Microeconomic Theory

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Quiz 8 :

Production Functions

Quiz 8 :

Production Functions

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Power Goat Lawn Company uses two sizes of mowers to cut lawns. The smaller mowers have a 22-inch deck. The larger ones combine two of the 22-inch decks in a single mower. For each size of mower, Power Goat has a different production function, given by the rows of the following table. img a. Graph the q = 40,000 square feet isoquant for the first production function. How much k and l would be used if these factors were combined without waste? b. Answer part (a) for the second function. c. How much k and l would be used without waste if half of the 40,000-square-foot lawn were cut by the method of the first production function and half by the method of the second? How much k and l would be used if one fourth of the lawn were cut by the first method and three fourths by the second? What does it mean to speak of fractions of k and l? d. Based on your observations in part (c), draw a q = 40,000 isoquant for the combined production functions.
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Production function shows the functional relationship between the levels of output obtained from each feasible combination of inputs.
a)
The production function characterized by constant
img is a fixed-proportions production function.
Here capital and labor must always be used in a fixed ratio. The isoquant for this production function are L-shaped.
The mathematical form of the fixed-proportions production function is as follows:
img The first production function is as follows:
img Here, hold
img and
img Then the production function for
img is as follows:
img As here
img img img img Then the total capital requirement would be
img units and labor requirement would be
img units.
The isoquant for the production function is as follows:
img Figure 1
The X axis shows the quantity of labor employed and Y axis shows the quantity of capital and the isoquant curve shows the combination of capital and labor that is used to produce 40,000 units.
b.
The second production function is as follows:
img Here, hold
img and
img Then the production function for
img is given below:
img As here,
img img img img img Then the total capital requirement would be
img units and labor requirement would be
img units.
Represent the graph showing the isoquant as follows:
img Figure 2
The X axis shows the quantity of labor employed and Y axis shows the quantity of capital and the isoquant curve shows the combination of capital and labor that is used to produce 40,000 units.
c)
The two production functions are as follows:
img img If half of the lawn is cut by first method, then the required capital and labor would be calculated as shown below:
img If half of the lawn is cut by second method, then the required capital and labor would be calculated as shown below:
img img Then the total capital requirement would be
img units and labor requirement would be
img units.
The two production functions are given below:
img img If one-fourth of the lawn is cut by first method, then the required capital and labor would be calculated as shown below:
img If three-fourth of the lawn is cut by second method, then the required capital and labor would be calculated as shown below:
img img Then the total capital requirement would be
img units and labor requirement would be
img units.
The fraction of K and l means the amount of capital input used per unit of labor.
d)
Let the fraction p of the 40,000-square-foot lawn or
img -square-foot lawn is cut by first method, then the required capital and labor would be calculated as shown below:
img If
img of the 40,000-square-foot lawn or
img -square-foot of the lawn is cut by second method, then the required capital and labor would be calculated as shown below:
img img Then the total labor requirement is calculated below:
img The total capital requirement is calculated below:
img Therefore, the two equations derived above are given below:
img Solving first equation for p and putting the value of p in second as shown below:
img img This is an equation of a negatively sloping straight line. Thus, the combined isoquant will be negative sloping straight line giving a fixed output
img .
Represent the graph as follows:
img Figure 3
The X axis shows the quantity of labor employed and Y axis shows the quantity of capital and the isoquant curve shows the combination of capital and labor that is used to produce 40,000 units by the combined production method.

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Suppose the production function for widgets is given by img where q represents the annual quantity of widgets produced, k represents annual capital input, and l represents annual labor input. a. Suppose k = 10; graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point? b. Again assuming that k = 10, graph the MP l curve. At what level of labor input does MP l = 0? c. Suppose capital inputs were increased to k = 20. How would your answers to parts (a) and (b) change? d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale?
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a.
The production function shows all the combinations of labor and capital inputs that can be used to produce the maximum amount of the good.
Production function is given below:
img The average product of labor is defined as the quantity of output produced by per unit of labor inputof the firm.
Average product of labor can be shown as given below:
img The production function for widgets is given below:
img Where,
img quantity of widgets produced annually.
img Annual capital input
img Annual labor input
The production function for
img is given below:
img This gives the total productivity of the labor.
Total value of output at different level of labor input is calculated as shown below:
img The total productivity of labor curve is given below:
img Figure 1
X axis shows the quantity of labor and Y axis shows the total productivity of labor.
The average productivity of labor is given below:
img This gives the average productivity of the labor.
Average productivity of labor is calculated at different level of labor input is shown below:
img The average productivity of labor curve is given below:
img Figure 2
X axis shows the quantity of labor and Y axis shows the average productivity of labor.
For average product to reach the maximum the partial derivative of AP with respect to l must be zero.
The derivative of average productivity of labor is given below:
img Evaluate it to zero to get the value of l as shown below:
img img That is at labor input level
img average productivity reach a maximum. At this level the widgets production would be calculated below:
img 40 widgets will be produced at that point where average productivity of labor reaches a maximum.
b.
The marginal productivity of labor is given below:
img Marginal productivity of labor is calculated at different level of labor input is shown below:
img The marginal productivity of labor curve is given below:
img Figure 3
X axis shows the quantity of labor and Y axis shows the marginal productivity of labor.
The level at which marginal labor of productivity is zero is calculated below:
img c.
The production function for widgets is given below:
img Where,
img quantity of widgets produced annually.
img Annual capital input
img Annual labor input
The production function for
img is given below:
img This gives the total productivity of the labor.
Total value of output at different level of labor input is calculated as shown below:
img The total productivity of labor curve is given below:
img Figure 4
X axis shows the quantity of labor and Y axis shows the total productivity of labor.
The average productivity of labor is given below:
img This gives the average productivity of the labor.
Average productivity of labor is calculated at different level of labor input is shown below:
img The average productivity of labor curve is given below:
img Figure 5
X axis shows the quantity of labor and Y axis shows the average productivity of labor.
For average product to reach the maximum the partial derivative of AP with respect to l must be zero.
The derivative of average productivity of labor is given below:
img Evaluate it to zero to get the value of l as shown below:
img That is at labor input level
img average productivity reach a maximum. At this level the widgets production would be
img 160 widgets will be produced at that point where average productivity of labor reaches a maximum.
The marginal physical product (MP) of an input is the additional output that can be produced by employing one more unit of that input while holding all other inputs constant.
The marginal productivity of labor is given below:
img Marginal productivity of labor is calculated at different level of labor input is shown below:
img The marginal productivity of labor curve is given below:
img Figure 6
X axis shows the quantity of labor and Y axis shows the marginal productivity of labor.
The level at which marginal labor of productivity is zero is calculated below:
img d.
The production function exhibits constant returns, when a proportionate increase in input leads to increase in output by the same proportion.
The function exhibits diminishing returns to scale if output increases less than proportionately. However, if output increases more than proportionately, there are increasing returns to scale.
If
img is production function and if all inputs are multiplied by the same positive constant t (where t 1), then return to scale of the production function will be:
img Constant returns to scale
img Diminishing returns to scale
img Increasing returns to scale
img The production function for widgets is given below:
img Multiply the inputs by the some positive constant t as shown below:
img As
img ; thus, output increases more than proportionately, the function exhibits increasing returns to scale.

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Sam Malone is considering renovating the bar stools at Cheers. The production function for new bar stools is given by img where q is the number of bar stools produced during the renovation week, k represents the number of hours of bar stool lathes used during the week, and l represents the number of worker hours employed during the period. Sam would like to provide 10 new bar stools, and he has allocated a budget of $10,000 for the project. a. Sam reasons that because bar stool lathes and skilled bar stool workers both cost the same amount ($50 per hour), he might as well hire these two inputs in equal amounts. If Sam proceeds in this way, how much of each input will he hire and how much will the renovation project cost? b. Norm (who knows something about bar stools) argues that once again Sam has forgotten his microeconomics. He asserts that Sam should choose quantities of inputs so that their marginal (not average) productivities are equal. If Sam opts for this plan instead, how much of each input will he hire and how much will the renovation project cost? c. On hearing that Norm's plan will save money, Cliff argues that Sam should put the savings into more bar stools to provide seating for more of his USPS colleagues. How many more bar stools can Sam get for his budget if he follows Cliff 's plan? d. Carla worries that Cliff 's suggestion will just mean more work for her in delivering food to bar patrons. How might she convince Sam to stick to his original 10-bar stool plan?
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Sam Malone is considering renovating the bar stools at Cheers. The production function for new bar stools is given by
img Where q is the number of bar stools produced during the renovation week, k represents the number of hours of bar stool lathes used during the week, and l represents the number of worker hours employed during the period. Sam would like to provide 10 new bar stools, and he has allocated a budget of $10,000 for the project.
a. Sam reasons that because bar stool lathes and skilled bar stool workers both cost the same amount ($50 per hour), he might as well hire these two inputs in equal amounts. If Sam proceeds in this way, his production function would be
img Here
img ; putting this on the production function we have
img As he will hire two inputs in equal amounts then;
img as well.
Both the inputs cost the same amount ($50 per hour), the total cost is,
img Therefore, he will hire 100units of both the inputs and the cost of the project will be $10,000.
b. Norm asserts that Sam should choose quantities of inputs so that their marginal productivities are equal.
The marginal physical product (MP) of an input is the additional output that can be produced by employing one more unit of that input while holding all other inputs constant. Mathematically,
img img Here the production function is given as
img Therefore;
img img Sam should choose quantities of inputs so that their marginal productivities are equal. Thus at equilibrium
img Putting
img in the production function we get
img Here
img ; putting this on the production function we have
img Hence
img As both the input costs $50; the total cost of renovation is
img If Sam opts for this plan instead, 33.003 units of capital and 132.012 units of labor will he hire and the renovation project will cost $8250.75.
c. Upon hearing that Norm's plan will save money, Cliff argues that Sam should put the savings into more bar stools in order to provide seating to more of his USPS colleagues. If Sam chooses quantities of inputs so that their marginal productivities are equal, the total cost of renovation is $8250.75. Here, he will save $1749.75. In this method one tool will cost him $825.075. Then in the saved money he can make approximately 2 more tools.
Therefore, 2 more bar stools can Sam get for his budget if he follows Cliff's plan.
d. Carla worries that Cliff's suggestion will just mean more work for her in delivering food to bar patrons. This could decrease the marginal product of Cliff. She can prove unprofitable for the bar, and add cost to bar renovation. This could make Sam think twice about adding more stool at bar. In this way she can convince Sam to stick to his original 10-bar stool plan.

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Suppose that the production of crayons (q) is conducted at two locations and uses only labor as an input. The production function in location 1 is given by img a. If a single firm produces crayons in both locations, then it will obviously want to get as large an output as possible given the labor input it uses. How should it allocate labor between the locations to do so? Explain precisely the relationship between l 1 and l 2. b. Assuming that the firm operates in the efficient manner described in part (a), how does total output (q) depend on the total amount of labor hired ( l )?
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As we have seen in many places, the general Cobb-Douglas production function for two inputs is given by img where the expression is to be evaluated at t = 1. Show that, for this Cobb-Douglas function img . Hence in this case the scale elasticity and the returns to scale of the production function agree (for more on this concept see Problem 9.9). d. Show that this function is quasi-concave. e. Show that the function is concave for img img
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Suppose we are given the constant returns-to-scale CES production function img img Note: The latter equality is useful in empirical work because we may approximate img by the competitively determined wage rate. Hence ? can be estimated from a regression of ln( q / l ) on ln w.
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Consider a generalization of the production function in Example 9.3: img where img a. If this function is to exhibit constant returns to scale, what restrictions should be placed on the parameters img b. Show that, in the constant returns-to-scale case, this function exhibits diminishing marginal productivities and that the marginal productivity functions are homogeneous of degree 0. c. Calculate ? in this case. Although ? is not in general constant, for what values of the img
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Show that Euler's theorem implies that, for a constant returns-to-scale production function [ q = f ( k , l ) ], img Use this result to show that, for such a production function, if MP l AP l then MP k must be negative. What does this imply about where production must take place? Can a firm ever produce at a point where AP l is increasing?
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Local returns to scale A local measure of the returns to scale incorporated in a production function is given by the scale elasticity img evaluated at t = 1. a. Show that if the production function exhibits constant returns to scale, then e q , t =1. b. We can define the output elasticities of the inputs k and l as img Show that img c. A function that exhibits variable scale elasticity is img Show that, for this function, img d. Explain your results from part (c) intuitively. Hint: Does q have an upper bound for this production function?
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Returns to scale and substitution Although much of our discussion of measuring the elasticity of substitution for various production functions has assumed constant returns to scale, often that assumption is not necessary. This problem illustrates some of these cases. a. In footnote 6 we pointed out that, in the constant returns-to-scale case, the elasticity of substitution for a two-input production function is given by img where f (k, l) is a constant returns-to-scale production function and img is a positive exponent. Show that the elasticity of substitution for this production function is the same as the elasticity of substitution for the function f. b. Show how this result can be applied to both the Cobb-Douglas and CES production functions.
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More on Euler's theorem Suppose that a production function f ( x 1 , x 2 ,... , x n ) is homogeneous of degree k. Euler's theorem shows that P , and this fact can be used to show that the partial derivatives of f are homogeneous of degree k - 1. a. Prove that img . b. In the case of n = 2 and k = 1, what kind of restrictions does the result of part (a) impose on the second-order partial derivative f 12 ? How do your conclusions change when k 1 or k 1? c. How would the results of part (b) be generalized to a production function with any number of inputs? d. What are the implications of this problem for the parameters of the multivariable Cobb-Douglas production function img
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