The following cost-output data were obtained as part of a study of the economies of scale in operating a charter high school in Wisconsin: Students in Average Daily Attendance Midpoint of Values in Column A Operating Expenditure per Student Number of Schools in Sample (A) (B) (C) (D) 143-200 171 $531.9 6 201-300 250 480.8 12 301-400 350 446.3 19 401-500 450 426.9 17 501-600 550 442.6 14 601-700 650 413.1 13 701-900 800 374.3 9 901-1100 1000 433.2 6 1101-1600 1350 407.3 6 1601-2400 2000 405.6 7 Plot the data in columns B and C in an output (enrollment-) cost graph and sketch a smooth curve that would appear to provide a good fit to the data. Based on the scatter diagram in Question 1, what kind of mathematical relationship would appear to exist between enrollment and operating expenditures per student? In other words, do operating expenditures per student appear to (i) be constant (and independent of enrollment), (ii) follow a linear relationship as enrollment increases, or (iii) follow some sort of nonlinear U-shape (possibly quadratic) relationship as enrollment increases? As part of this study, the following cost function was developed: C = f(Q, X1, X2, X3, X4, X5) Where C = operating expenditures per student in average daily attendance (measured in dollars) Q = enrollment (number of students in average daily attendance) X1 = average teacher salary X2 = number of credit units (“courses”) offered X3 = average number of courses taught per teacher X4 = change in enrollment between 1957 and 1960 X5 = percentage of classrooms built after 1950 Variables X1, X2, and X3 were considered measures of teacher qualifications, breadth of curriculum, and the degree of specialization in instruction, respectively. Variable X4 measured changes in demand for school services that could cause some lagging adjustments in cost. Variable X5 was used to reflect any differentials in the costs of maintenance and operation due to the varying ages of school properties. Statistical data on 109 selected high schools yielded the following regression equation: C = 10.31 − .402Q + .00012Q2 + .107X1 + .985X2 − 15.62X3 + .613X4 − .102X5 (.063)* (0.000023)* (.013)* (.640) (11.95) (.189)* (.109) r2=.557* Notes: The numbers in parentheses are the standard deviations of each of the respective (b) coefficients. An asterisk (*) indicates that the result is statistically significant at the 0.01 level. What type of cost-output relationship (linear, quadratic, cubic) is suggested by these statistical results? What variables (other than enrollment) would appear to be most important in explaining variations in operating expenditures per student? Holding constant the effects of the other variables (X1 through X5), determine the enrollment level (Q) at which average operating expenditures per student are minimized. (Hint: Find the value of Q that minimizes the (∂C/∂Q function.) Again, holding constant the effects of the other variables, use the ∂C/∂Q function to determine, for a school with 500 students, the reduction in per-student operating expenditures that will occur as the result of adding one more student. Again, holding the other variables constant, what would be the saving in per student operating expenditures of an increase in enrollment from 500 to 1,000? Based on the results of this study, what can we conclude about the existence of economies or diseconomies in operating a public high school?

The following cost-output data were obtained as part of a study of the economies of scale in operating a charter high school in Wisconsin: Students in Average Daily Attendance Midpoint of Values in Column A Operating Expenditure per Student Number of Schools in Sample (A) (B) (C) (D) 143-200 171 $531.9 6 201-300 250 480.8 12 301-400 350 446.3 19 401-500 450 426.9 17 501-600 550 442.6 14 601-700 650 413.1 13 701-900 800 374.3 9 901-1100 1000 433.2 6 1101-1600 1350 407.3 6 1601-2400 2000 405.6 7 Plot the data in columns B and C in an output (enrollment-) cost graph and sketch a smooth curve that would appear to provide a good fit to the data. Based on the scatter diagram in Question 1, what kind of mathematical relationship would appear to exist between enrollment and operating expenditures per student? In other words, do operating expenditures per student appear to (i) be constant (and independent of enrollment), (ii) follow a linear relationship as enrollment increases, or (iii) follow some sort of nonlinear U-shape (possibly quadratic) relationship as enrollment increases? As part of this study, the following cost function was developed: C = f(Q, X1, X2, X3, X4, X5) Where C = operating expenditures per student in average daily attendance (measured in dollars) Q = enrollment (number of students in average daily attendance) X1 = average teacher salary X2 = number of credit units (“courses”) offered X3 = average number of courses taught per teacher X4 = change in enrollment between 1957 and 1960 X5 = percentage of classrooms built after 1950 Variables X1, X2, and X3 were considered measures of teacher qualifications, breadth of curriculum, and the degree of specialization in instruction, respectively. Variable X4 measured changes in demand for school services that could cause some lagging adjustments in cost. Variable X5 was used to reflect any differentials in the costs of maintenance and operation due to the varying ages of school properties. Statistical data on 109 selected high schools yielded the following regression equation: C = 10.31 − .402Q + .00012Q2 + .107X1 + .985X2 − 15.62X3 + .613X4 − .102X5 (.063)* (0.000023)* (.013)* (.640) (11.95) (.189)* (.109) r2=.557* Notes: The numbers in parentheses are the standard deviations of each of the respective (b) coefficients. An asterisk (*) indicates that the result is statistically significant at the 0.01 level. What type of cost-output relationship (linear, quadratic, cubic) is suggested by these statistical results? What variables (other than enrollment) would appear to be most important in explaining variations in operating expenditures per student? Holding constant the effects of the other variables (X1 through X5), determine the enrollment level (Q) at which average operating expenditures per student are minimized. (Hint: Find the value of Q that minimizes the (∂C/∂Q function.) Again, holding constant the effects of the other variables, use the ∂C/∂Q function to determine, for a school with 500 students, the reduction in per-student operating expenditures that will occur as the result of adding one more student. Again, holding the other variables constant, what would be the saving in per student operating expenditures of an increase in enrollment from 500 to 1,000? Based on the results of this study, what can we conclude about the existence of economies or diseconomies in operating a public high school?

Managerial Economics: Applications, Strategies and Tactics (MindTap Course List)

14th Edition

ISBN:9781305506381

Author:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris

Publisher:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris

Chapter9: Applications Of Cost Theory

Section: Chapter Questions

Problem 1.2CE

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Question

The following cost-output data were obtained as part of a study of the economies of scale in operating a charter high school in Wisconsin:

Students in Average Daily Attendance	Midpoint of Values in Column A	Operating Expenditure per Student	Number of Schools in Sample
(A)	(B)	(C)	(D)
143-200	171	$531.9	6
201-300	250	480.8	12
301-400	350	446.3	19
401-500	450	426.9	17
501-600	550	442.6	14
601-700	650	413.1	13
701-900	800	374.3	9
901-1100	1000	433.2	6
1101-1600	1350	407.3	6
1601-2400	2000	405.6	7

Plot the data in columns B and C in an output (enrollment-) cost graph and sketch a smooth curve that would appear to provide a good fit to the data.
Based on the scatter diagram in Question 1, what kind of mathematical relationship would appear to exist between enrollment and operating expenditures per student? In other words, do operating expenditures per student appear to (i) be constant (and independent of enrollment), (ii) follow a linear relationship as enrollment increases, or (iii) follow some sort of nonlinear U-shape (possibly quadratic) relationship as enrollment increases?

As part of this study, the following cost function was developed:

C = f(Q, X₁, X₂, X₃, X₄, X₅)

Where C = operating expenditures per student in average daily attendance (measured in dollars)

Q = enrollment (number of students in average daily attendance)

X₁ = average teacher salary

X₂ = number of credit units (“courses”) offered

X₃ = average number of courses taught per teacher

X₄ = change in enrollment between 1957 and 1960

X₅ = percentage of classrooms built after 1950

Variables X₁, X₂, and X₃ were considered measures of teacher qualifications, breadth of curriculum, and the degree of specialization in instruction, respectively. Variable X₄ measured changes in demand for school services that could cause some lagging adjustments in cost. Variable X₅ was used to reflect any differentials in the costs of maintenance and operation due to the varying ages of school properties. Statistical data on 109 selected high schools yielded the following regression equation:

C = 10.31 − .402Q + .00012Q² + .107X₁ + .985X₂ − 15.62X₃ + .613X₄ − .102X₅

(.063)* (0.000023)* (.013)* (.640) (11.95) (.189)* (.109)

r²=.557*

Notes: The numbers in parentheses are the standard deviations of each of the respective (b) coefficients. An asterisk (*) indicates that the result is statistically significant at the 0.01 level.

What type of cost-output relationship (linear, quadratic, cubic) is suggested by these statistical results?
What variables (other than enrollment) would appear to be most important in explaining variations in operating expenditures per student?
Holding constant the effects of the other variables (X₁ through X₅), determine the enrollment level (Q) at which average operating expenditures per student are minimized. (Hint: Find the value of Q that minimizes the (∂C/∂Q function.)
Again, holding constant the effects of the other variables, use the ∂C/∂Q function to determine, for a school with 500 students, the reduction in per-student operating expenditures that will occur as the result of adding one more student.
Again, holding the other variables constant, what would be the saving in per student operating expenditures of an increase in enrollment from 500 to 1,000?
Based on the results of this study, what can we conclude about the existence of economies or diseconomies in operating a public high school?

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