(K,L) = K.5L.5

K is fixed at 16 units.Short run production function:

Q = (16).5L.5 = 4L.5

Production when 100 units of labor are used?Q = 4 (100).5 = 4(10) = 40 units

Revenue Concepts that are a fn of the level of input usage.

1. Total Revenue Product = TRP= TP x P= paired observations on

the S value of output and physical units of a variable input

2. Average Revenue Product = ARP= AP x P= revenue per unit of input

3. Marginal Revenue Product = MRP= MP x MR

(note MR=P for price-taking firms)

= add’l revenue per unit of add’l input

SR Revenue Product Function Example

Q = K1/2L1/2

where = 16

q = 4L1/2 = TP

AP =

K

TP

L

L

L L 4 41 2/

MPTP

LL

L

( / )( ) /1 2 421 2 1

Revenue Products if P of output = 50

TRP = P • TP = 50 (4L1/2) = 200L1/2

ARP = P • AP = 50 ( ) =

MRP = P • MP = 50 ( ) =

2 0 0

L

1 0 0

L

4

L

2

L

Total ‘Cost’ Concepts as Functions of Input Level

TVC = total variable costs

= wL

TFC = total fixed costs

= r

TC = total costs

= TVC + TFC

= r + wL

K

K

Graphs of Total ‘Cost’ Concepts as Functions of Input Level

Marginal Factor Cost (MFC)= the additional cost per unit of

additional input

= the wage rate (w) if the additional

input is an additional unit of labor

=> MFC = w = the price of labor$ $

S

MFC=w

D

L LFirm Mkt

‘Optimal’ input level (usage)

Profit-maximizing input level

A manager should keep using additional Qs of an input up to the point where the additional income equals the additional cost from the last input unit (sometimes called MFC = marginal factor cost)

e.g. labor, MRP = W (= MFC)

Profit-maximizing L ifMRP

L

MFC

MRP MFC

L

L

L

1 0 0

1 0

1 0 01 0

1 0

1 0 0*

SR Profit-Maximizing Input Level (Solution Procedure)

Given: prod fn, r, w, PDerive: product concept equations as fns of L

(e.g. TP, AP, MP)

Derive: revenue concept equations as fns of L TRP = TP x P ARP = AP x P MRP = MP x MR (MR = P for P taking firm)

Find optimal L* Set MRP = MFC (=w) and solve for L

Calculate optimal profit at L*= TRP – TFC – wL*

Profit-Maximizing Input Level

Firm D for Variable Input (e.g. L)

Input D Factors

1. Input price

2. Output price

3. Input productivity

the demand for an input is a

‘derived’ demand (i.e. derived from factors that

determine

the profitability of using that input)

Increased D for Labor (examples)

LR Input P Disequilibrium

LR Equilibrium Competitive Input P