Order 1369245: Long Run Average Cost
Chapter Six
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Chapter 6
The Theory
and
Estimation of Production
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Chapter Six
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Overview
- The production function
- Short-run analysis of average and marginal product
- Long-run production function
- Importance of production function in managerial decision making
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Chapter Six
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Learning objectives
- define the production function
- explain the various forms of production functions
- provide examples of types of inputs into a production function for a manufacturing or service company
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Chapter Six
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Learning objectives
- understand the law of diminishing returns
- use the Three Stages of Production to explain why a rational firm always tries to operate in Stage II
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Chapter Six
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Production function (p. 187)
- Production function:
It defines the relationship between inputs and the maximum amount that can be produced within a given period of time with a given level of technology
Q=f(X1, X2, ..., Xk)
Q = level of output
X1, X2, ..., Xk = inputs used in
production
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Chapter Six
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Production function
- For simplicity we will often consider a production function of two inputs:
Q=f(X, Y)
Q = output
X = labor
Y = capital
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Chapter Six
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Production function
- Short-run production function shows the maximum quantity of output that can be produced by a set of inputs, assuming that the amount of at least one of the inputs used remains unchanged.
- Long-run production function shows the maximum quantity of output that can be produced by a set of inputs, assuming that the firm is free to vary the amount of all the inputs being used.
- Fixed inputs ….. Variable inputs
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product (p. 189)
- Alternative terms in reference to inputs
- ‘inputs’
- ‘factors’
- ‘factors of production’
- ‘resources’
- Alternative terms in reference to outputs
- ‘output’
- ‘quantity’ (Q)
- ‘total product’ (TP)
- ‘product’ (goods or services)
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product
- Marginal product (MP) = change in output (Total Product) resulting from a unit change in a variable input
- Average product (AP) = Total Product per unit of input used
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product
- if MP > AP then AP is rising
- if MP < AP then AP is falling
- MP=AP when AP is maximized
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product
- Law of diminishing returns: as additional units of a variable input are combined with a fixed input, after some point the additional output (i.e., marginal product) starts to diminish (decrease).
all inputs added to the production process have the same productivity.
nothing in theory tells when diminishing returns will start to take effect. Technology? Can it be estimated?
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product
- The Three Stages of Production in the short run: (p. 194)
- Stage I: from zero units of the variable input to where AP is maximized (where MP=AP)
- Stage II: from the maximum AP to where MP=0
- Stage III: from where MP=0 on
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product
- In the short run, rational firms should be operating only in Stage II
Q: Why not Stage III? firm uses more
variable inputs to produce less output
Q: Why not Stage I? underutilizing fixed capacity, so can increase output per unit by increasing the amount of the variable input
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product
- What level of input usage within Stage II is best for the firm?
answer depends upon:
- how many units of output the variable input can add to the firm
- the price of the product (output)
- the monetary costs of employing the
variable input
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product
- We develop here the rule that helps us answer the question raised in last page.
- We have defined before (in chapter 3) the total revenue as the value of output, i.e. multiplying the total product by the market price of it. Your textbook calls this the total revenue product. No need to get confused.
TRP = Q · P
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product
- Marginal revenue product (MRP) = change in the firm’s TRP resulting from a unit change in the number of inputs used.
MRP = MP · P =
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product
- Total labor cost (TLC) = total cost of using the variable input (labor), computed by multiplying the unit price of labor (wage rate) by the number of units of the variable inputs (labor) employed
TLC = w · X
- Marginal labor cost (MLC) = change in total labor cost resulting from the change in the employment of the variable input by one unit. This of course will equal to the wage rate.
MLC = w
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product
- Summary of relationship between demand for output and demand for a single input:
A profit-maximizing firm operating in perfectly competitive output and input markets will be using the optimal amount of an input at the point at which the monetary value of the input’s marginal product is equal to the additional cost of using that input
MRP = MLC
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Chapter Six
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Short-run analysis of Total,
Average, and Marginal product
- Multiple variable inputs
- Consider the relationship between the ratio of the marginal product of one input and its cost to the ratio of the marginal product of the other input(s) and their cost
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Chapter Six
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Long-run production function (p.199)
- In the long run, a firm has enough time to change the amount of all its inputs
- The long run production process is described by the concept of returns to scale
Returns to scale = the resulting increase
in total output as all inputs increase
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Chapter Six
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Long-run production function
- If all inputs into the production process are doubled, three things may happen:
- output may more than double
‘increasing returns to scale’ (IRTS)
- output may exactly double
‘constant returns to scale’ (CRTS)
- output may less than double
‘decreasing returns to scale’ (DRTS)
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Chapter Six
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Long-run production function
- One way to measure returns to scale is to use a coefficient of output elasticity:
if EQ > 1 then IRTS
if EQ = 1 then CRTS
if EQ < 1 then DRTS
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Chapter Six
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Long-run production function
- Returns to scale can also be described using the following equation [when we know the production function Q = f(X, Y)]
hQ = f(kX, kY)
if h > k then IRTS
if h = k then CRTS
if h < k then DRTS
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Chapter Six
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Estimation of production functions (p. 202)
- Examples of production functions
- short run: one fixed factor, one variable factor
Q = f(L)K
- cubic: increasing marginal returns followed by decreasing marginal returns
Q = a + bL + cL2 – dL3
- quadratic: diminishing marginal returns but no Stage I
Q = a + bL - cL2
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Chapter Six
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Estimation of production functions
- Examples of production functions
- power function: exponential for one input
Q = aLb
if b > 1, MP increasing
if b = 1, MP constant
if b < 1, MP decreasing
Advantage: can be transformed into a linear
(regression) equation when expressed in log terms
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Chapter Six
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Estimation of production functions
- Examples of production functions
- Cobb-Douglas function: exponential for two inputs
Q = aLbKc
if b + c > 1, IRTS
if b + c = 1, CRTS
if b + c < 1, DRTS
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Chapter Six
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Estimation of production functions
Cobb-Douglas production function
Advantages:
- can investigate MP of one factor holding others fixed
- elasticities of factors are equal to their exponents
- can be estimated by linear regression
- can accommodate any number of independent variables
- does not require constant technology
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Chapter Six
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Estimation of production functions
Cobb-Douglas production function
Shortcomings:
- cannot show MP going through all three stages in one specification
- cannot show a firm or industry passing through increasing, constant, and decreasing returns to scale
- specification of data to be used in empirical estimates
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Chapter Six
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Estimation of production functions
- Statistical estimation of production functions
- inputs should be measured as ‘flow’ rather than ‘stock’ variables, which is not always possible
- usually, the most important input is labor
- most difficult input variable is capital
- must choose between time series and cross-sectional analysis
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Chapter Six
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Estimation of production functions
- Aggregate production functions: whole industries or an economy
gathering data for aggregate functions can be difficult:
- for an economy … GDP could be used
- for an industry … data from Census of Manufactures or production index from Federal Reserve Board
- for labor … data from Bureau of Labor Statistics
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Chapter Six
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Importance of production functions in managerial decision making
- Capacity planning: planning the amount of fixed inputs that will be used along with the variable inputs
Good capacity planning requires:
- accurate forecasts of demand
- effective communication between the production and marketing functions
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Chapter Six
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Importance of production functions in managerial decision making
- Example: cell phones
- Asian consumers want new phone every 6 months
- demand for 3G products
- Nokia, Samsung, SonyEricsson must be speedy and flexible
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Chapter Six
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Importance of production functions in managerial decision making
- Example: Zara
- Spanish fashion retailer
- factories located close to stores
- quick response time of 2-4 weeks
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Chapter Six
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Importance of production functions in managerial decision making
- Application: call centers
- service activity
- production function is
Q = f(X,Y)
where Q = number of calls
X = variable inputs
Y = fixed input
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Chapter Six
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Importance of production functions in managerial decision making
- Application: China’s workers
- is China running out of workers?
- industrial boom
- eg bicycle factory in Guangdong Provence
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