...The Cobb-Douglas Production Function Introduction A production function is a function which relates the factors of production, capital and labour, to output given the available technology. Capital here refers to things workers use in producing goods and services such as computers. Labour refers to the time that people spend working. The production function can thus be written as: Y = F(K, L) where Y denotes output, K denotes capital, L denotes labour and F is the relationship between the factors of production and output. Knowing the specific form of the production is important as it can explain the incomes of capital and labour, i.e. how much of output is compensated to workers and how much is compensated to owners of capital. This may seem a daunting task given modern economies’ vast array of industries which use labour and capital differently but there is a particular form which has proven to be resilient since its discovery. In 1927, Paul Douglas, a professor of economics in the United States (U.S.), discovered a pattern in data he had constructed from 1899 to 1922. Labour share of total income was relatively constant over this long period. This meant that even as the economy grew, total income of capital owners and workers grew at almost exactly the same rate. He consulted Charles Cobb, a mathematician and colleague, on what sort of production function would yield constant factor shares. Cobb showed that the function with this property was: Y = F(K, L) = AKαL1-α, where...
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...Lauren Liwen MBA 608 Dr. Turek Cranston Coils Regression Case Executive Summary The Cobb-Douglas cost function of Cranston Coils was found using output, capital, and labor data from their eighteen plants. The cost function, Q = (0.40692) K0.32477 L0.79466, was used to determine the short-run cost equations of total cost, average cost, average variable cost and marginal cost. Calculations using these equations gave rise to Cranston Coils cost structure, which predicts cash flow within the company. Cranston Coils’ cost function was also used to determine if a contract between Sleep Easy and Cranston Coils should be accepted. After determining marginal cost and revenue (see Appendix 5: Sleep Easy Contract Costs at Connecticut Plant), the contract should be accepted. Problem Definition The short-run cost structure of the new Cranston Coils facility in Connecticut needs to be determined in order to predict the cash flow of the new plant. Once the cost structure is defined, the cost function can be used to evaluate whether or not certain contracts should be accepted, such as the Sleep Easy Company’s proposed contract of fifty units at $70.00 per unit. Identification of Possible Solutions The short-run cost equations of Cranston Coils indicate that the cost structure varies between the plants because of the significant discrepancy between the marginal costs of each plant. One possible solution would be to reevaluate each plant’s marginal product of labor and marginal...
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...Cobb–Douglas Production Functions 1 mathematical tricks • the derivative of αxβ with respect to x is αβxβ−1 • xα xβ = xα+β (for any α and β) • 1 xα = x−α • if m = nB , then n = m1/B 2 the production function A production function y = f (x1 , x2 ) is a Cobb–Douglas production function if it can be written in the form y = Axa xb 1 2 (1) where A, a and b are positive constants. So the partial derivatives of a Cobb–Douglas production function are : M P1 = M P2 = ∂y = aAxa−1 xb 1 2 ∂x1 (2) ∂y = bAxa xb−1 (3) 1 2 ∂x2 The absolute value of the slope of an isoquant is the technical rate of substitution, or T RS. This T RS equals M P1 /M P2 so that (2) and (3) imply that T RS = M P1 ax1 = M P2 bx2 1 (4) Equations (2) and (3) imply that the Cobb–Douglas technology is monotonic, since both partial derivatives are positive. Equation (4) demonstrates the technology is convex, since the (absolute value) of the T RS falls as x1 increases and x2 decreases. 3 returns to scale Suppose that all inputs are scaled up by some factor t. The new level of output is (5) f (tx1 , tx2 ) = A(tx1 )a (tx2 )b = ta+b Axa xb 1 2 Notice from equation (5) that f (tx1 , tx2 ) = ta+b f (x1 , x2 ). We have increasing returns to scale if f (tx1 , tx2 ) > tf (x1 , x2 ) whenever t > 1. So here we have increasing returns to scale if ta+b > t, which is the same thing as a + b > 1. Similarly, decreasing returns to scale arise if f (tx1 , tx2 ) < tf (x1 , x2 ) whenever t > 1. Here...
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...[pic] OUM BUSINESS SCHOOL ASSIGNMENT SUBMISSION AND ASSESSMENT _________________________________________________________________________ BMME5103 MANAGERIAL ECONOMICS JANUARY 2015 _________________________________________________________________________ INSTRUCTIONS TO STUDENTS 1. This assignment contains question that is set in English. 2. Answer in English only. 3. Your assignment should be typed using 12 point Times New Roman font and 1.5 line spacing. 4. You must submit your hardcopy assignment to your Fasilitator and ON-LINE via the MyVLE. Refer to the portal for instructions on the procedures to submit your assignment on-line. You are advised to keep a copy of your submitted assignment and proof of the submission for personal reference. Your assignment must be submitted on 21st – 29th March 2015. 5. Your assignment should be prepared individually. You should not copy another person’s assignment. You should also not plagiarise another person’s work as your own. EVALUATION This assignment accounts for 60% of the total marks for the course. ASSIGNMENT QUESTION PART 1 PURPOSE: The purpose of PART 1 of this assignment is to enable the students to understand the fundamental economic concept and to apply his/her knowledge on factors affecting demand; in particular when more than one factors change simultaneously. REQUIREMENT: 1. The concept of a marginal use value that declines as the rate of consumption...
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...besa44438_ch06.qxd 09/23/2004 04:18 PM Page 183 6 C H A P T E R INPUTS AND PRODUCTION FUNCTIONS 6.1 INTRODUCTION TO INPUTS AND PRODUCTION FUNCTIONS APPLICATION 6.1 Competition Breeds Efficiency 6.2 PRODUCTION FUNCTIONS WITH A SINGLE INPUT 6.3 PRODUCTION FUNCTIONS WITH MORE THAN ONE INPUT High-Tech Workers versus Low-Tech Workers APPLICATION 6.2 6.4 S U B S T I T U TA B I L I T Y A M O N G INPUTS Elasticities of Substitution in German Industries APPLICATION 6.3 APPLICATION 6.4 Measuring Productivity APPLICATION 6.5 Returns to Scale in Electric Power 6.5 RETURNS TO SCALE Generation APPLICATION 6.6 Returns to Scale in Oil Pipelines APPLICATION 6.7 Technological Progress in the U.K. 6.6 TECHNOLOGICAL PROGRESS Appendix THE ELASTICITY OF SUBSTITUTION FOR A COBB–DOUGLAS PRODUCTION FUNCTION besa44438_ch06.qxd 09/23/2004 04:19 PM Page 184 Can They Make It Better and Cheaper? The production of semiconductor chips—thin, glasslike wafers that are used to store information in digital equipment—is costly, complex, and delicate.1 Production involves hundreds of steps and takes place in facilities called fabs, expensive factories that can cost more than $1 billion to construct. To avoid contaminating chips, fabs must be 1000 times cleaner than a hospital operating room. Because the manufacturing process is so expensive and because a typical fab is obsolete in 3 to 5 years (and...
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...หรือบางครั้งเรียกว่ามังกรน้อยทั้งสี่(มีประเทศใต้หวัน เกาหลีใต้ สิงคโปร์ ฮ่องกง) แม้ว่าในปัจจุบันประเทศไทยได้ถูกจัดเข้าในกลุ่มประเทศอุตสาหกรรมใหม่ และกลุ่มเสือ แล้วแต่ก็ยังเข้ากลุ่มดังกล่าวช้ากว่าประเทศเกาหลีใต้ เพราะเหตุใดประเทศไทยจึงพัฒนาได้ช้ากว่าประเทศเกาหลีใต้ทั้งที่ประเทศไทยเคยมีรายได้ประชาชาติต่อหัวสูงกว่าประเทศเกาหลีซึ่งน่าจะแสดงให้เห็นว่าไทยเคยมีรากฐานทางเศรษฐกิจที่ดีกว่าเกาหลีใต้ แต่ทำไมในปัจจุบันประเทศเกาหลีใต้จึงพัฒนาเศรษฐกิจจนแซงไทยไปแล้วหลายก้าว ในterm paper นี้จะแบ่งออกเป็น 2 ส่วนในการศึกษา ส่วนแรกจะกล่าวถึงการพัฒนาเศรษฐกิจของประเทศประเทศเกาหลีใต้เปรียบเทียบกับไทย ตั้งแต่เริ่มมีแผนพัฒนาเศรษฐกิจในปี 1960 และส่วนที่สองจะกล่าวการวิเคราะห์ความเจริญเติบโตทางเศรษฐกิจระหว่างประเทศเกาหลีใต้กับประเทศไทย ทางด้านอุปทาน โดยใช้ cobb-douglas production function นอกจากนี้จะมีการวิเคราะห์ทางด้านอุปสงค์ ซึ่งจะทำให้ได้ทราบถึงกระบวนการพัฒนาเศรษฐกิจโดยยึดประเทศเกาหลีใต้เป็นแบบแผน เพี่อพิจารณาข้อดี ข้อเสียจากบทเรียนของประเทศเกาหลีใต้ เพื่อให้ในการปรับปรุงและพัฒนาการวางแผนเศรษฐกิจของประเทศไทยให้เจริญก้าวหน้าได้ต่อไป การพัฒนาเศรษฐกิจของประเทศประเทศเกาหลีใต้เปรียบเทียบกับไทย ประเทศเกาหลีและประเทศไทยเริ่มมีการวางแผนการพัฒนาเศรษฐกิจแห่งชาติในปี 1960 ปีเดียวกัน...
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...The returns to scale in the production function Y = K0.5L0.5 are: constant If a production function has two inputs and exhibits constant returns to scale, then doubling both inputs will cause the output to: double If the supplies of capital and labor are fixed and technology is unchanging, then real output is: fixed If a production function has the property of diminishing marginal product, then doubling: one of the inputs will reduce its marginal product. Consider the following production table: Assuming that the production function displays constant returns to scale, what is the marginal product of labor when labor and capital are both equal to 1,000? 5 Consider the following production table: By how much does the marginal product of labor decrease as labor input increases from 1 to 2 and from 2 to 3? 1 Euler's theorem implies that if a production function exhibits constant returns to scale, then: economic profit is zero. If a firm with a constant returns to scale production function pays all factors their marginal products, then: economic profit is zero and accounting profit is positive. A competitive firm hires labor until the marginal product of labor equals the: real wage. A competitive firm rents capital until the marginal product of capital equals the: real rental price of capital. Suppose that a major natural disaster destroys a large part of a country's capital stock but miraculously does not cause anybody bodily harm. What will happen...
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...CHAPTER I INTRODUCTION “Technology... is a queer thing. It brings you great gifts with one hand, and it stabs you in the back with the other.” Carrie P. Snow With remarkable technological advancements in third world countries like the Philippines, modern electronic gadgets play a vital role in everyone’s lives. Virtually every day, we find ourselves in situations that require us to use gadgets and devices to boost up performance in tasks we ought to do. Because of the advent of modern technology, the incorporation of electronic gadgets in student’s education continuously remain as one of the “educational battlegrounds” that is fought over and over again. Ever since, the introduction of technology in the country has evidently improved our performances in the various facets of our lives, not to mention the improvements it has done in the Philippine economy. Thus, the country is placing a high priority in technological advancements, with the hope of bringing the name of the Philippines in the list of “developed” countries in the world. Background of the Study In the Ateneo de Zamboanga University which is known for its innovative culture of technological revolution, students relatively have the capacity to acquire and own modern electronic gadgets. Because of these technological advancements, college professors tend to make their requirements or projects technically “modern”. Thus, electronic gadgets essentially have an impact most especially to the students for these...
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...The Production Function for Wilson Company By using the EViews software, we get the result below by using Least Square method: Dependent Variable: Y | | | Method: Least Squares | | | Date: 06/18/12 Time: 03:24 | | Sample: 1 15 | | | Included observations: 15 | | | | | | | | | | | | Variable | Coefficient | Std. Error | t-Statistic | Prob. | | | | | | | | | | | C | -130.0086 | 129.8538 | -1.001192 | 0.3365 | L | 0.359450 | 0.245593 | 1.463601 | 0.1690 | K | 0.027607 | 0.006051 | 4.562114 | 0.0007 | | | | | | | | | | | R-squared | 0.838938 | Mean dependent var | 640.3800 | Adjusted R-squared | 0.812094 | S.D. dependent var | 227.9139 | S.E. of regression | 98.79645 | Akaike info criterion | 12.20086 | Sum squared resid | 117128.9 | Schwarz criterion | 12.34247 | Log likelihood | -88.50643 | F-statistic | 31.25263 | Durbin-Watson stat | 1.458880 | Prob(F-statistic) | 0.000017 | | | | | | | | | | | 1. In standard form the estimated Cobb-Douglas equation is written as: Q= α Lβ1 Kβ2 The multiplicative exponential Cobb-Douglass Function can be estimated as a linear regression relation by taking logarithm: Log Q = log α + β1 log L + β2 log K Therefore: log(y) = -130.0086 + 0.359450*log(L) + 0.027607*log(K) The output elasticity of capital is 0.027607 and the output elasticity of labor is 0.359450. 2...
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...nationalized banks and SBI and its associates. We have collated the data for these two groups of banks and used them as one. We can consider multiple output and input measures, which are in particular applicable to the Indian Banking industry. However for this case we have considered output as deposits and inputs as capital and labour costs. Following this approach, we have defined output as the rupee value of total deposits at the end of the year. We have measured labour by the number of employees which comprises of all the employees viz., officers and other employees i.e., clerks and sub staff. Capital input is a crucial input in the production process and it is the most complex of all the inputs to measure and has been taken as the value of the net fixed assets of the banks. We already have the formula for Cobb-Douglas production function: F (K, L) = A.Kα.Lβ Here K represents capital, L represents labour and A, α, β are Positive constants. In order to calculate the value of A, α, β we need to convert this equation into a linear one. Taking the natural logarithm of both the sides we get, logeF = logeA + α logeK+ β logeL Where logeF = Y, logeA = C, logeK = x1, logeL = x2, α= A, β= B Now we plot the multiple regression line for the variables F, K and L through the data obtained for the public sector banks over the range of years , and we obtain the values for the coefficients A, B and C in the standard line equation. To get the value of A, α, β we take the anti log that...
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...capital productivity increases and its real rental price increases b. An earthquake destroys some of the capital stock (assuming labor fixed) Answer: Decrease in the capital stock decreases the amount of the capital available for production, marginal productivity of the capital increases when the usage of capital decreases and its real rental price increases. c. A. technological advance improves production function Answer: A technological change increases the productivity of labor and capital, since the inputs are paid their respective marginal products, real wage and real rental price of capital increase. 2. Suppose that an economy” production function is Cobb-Douglas with parameter a = 0.3. ( a = parameter α in the textbook) ( 2 points) a. What fraction of income do capital and labor receive? Answer: assuming constant returns to scale and given the size of the a parameter, capital receives 1/3 of the total income while labor receives b= 1- a = 1- 1/3 = 2/3 of income Or 30 percent of the income goes to the capital and 70 percent to the labor b. Suppose that immigration increases the labor force by 10 percent. What happens to the output (in percent)? Answer: Given the production function. the exponent to K measures how responsive (elastic) is output (Y) to the change in the amount of capital while holding the amount of labor constant. The exponent b = 1- a measures how responsive (elastic) is output (Y) to the...
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...Mankiw, Macroeconomics 8e Chapter 3 1. The returns to scale in the production function Y = K0.5L0.5 are: A. decreasing. B. constant. C. increasing. D. subject to wide fluctuations. 2. If a production function has two inputs and exhibits constant returns to scale, then doubling both inputs will cause the output to: A. reduce by half. B. stay the same. C. double. D. quadruple. 3. If the supplies of capital and labor are fixed and technology is unchanging, then real output is: A. fixed. B. determined by demand. C. uncertain. D. subject to wide fluctuations. 4. If a production function has the property of diminishing marginal product, then doubling: A. all of the inputs will less than double the output. B. all of the inputs will double the output. C. all of the inputs will more than double the output. D. one of the inputs will reduce its marginal product. 5. Consider the following production table: Assuming that the production function displays constant returns to scale, what is the marginal product of labor when labor and capital are both equal to 1,000? A. 1 B. 5 C. 10 D. 20 6. Consider the following production table: By how much does the marginal product of labor decrease as labor input increases from 1 to 2 and from 2 to 3? A. 0 B. 1 C. 2 D. 3 7. Euler's theorem implies that if a production function exhibits constant returns to scale, then: A. economic profit is zero. ...
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...Efficiency Analysis of Container Ports and Terminals Qianwen Liu A thesis submitted for the degree of Doctor of Philosophy of University College London Centre for Transport Studies Department of Civil, Environmental and Geomatic Engineering University College London 2010 1 Declaration I, Qianwen Liu, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Candidate’s signature ………………………………….. Qianwen Liu 2 Abstract In the past two decades the steady growth of seaborne trade has resulted in the increase of container ships, container ports and their terminals. The structure of the shipping market is, moreover, continuously evolving. On the carrier side, shipping companies form consortia and alliances; on the port side, global terminal operators and dedicated container terminals are emerging. The aim of this research is to evaluate the efficiency of container ports and terminals and to study how to improve the scale efficiency of any particular port/terminal. In particular we study how certain factors influence the efficiency of container ports and terminals. Regional container ports and global container terminals are examined based on the econometrics benchmarking method Stochastic Frontier Analysis (SFA). Two datasets are used, a panel dataset for 32 container ports in the North Mediterranean Sea over a nine-year period, and a cross-sectional...
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...second-order condition.) 1. Consider the function: y = 2 x2. a. Determine the average rate of change of the function in the closed interval [1,4] of the argument. b. Determine whether the function is concave or convex, using geometric test and a specific value of ( = 0.3. c. Do the concavity/convexity test using the derivative conditions. 2. Consider the function: y = 2x. a. Determine the average rate of change of the function in the closed interval [1,4] of the argument. b. Determine whether the function is concave or convex, using geometric test and a specific value of ( = 0.3. c. Do the concavity/convexity test using the derivative conditions. 3. Use the derivative condition to test whether the function y = 8 + 10x - x2 is concave or convex over the domain [0,7]. 4. In a cross-country study of the relationship between income per-capita (Y) and pollution (S), Grossman and Krueger estimate a cubic relationship as S = 0.083Y3 - 2.2Y2 + 13.5Y + X, where X represents other factors not linked to income. For this exercise, consider X as a fixed quantity for a country. Identify and characterize the extreme values of this function. 5. In the model of perfect competition, all firms are price-takers since they treat price (P) as a market-determined constant. Assume that P = 12. A firm's total revenue (TR) function is TR(Q) = P ( Q, where Q is the quantity of output of the firm. The total cost (TC) function of the firm is TC(Q) = Q3 - 4.5Q2 + 18Q -...
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...STOCHASTIC FRONTIER ANALYSIS 1 MOTIVATION • Usual textbook presentations treat producers as successful optimizers. They maximize production, minimize cost, and maximize profits. • Conventional econometric techniques build on this paradigm to estimate production/cost/profit function parameters using regression techniques where deviations of observed choices from optimal ones are modeled as statistical noise. • However though every producer may attempt to optimize, not all of them may succeed in their efforts. For example, given the same inputs, and the same technology, some will produce more output than others, i.e., some producers will be more efficient than others. • Econometric estimation techniques should allow for the fact that deviations of observed choices from optimal ones are due to two factors: failure to optimize i.e., inefficiency due to random shocks • Stochastic Frontier Analysis or SFA is one such technique to model producer behavior. 2 USEFULNESS OF SFA • SFA produces efficiency estimates or efficiency scores of individual producers. Thus one can identify those who need intervention and corrective measures. Since efficiency scores vary across producers, they can be related to producer characteristics like size, ownership, location, etc. Thus one can identify source of inefficiency. SFA provides a powerful tool for examining effects of intervention. For example, has efficiency of the banks changed after deregulation? Has this change varied across...
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