I have completed 1-3, hopefully correctly. If correct, need help with questions 4-9.

Need to confirm I did the regression correctly, using the natural log of the data. the data can be found towards the end.

After completing his bachelor's degree in business in Texas State University in December, 1994, Allen Bond used a $100,000 inheritance to open his own construction business in January, 1995. The firm specializes in building a standard-design utility (or storage) building for residential homes.

From a small beginning in 1995 when it employed only four carpenters, leased only one unit of capital, and built only 25 of its standard-design utility buildings, the firm has grown substantially. In 2020, the firm employs 75 carpenters part-time, leases 25 units of capital, and, by the end of the year, will have built 823 of its utility buildings which sell for $2,750 each.

The 75 part-time carpenters are paid $25,000 per year and constitute the entire labor component of the firm since all managerial, accounting, and clerical functions are handled by Mr. Bond. Mr. Bond has developed a measure for a unit of capital that includes one truck and a specified amount of other equipment, including a ladder, several power tools, an air compressor, etc. Mr. Bond currently leases 25 units of capital at $5,000 per unit. Labor and capital inputs can be varied daily and are the only significant explicit expenses incurred by the firm since building materials for each utility building are supplied by the customer.

Mr. Bond has just completed an MBA managerial economics course at Texas State University. Although he is not sure he understood everything, the class did make him aware of many problems he had not considered before. In particular, Mr. Bond is concerned about whether the current mix of labor and capital is optimal for the production process employed by Bond, Inc. Additionally, Mr. Bond now realizes that the business environment in which he operates is much more complex than he had previously considered. For example, skilled carpenters have recently organized and their salaries will increase to $30,000 per year beginning in 2021, the cost of capital will increase to $7,500 per unit, and the demand for utility buildings is expected to slow because new builders are including plenty of storage space in new structures. Finally, the city commission has just passed an ordinance restricting the construction of utility buildings after 2021 that will essentially eliminate the industry in the city that Bond, Inc., serves. For all practical purposes, with no assets and a "doomsday" future ahead of it, Bond, Inc., has no market value.

Mr. Bond anticipates retiring at the end of 2025, but the pressures of managing his business are beginning to bother him. His own projections are that because of the developments, growth will slow to 5% per year over the remaining five years the industry can exist. Increasingly, he is wondering if he would not be happier simply accepting a job offer from a local firmBobcats, Inc., a very large construction firm. Last year the personnel manager offered Mr. Bond an annual salary of $180,000 and has already called again this year to offer Mr. Bond a five-year contract, beginning in January of 2021, at a beginning annual salary of $180,000 with 5% annual increases.

Mr. Bond has hired your consulting group to make a thorough economic analysis of the firm's operation and to help him decide how to spend the next five years. Unfortunately, he has kept very few records that might be used for economic analysis. Records were maintained on output and inputs of capital and labor for each of the 26 years of operation. In the attached table, Q represents the number of utility buildings constructed each year, L is the average number of carpenters employed during the year, and K is the average number of units of capital leased during the year.

After completing the managerial economics class, Mr. Bond thinks his process for constructing utility buildings can be described by a Cobb-Douglas production function of the form Q = ALK. He's very interested in using the available data to estimate the parameters (i.e., A, , and ) of the Cobb-Douglas production function. He wonders if his "seat-of-the pants" intuition about the labor-capital mix has been on target, or would his business have been much more profitable over the years if he had used more sophisticated analytical tools to determine the optimal labor-capital mix. From experience, he thinks building utility buildings is an increasing returns to scale operation, but he's anxious to discover what the Cobb-Douglas production function will reveal. He knows his firm has grown rapidly, but he's never really thought much about the actual compound rate of growth until he learned how to make such computations while studying time value of money. These are just a few of the questions that Mr. Bond has been thinking about. And now he has hired your consulting team to help him with the answers. Let's get to work!

Rather than a narrative report, Mr. Bond wants your consulting team to respond directly to each of the following questions. Please answer in complete, grammatically correct, sentences. Attach supporting regression analyses, spreadsheets, computations, etc., as appendices to your responses to the questions.

1. What has been the firm's average annual compound growth rate?

Mr. Bond thinks that the firm will only grow at an annual rate of 5% over the next five years. If he is correct, then how many utility buildings (to the nearest whole number) will the firm build during each the next five years (i.e., 2021 through 2025)?

Answer: firms' average annual compound rate?

CAGR = [(Ending Value / Beginning Value) ^ (1 / Number of Years)] - 1

CAGR = [(823 / 25) ^ (1 / 26)] - 1

CAGR 0.2933 or 29.33%

Answer: Mr. Bond thinks firm will grow at annual rate of 5%(not compound rate) if he is correct, how many utility buildings to nearest whole number will firm build during each next 5 years 2021 thru 2025

In 2021, the number of buildings constructed will be approximately 823 * 1.05 864.

In 2022, it will be 864 * 1.05 907

In 2023, it will be 907 * 1.05 952

In 2024, it will be 952 * 1.05 1,000

In 2025, it will be 1,000 * 1.05 1,050

2. Based on the records in the attachment, what were the firm's annual pre-tax profits from 1995 through 2020? How much pre-tax income has the firm earned during its life?

Answer: pre-tac 1995 thru 2020

Pre-tax profit = (Q * Price per unit - Total labor cost - Total capital cost)

Table with the calculated pre-tax profits for each year:

Year	Pretax
1995	-36250
1996	-36250
1997	-67500
1998	-45000
1999	23750
2000	23750
2001	-23750
2002	-35000
2003	-7500
2004	1250
2005	56250
2006	92500
2007	75000
2008	25000
2009	38750
2010	61250
2011	33750
2012	61250
2013	45000
2014	180000
2015	168750
2016	153750
2017	101250
2018	177500
2019	136250
2020	263250

Year Pre-tax Profit Calculations in excel...

1995 (25 * 2,750) - (4 * 25,000) - (1 * 5,000) = -36250

1996 (25 * 2,750) - (4 * 25,000) - (1 * 5,000) = -36250

Thru in excel, results shown above

2020 (823 * 2,750) - (75 * 25,000) - (25 * 7,500) = 263250

Answer: Pretax income during Firm's life - sum up the pretax/year

Total Pretax = $1,467,000

3. Based on the data, what is the estimated Cobb-Douglas production function for Bond, Inc.? Is this an increasing returns to scale industry as Mr. Bond suspected? Why or why not?

Answer:

Given the coefficients from the regression data:

- Intercept: 1.040242546

- Coefficient for log(L): 0.754868383

- Coefficient for log(K): 0.347140371

A is the intercept term.

is the coefficient for L (Labor).

is the coefficient for K (Capital).

Q = ALK

The estimated Cobb-Douglas production function for Bond, Inc. is:

Q = 1.04* L^0.75* K^0.35

(Q=A *L^0.754868383*K^0.347140371)

This implies that the production output (Q) is related to labor (L) and capital (K) in a multiplicative manner, with the respective exponents indicating the elasticity of output with respect to each input.

Increasing Returns to Scale:

To determine if this industry exhibits increasing returns to scale, check the sum of the exponents:

Sum of Exponents = 0.754868383 + 0.347140371 = 1.102008754

If the sum is greater than 1, it indicates increasing returns to scale. In this case:

Answer: Since the sum is greater than 1, this suggests that Bond, Inc. operates in an industry with increasing returns to scale. In such industries, an increase in both labor and capital leads to a more than proportional increase in output.

Therefore, Mr. Bond's suspicion is correct; it is an increasing returns to scale industry.

4. Using the estimated Cobb-Douglas production function, and given the costs of labor and capital, what is the expansion path (i.e., what is the optimal mix of labor and capital)? Casual observation suggests that over the years Mr. Bond has employed on average about 3 carpenters for every unit of capital (i.e., one unit of capital supports about three carpenters). Does it appear that his "seat-of-the-pants" intuition about the mix of labor and capital was correct? Why or why not?

Answer:

The estimated Cobb-Douglas production function is given as:

\[ Q = 1.04 \times L^{0.75} \times K^{0.35} \]

Q = 1.04* L^0.75* K^0.35

5. Using the estimated production function, determine how much labor and capital Bond, Inc., should have used each year in building the utility buildings. Assuming the optimal labor-capital mix had been used, determine annual pre-tax profits from 1995 to 2020. How much pre-tax income would the firm have earned during its life if it had used the optimal labor-capital mix? How does this lifetime pre-tax income compare to the lifetime pre-tax income determined in #2.

6. Next, let's examine the future. Recall that the cost of carpenters is going to increase from $25,000 to $30,000 per year beginning in January of 2021 and cost of capital is expected to increase from $5,000 to $7,500 per unit. How should Mr. Bond respond to this change in the relative cost of inputs (i.e., what is the new expansion path)?

7. Determine the optimal amount of labor and capital to use in building the utility buildings projected for 2021 through 2025.

8. Because new construction has slowed, a home builder in the city has now begun to build utility buildings similar to those that Mr. Bond builds. To be competitive, Mr. Bond must reduce his price from $2,750 to $2,400 during 2021-2025. What are Mr. Bond's projected annual pre-tax profits (or losses) over the next five years if he continues to operate his firm?

9. Recall, that Mr. Bond could shut down operations (there are no assets to dispose of) and go to work for Bobcats, Inc., at a salary, beginning in January of 2021, of $180,000 with 5% annual increases. Mr. Bond must make a decision in December of 2020. Does he continue to operate Bond, Inc., for the next five years, or does he close down the business and go to work for Bobcats, Inc.? Using a 10% discount rate, he will make this decision solely on the basis of the present value of the pre-tax earnings from these two employment opportunities. What is the present value of each of these income streams as of the end of December, 2020? [For purposes of simplification, you may assume that all cash flows (earnings and salary) occur at the end of the year.]

10. Conclude by writing a brief memo (no more than a page) summarizing your findings. Remember, such a brief summary gets to the point immediately. The opening sentence should make clear to Mr. Bond what he should do. The remaining sentences can summarize important particulars that led to your group's recommendation.

Bond, Inc.
Year	Q	L	K
1995	25	4	1
1996	25	4	1
1997	50	8	1
1998	60	8	2
1999	85	8	2
2000	85	8	2
2001	115	13	3
2002	120	14	3
2003	150	16	4
2004	155	16	5
2005	175	16	5
2006	190	16	6
2007	220	20	6
2008	220	22	6
2009	245	24	7
2010	255	24	8
2011	285	28	10
2012	315	30	11
2013	320	31	12
2014	380	32	13
2015	385	33	13
2016	465	42	15
2017	475	45	16
2018	610	56	20
2019	635	60	22
2020	823	75	25

The natural log of data for regression used is:

Year	Q	NL 0f Q	L	LN of L	K	LN of K
1995	25	3.218876	4	1.3862944	1	0
1996	25	3.218876	4	1.3862944	1	0
1997	50	3.912023	8	2.0794415	1	0
1998	60	4.094345	8	2.0794415	2	0.6931472
1999	85	4.442651	8	2.0794415	2	0.6931472
2000	85	4.442651	8	2.0794415	2	0.6931472
2001	115	4.744932	13	2.5649494	3	1.0986123
2002	120	4.787492	14	2.6390573	3	1.0986123
2003	150	5.010635	16	2.7725887	4	1.3862944
2004	155	5.043425	16	2.7725887	5	1.6094379
2005	175	5.164786	16	2.7725887	5	1.6094379
2006	190	5.247024	16	2.7725887	6	1.7917595
2007	220	5.393628	20	2.9957323	6	1.7917595
2008	220	5.393628	22	3.0910425	6	1.7917595
2009	245	5.501258	24	3.1780538	7	1.9459101
2010	255	5.541264	24	3.1780538	8	2.0794415
2011	285	5.652489	28	3.3322045	10	2.3025851
2012	315	5.752573	30	3.4011974	11	2.3978953
2013	320	5.768321	31	3.4339872	12	2.4849066
2014	380	5.940171	32	3.4657359	13	2.5649494
2015	385	5.953243	33	3.4965076	13	2.5649494
2016	465	6.142037	42	3.7376696	15	2.7080502
2017	475	6.163315	45	3.8066625	16	2.7725887
2018	610	6.413459	56	4.0253517	20	2.9957323
2019	635	6.453625	60	4.0943446	22	3.0910425
2020	823	6.712956	75	4.3174881	25	3.2188758

	df	SS	MS	F	Significance F
Regression	2	21.3500573	10.6750286	775.959459	7.7875E-22
Residual	23	0.31641558	0.0137572
Total	25	21.6664729
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	2.39524698	0.25075481	9.55214772	1.8018E-09	1.87652113	2.91397282
LN of L	0.75486838	0.15505687	4.86833233	6.4674E-05	0.4341088	1.07562796
LN of K	0.34714037	0.1242049	2.79490088	0.01028985	0.09020297	0.60407777
RESIDUAL OUTPUT
Observation	Predicted NL 0f Q	Residuals	Standard Residuals
1	3.44171676	-0.2228409	-1.9807791
2	3.44171676	-0.2228409	-1.9807791
3	3.96495165	-0.0529286	-0.4704699
4	4.20557102	-0.1112265	-0.9886651
5	4.20557102	0.23708024	2.10734881
6	4.20557102	0.23708024	2.10734881
7	4.71281883	0.0321133	0.28544734
8	4.76876059	0.01873115	0.16649664
9	4.96942528	0.04121001	0.36630582
10	5.04688742	-0.0034623	-0.0307756
11	5.04688742	0.11789856	1.04797172
12	5.11017859	0.13684548	1.21638636
13	5.2786226	0.11500494	1.02225111
14	5.35056924	0.0430583	0.38273483
15	5.4697633	0.03149491	0.27995059
16	5.51611744	0.0251461	0.22351762
17	5.70994305	-0.0574539	-0.5106936
18	5.7951096	-0.042537	-0.3781008
19	5.85006676	-0.0817458	-0.7266183
20	5.90181891	0.03835235	0.34090472
21	5.92504746	0.02819588	0.25062631
22	6.15676915	-0.0147317	-0.1309469
23	6.23125361	-0.0679388	-0.6038916
24	6.47379731	-0.0603384	-0.536333
25	6.55896386	-0.1053389	-0.9363316
26	6.771784	-0.0588278	-0.522906