ITSF 4151. Special Topics in the Economics of Education Dr. Manuel Salas-Velasco
ITSF 4151. Special Topics in the Economics of Education Dr. Manuel Salas-Velasco
ITSF 4151. Special Topics in the Economics of Education Dr. Manuel Salas-Velasco
ITSF 4151. Special Topics in the Economics of Education Dr. Manuel Salas-Velasco With labor time continuously divisible, we can smooth TP, MP and AP curves.
ITSF 4151. Special Topics in the Economics of Education Dr. Manuel Salas-Velasco
Production refers to the transformation of inputs into outputs (or products)
An input is a resource that a firm uses in its production process for the purpose of creating a good or service
A production function indicates the highest output (Q) that a firm can produce for every specified combinations of inputs (physical relationship between inputs and output), while holding technology constant at some predetermined state
Mathematically, we represent a firm’s production function as:
Dr. Manuel Salas Velasco Q = f (L, K) Assuming that the firm produces only one type of output with two inputs, labor (L) and capital (K)
The short run is a time period in which the quantity of some inputs, called fixed factors, cannot be increased. So, it does not correspond to a specific number of months or years
A fixed factor is usually an element of capital (such as plant and equipment). Therefore, in our production function capital is taken to be the fixed factor and labor the variable one
Total product (TP) is the total amount that is produced during a given period of time
Total product will change as more or less of the variable factor is used in conjunction with the given amount of the fixed factor
Average product (AP) is the total product divided by the number of units of the variable factor used to produce it
Marginal product (MP) is the change in total product resulting from the use of one additional unit of the variable factor
Dr. Manuel Salas Velasco
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Output with Fixed Capital and Variable Labor Dr. Manuel Salas Velasco Quantity of labor (L) Total product (TP) Marginal product (MP) Average product (AP) 0 0 1 50 50 50.00 2 110 60 55.00 3 390 280 130.00 4 520 130 130.00 5 580 60 116.00 6 630 50 105.00 7 650 20 92.86 8 650 0 81.25 9 640 -10 71.11
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Total Product, Average Product, and Marginal Product Curves Dr. Manuel Salas Velasco TP MP L AP L
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Relationships among Total, Marginal, and Average Products of Labor Dr. Manuel Salas Velasco Total product Labor Labor Marginal product Average product AP L MP L TP L A L B L C A A B B C C
With labor time continuously divisible, we can smooth TP, MP L , and AP L curves
The MP L rises up to point A, becomes zero at C, and is negative thereafter
The AP L raises up to point B and declines thereafter (but remains positive as long TP is positive)
The TP curve increases at an increasing rate up to point A; past this point, the TP curve rises at a decreasing rate up to point C (and declines thereafter)
A ray drawn from the origin A = inflection point
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Law of Diminishing Returns Dr. Manuel Salas Velasco Total product Labor Marginal product Average product AP L MP L TP L A L B L C A A B C
This law states that as additional units of an input are used in a production process, while holding all other inputs constant, the resulting increments to output (or total product) begin to diminish beyond some point (after A, in the bottom graph)
As the firm uses more and more units of the variable input with the same amount of the fixed input, each additional unit of the variable input has less and less of the fixed input to work and, after this point, the marginal product of the variable input declines
Labor C B
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Stages of Production Dr. Manuel Salas Velasco Total product Labor Marginal product Average product AP L MP L Stage III of labor Stage I of labor Stage II of labor TP 3 4 8 A A B B C C The relationship between the MP L and AP L curves can be used to define three stages of production of labor (the variable input) Is the range of production for which increases in the use of a variable input cause increases in its average product Is the range for which increases in the use of a variable input causes decreases in its average product, while values of its associated marginal product remain nonnegative Is the range for which the use of a variable input corresponds to negative values for its marginal product Marginal product Average product B A Labor
Marginal product of labor = change in output/change in labor input = Δ Q/ Δ L
Dr. Manuel Salas Velasco Example. Let’s consider the following production function: Q = 8 K 1/2 L 1/2 If K = 1 Cobb-Douglas production function
If we assume that inputs and outputs are continuously or infinitesimally divisible (rather than being measured in discrete units), then the marginal product of labor would be:
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The Production or Output Elasticity of Labor Dr. Manuel Salas Velasco The elasticity of output with respect to the labor input measures the percentage change in output for a 1 percent change in the labor input, holding the capital input constant
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Production Theory The Production Function in the Long Run Dr. Manuel Salas Velasco
The long run is a time period in which all inputs may be varied but in which the basic technology of production cannot be changed
The long run corresponds to a situation that the firm faces when is planning to go into business (to expand the scale of its operations)
Like the short run, the long run does not correspond to a specific length of time
We can express the production function in the form:
Dr. Manuel Salas Velasco
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Production Isoquants Dr. Manuel Salas Velasco L K Units per time period Units per time period K 1 K 2 K 3 L 1 L 2 L 3 An isoquant is a set of input combinations that can be used to produce a given level of output This curve indicates that a firm can produce the specified level of output from input combinations (L 1 , K 1 ), (L 2 , K 2 ), (L 3 , K 3 ), … a b c As we move down from one point on an isoquant to another, we are substituting one factor for another while holding output constant
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Marginal Rate of Technical Substitution Dr. Manuel Salas Velasco L K Units per time period Units per time period K 1 K 2 K 3 L 1 L 2 L 3 a b c
The marginal rate of technical substitution (MRTS) measures the rate at which one factor is substituted for another with output being held constant
We multiply the ratio by -1 in order to express the MRTS as a positive number
Since we measure K on the vertical axis, the MRTS represents the amount of capital that must be sacrificed in order to use more labor in the production process, while producing the same level of output: Δ K/ Δ L (the slope of the isoquant which is negative)
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MRTS When Labor and Capital Are Continuously Divisible Dr. Manuel Salas Velasco Let’s take the total differential of the production function: Setting this total differential equal zero (since output does not change along a given isoquant): In production theory, the marginal rate of technical substitution is equal to the ratio of marginal products (in consumer theory, the marginal rate of substitution is equal to the ratio of marginal utilities)
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A Numerical Example Dr. Manuel Salas Velasco Assume the production function is: The marginal product functions are: If we specify the level of output as Q = 9 units, and the firm uses 3 units of labor, then the amount of capital used is: This result indicates that the firm can substitute 2 units of capital for 1 unit of labor and still produce 9 units of output
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Production Theory Econometric Analysis of Production Theory Dr. Manuel Salas Velasco
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The Cobb-Douglas Production Function Dr. Manuel Salas Velasco
Q = output
L = labor input
K = labor capital
u = stochastic disturbance term
e = base of natural logarithm
The parameter A measures, roughly speaking, the scale of production: how much output we would get if we used one unit of each input
The parameters beta measure how the amount of output responds to changes in the inputs
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The Cobb-Douglas Production Function Dr. Manuel Salas Velasco
Our problem is to obtain an estimated function:
However, if we take logarithms in both sides, we have:
β 0 This is the log-log, double-log, or log-linear model
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The Properties of the Cobb-Douglas Production Function Dr. Manuel Salas Velasco
The estimated coefficient β 1 is the elasticity of output with respect to the labor input; that is, it measures the percentage change in output for a 1 percent change in the labor input, holding the capital input constant
Likewise, the estimated coefficient β 2 is the elasticity of output with respect to the capital input, holding the labor input constant
Fitting this equation by the method of least squares, we have:
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The Properties of the Cobb-Douglas Production Function Dr. Manuel Salas Velasco
The sum of the estimated coefficients β 1 and β 2 gives information about the returns to scale
If this sum is 1, then there are constant returns to scale ; that is, doubling the inputs will double the output, tripling the inputs will triple the output, and so on
If the sum is less than 1, there are decreasing returns to scale ; e.g. doubling the inputs will less than double the output
If the sum is greater than 1, there are increasing returns to scale ; e.g. doubling the inputs will more than double the output
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Cobb-Douglas Production Function: The Agricultural Sector in Taiwan (1958-1972)
The log-linear model:
Dr. Manuel Salas Velasco
Regression by the OLS method:
output elasticity of labor output elasticity of capital 1.989 increasing returns to scale d = 0.891 ** ** ** Significant at 5%-level
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Cobb-Douglas Production Function: The Agricultural Sector in Taiwan (1958-1972) Dr. Manuel Salas Velasco 2 Zone of indecision Zone of indecision 0 4 d L d U No autocorrelation 4 – d U Evidence of positive autocorrelation Evidence of negative autocorrelation 4 – d L d = 0.891 0.946 1.543 2.457 3.054 Regression including a time variable: 0.814 1.750 2.250 3.186 d = 1.939 ** ** * ** Significant at 5%-level * Significant at 10%-level d = 1.939
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Cobb-Douglas Production Function: The U.S. Bell Companies (1981-82) Dr. Manuel Salas Velasco
The log-linear model:
Regression by the OLS method:
d = 1.954 (no autocorrelation) ** Significant at 5%-level ** ** ** ** output elasticity of labor A 1 percent increase in the labor input led on the average to about a 0.2 percent increase in the output