*This post is concerned only with the career average real salary of specialists such as anesthesiologists, dermatologists, and ophthalmologists. A later post may consider traditional general practitioners, whose career average real compensation levels are likely to be even lower than for specialists.*

An article recently appeared online claiming that, given the enormous amount of time and expense required to enter the profession, medical doctors, contrary to popular perception, really only receive an overall career average of $33.03 per hour of study and work.^{1}“The Deceptive Salary of Doctors” Best Medical Degrees, 2019, https://www.bestmedicaldegrees.com/salary-of-doctors/ However, in making his calculations, the anonymous author of this article fails to recognize the time value of money.^{2}This anonymous author also fails to recognize the existence of nonmonetary compensation for work. Money is not the only reward for work. There is also psychic income such as, for example, enjoyment of the work itself or professional prestige. Many who enter the medical profession do so not only for the high pay but because they have a genuine desire to cure disease and to lessen the suffering of their fellow human beings. Successfully helping people in this regard gives them great satisfaction. Prestige is certainly also a compensation for becoming a physician. Although most of the prestige of being a medical doctor is derived from its perceived high monetary rewards, some of that prestige also comes from the fact that much effort and intelligence is required to surmount the high barriers to entry into this profession. Even if there is no inflation, a dollar to be received in the future is worth less than a dollar now. This is because we can receive a dollar in the future if we invest less than a dollar now at interest. This time value of money will be accounted for in the calculations below.

A doctor’s career is usually composed of four phases. These are undergraduate college education, medical school education, residency training (or medical apprenticeship), and licensed practice. I will attempt here to calculate the overall career average real hourly rate of compensation, using continuous compounding of interest (with taxes), of a specialist physician over all four phases of his career. The hours spent in undergraduate and medical school study are properly considered as working hours since this study is required in order to earn income later as a specialist physician.^{3}It is assumed here that each hour of study and work represents the same level of sacrifice. However, if it were possible to buy and sell the hours of one’s life in a marketplace, it is likely that the hours of youth would attract a higher price than the hours of middle age. And it should be kept in mind that one of the most important costs of having a medical career is that the long hours needed to prepare and train for it are spent during the best years of a young medical student’s life. Likewise, the money costs of required undergraduate and medical school study are properly considered expenses of a career as a specialist physician.

Be aware that in order to obtain the following results many estimates and simplifying assumptions needed to be made. The reader should regard any conclusions made in this article as both approximate and tentative. And, of course, each individual situation is different and may require a different model or methodology than those used here.^{4}I will not attempt here to place a monetary value on any imputed, or psychic, income associated with the work of specialist physicians. Since I do regard psychic income as an important component of total compensation, I may address it in one of my future blog posts. Indeed, these psychic rewards may, depending on the particular situation, be more important to the physician worker than money wages and other tangible benefits. But, here I will confine myself to roughly estimating, in terms of real purchasing power, the overall average hourly compensation to all phases of a specialist doctor’s career.

By the way, in case you want to get into medicine just for the money, there are more efficient ways of making far more money than a doctor! The income of physicians, **when they act as physicians**, is limited. Firstly, there are only 24 hours in a day. If we subtract 8 hours for sleep, 3 hours for meals, 1 hour for transportation, and 2 hours for personal hygiene and miscellaneous household chores, we are left with, at most, 10 hours for paid work per day. Secondly, as professionals, physicians must personally perform their own work. They cannot delegate to others the performance of their professional services. And finally, the amount they can charge per hour of their work is limited by the supply and demand for medical services. Therefore, the total yearly income of a doctor, **when working as a doctor**, though it may be substantial, is always limited. Now, if a doctor hires other doctors (thinking he can multiply his income by doing so) and takes part of their earnings in return for the work of management, then the doctor is no longer working as a doctor. He is instead **acting as a businessman**. Or, if the doctor receives an income from his investments in securities or from the interest on his loan to a friend’s small business then, again, he is not acting in these cases as a physician but rather **as an investor or as a lender**. Businessmen (together with investors and lenders), unlike doctors and other professionals (e.g., lawyers, engineers, and architects), have unlimited income potential because they can own, hire, or manage unlimited amounts of income producing labor and capital. There have never been any billionaire doctors, **when they only work as doctors**. But, there are many billionaire businessmen.^{5}And, with recent advances in telecommunications technology, businessmen can work literally anywhere. But, physicians almost always are required to do their work in a hospital or medical office. Plus, there are no educational, training, or licensing requirements to become a businessman. All that is required is intelligence, creativity, ambition, and an appetite for work.

#### Calculations and results

Here, I will discount to present value all income (while working as a medical apprentice and as a licensed physician) and expenses (taxes on his income, plus the tuition costs of the education necessary to become a doctor) of a specialist physician over his career. Next, I will use this net present value to fund a **constant purchasing power continuous annuity** over the entire period of a specialist physician’s career (undergraduate study, medical school study, residency, and licensed practice).^{6}A worker should not be so concerned about how many units of money he gets per hour of his labor (because this money may lose its value through inflation or debasement), but rather with how much real purchasing power (i.e., the ability to buy gallons of milk, or loaves of bread) he will ultimately receive per hour of work. This will allow me to arrive at his overall career average real purchasing power compensation per hour of work and required study. Please see the appendix at the end of this article for definitions of the symbols used in the following equations.

\begin{align}

V &= \int_{w=0}^{W_{c}}c_{0}e^{g_{c}\phi_{c}w}e^{-\delta(1-\tau)\phi_{c}w}\phi_{c}dw\notag\\&\phantom{A=}\left.+\>e^{-\delta(1-\tau)\phi_{c}W_{c}}\int_{w=0}^{W_{m}}m_{0}e^{g_{m}\phi_{c}W_{c}}e^{g_{m}\phi_{m}w}\phi_{m}e^{-\delta(1-\tau)\phi_{m}w}dw\right.\notag\\&\phantom{A=}\left.+\>e^{-\delta(1-\tau)(\phi_{c}W_{c}+\phi_{m}W_{m})}\int_{w=0}^{W_{r}}r_{0}e^{g_{r}(\phi_{c}W_{c}+\phi_{m}W_{m})}e^{g_{r}\phi_{r}w}(1-\tau)\phi_{r}e^{-\delta(1-\tau)\phi_{r}w}dw\right.\notag\\&\phantom{A=}+\>e^{-\delta(1-\tau)(\phi_{c}W_{c}+\phi_{m}W_{m}+\phi_{r}W_{r})}\int_{w=0}^{W_{p}}p_{0}e^{g_{p}(\phi_{c}W_{c}+\phi_{m}W_{m}+\phi_{r}W_{r})}e^{g_{e}\phi_{p}w}(1-\tau)\phi_{p}e^{-\delta(1-\tau)\phi_{p}w}dw\label{eq:doctor 1} \\ &=\int_{w=0}^{W_{c}}c_{0}\phi_{c}e^{\left[g_{c}-\delta(1-\tau)\right]\phi_{c}w}dw\notag\\&\phantom{A=}\left.+\>e^{\left[g_{m}-\delta(1-\tau)\right]\phi_{c}W_{c}}\int_{w=0}^{W_{m}}m_{0}\phi_{m}e^{\left[g_{m}-\delta(1-\tau)\right]\phi_{m}w}dw\right.\notag\\&\phantom{A=}\left.+\>e^{\left[g_{r}-\delta(1-\tau)\right](\phi_{c}W_{c}+\phi_{m}W_{m})}\int_{w=0}^{W_{r}}r_{0}\phi_{r}\left(1-\tau\right)e^{\left[g_{r}-\delta(1-\tau)\right]\phi_{r}w}dw\right.\notag\\&\phantom{A=}+\>e^{\left[g_{p}-\delta(1-\tau)\right](\phi_{c}W_{c}+\phi_{m}W_{m}+\phi_{r}W_{r})}\int_{w=0}^{W_{p}}p_{0}\phi_{p}\left(1-\tau\right)e^{\left[g_{e}-\delta(1-\tau)\right]\phi_{p}w}dw\label{eq:doctor 2} \\ &=c_{0}\phi_{c}\frac{e^{\left[g_{c}-\delta(1-\tau)\right]\phi_{c}P_{c}K_{c}Y_{c}}-1}{\left[g_{c}-\delta(1-\tau)\right]\phi_{c}}\notag\\&\phantom{A=}\left.+\>e^{\left[g_{m}-\delta(1-\tau)\right]\phi_{c}P_{c}K_{c}Y_{c}}m_{0}\phi_{m}\frac{e^{\left[g_{m}-\delta(1-\tau)\right]\phi_{m}P_{m}K_{m}Y_{m}}-1}{\left[g_{m}-\delta(1-\tau)\right]\phi_{m}}\right.\notag\\&\phantom{A=}\left.+\>e^{\left[g_{r}-\delta(1-\tau)\right](\phi_{c}P_{c}K_{c}Y_{c}+\phi_{m}P_{m}K_{m}Y_{m})}r_{0}\phi_{r}\left(1-\tau\right)\frac{e^{\left[g_{r}-\delta(1-\tau)\right]\phi_{r}P_{r}K_{r}Y_{r}}-1}{\left[g_{r}-\delta(1-\tau)\right]\phi_{r}}\right.\notag\\&\phantom{A=}+\>e^{\left[g_{p}-\delta(1-\tau)\right](\phi_{c}P_{c}K_{c}Y_{c}+\phi_{m}P_{m}K_{m}Y_{m}+\phi_{r}P_{r}K_{r}Y_{r})}p_{0}\phi_{p}\left(1-\tau\right)\frac{e^{\left[g_{e}-\delta(1-\tau)\right]\phi_{p}P_{p}K_{p}Y_{p}}-1}{\left[g_{e}-\delta(1-\tau)\right]\phi_{p}}\label{eq:doctor 3} \\ &=\>\left(-\$26593\right)\left(0.000667\right)\frac{e^{\left[\left(0.04955\right)-\left(0.04879\right)(1-0.30)\right]\left(0.000667\right)\left(50\right)\left(30\right)\left(4\right)}-1}{\left[\left(0.04955\right)-\left(0.04879\right)(1-0.30)\right]\left(0.000667\right)}\notag\\&\phantom{A=}\left.+\>e^{\left[(0.06785)-\left(0.04879\right)(1-0.30)\right]\left(0.000667\right)\left(50\right)\left(30\right)\left(4\right)}\right.\notag\\&\qquad\qquad\left.\cdot\>\left(-\$50000\right)\left(0.000386\right)\frac{e^{\left[\left(0.06785\right)-\left(0.04879\right)(1-0.30)\right]\left(0.000386\right)\left(70\right)\left(37\right)\left(4\right)}-1}{\left[\left(0.06785\right)-\left(0.04879\right)(1-0.30)\right]\left(0.000386\right)}\right.\notag\\&\phantom{A=}\left.+\>e^{\left[(0.02235)-\left(0.04879\right)(1-0.30)\right]\left[\left(0.000667\right)\left(50\right)\left(30\right)\left(4\right)+\left(0.000386\right)\left(70\right)\left(37\right)\left(4\right)\right]}\right.\notag\\&\qquad\qquad\left.\cdot\>\left(\$61200\right)\left(0.00025\right)\left(1-0.30\right)\frac{e^{\left[\left(0.02235\right)-\left(0.04879\right)(1-0.30)\right]\left(0.00025\right)\left(80\right)\left(50\right)\left(5\right)}-1}{\left[\left(0.02235\right)-\left(0.04879\right)(1-0.30)\right]\left(0.00025\right)}\right.\notag\\&\phantom{A=}\left.+\>e^{\left[(0.04402)-\left(0.04879\right)(1-0.30)\right]\left[\left(0.000667\right)\left(50\right)\left(30\right)\left(4\right)+\left(0.000386\right)\left(70\right)\left(37\right)\left(4\right)+\left(0.00025\right)\left(80\right)\left(50\right)\left(5\right)\right]}\right.\notag\\&\qquad\qquad\cdot\>\left(\$329000\right)\left(0.000333\right)\left(1-0.30\right)\frac{e^{\left[\left(0.04402\right)-\left(0.04879\right)(1-0.30)\right]\left(0.000333\right)\left(60\right)\left(50\right)\left(34\right)}-1}{\left[\left(0.04402\right)-\left(0.04879\right)(1-0.30)\right]\left(0.000333\right)}\label{eq:doctor 4} \\ &= -86671.61-244947.88+189258.25+10564664.74\label{eq:doctor 5} \\ &=10422303.50\label{eq:doc avg} \\ &=\int_{0}^{W_{c}+W_{m}+W_{r}+W_{p}}qe^{\pi\phi_{q}w}(1-\tau)\phi_{q}e^{-\delta(1-\tau)\phi_{q}w}dw \label{eq:doctor 7} \\ &=q\phi_{q}(1-\tau)\frac{e^{\left[\pi-\delta(1-\tau)\right]\phi_{q}(W_{c}+W_{m}+W_{r}+W_{p})}-1}{\left[\pi-\delta(1-\tau)\right]\phi_{q}} \label{eq:doctor 8} \\ &=q\phi_{q}(1-\tau)\frac{e^{\left[\pi-\delta(1-\tau)\right]\phi_{q}(P_{c}K_{c}Y_{c}+P_{m}K_{m}Y_{m}+P_{r}K_{r}Y_{r}+P_{p}K_{p}Y_{p})}-1}{\left[\pi-\delta(1-\tau)\right]\phi_{q}} \label{eq:doctor 9} \\ &=q\phi_{q}(1-\tau)\frac{e^{\left[\pi-\delta(1-\tau)\right]\frac{Y_{c}+Y_{m}+Y_{r}+Y_{p}}{P_{c}K_{c}Y_{c}+P_{m}K_{m}Y_{m}+P_{r}K_{r}Y_{r}+P_{p}K_{p}Y_{p} }(P_{c}K_{c}Y_{c}+P_{m}K_{m}Y_{m}+P_{r}K_{r}Y_{r}+P_{p}K_{p}Y_{p})}-1}{\left[\pi-\delta(1-\tau)\right]\phi_{q}} \label{eq:doctor 10} \\ &=q\phi_{q}(1-\tau)\frac{e^{\left[\pi-\delta(1-\tau)\right](Y_{c}+Y_{m}+Y_{r}+Y_{p})}-1}{\left[\pi-\delta(1-\tau)\right]\phi_{q}} \label{eq:doctor 11} \\ &=q\phi_{q}(1-0.30)\frac{e^{\left[(0.0322)-(0.04879)(1-0.30)\right](4+4+5+34)}-1}{\left[(0.0322)-(0.04879)(1-0.30)\right](0.00033969)} \label{eq:doctor 12} \\ &=92540.84q\phi_{q} \label{eq:doctor 13}\end{align}

\begin{align}

\Rightarrow \quad q\phi_{q} &=\frac{10422303.50}{92540.84}\label{eq:doctor 14}\\ &=\mathbf{112.62}\label{eq:doctor 15}\end{align}

The result in line \eqref{eq:doctor 15} reveals that a specialist physician receives an average of **$112.62 in real purchasing power per hour of study and work over his entire course of education, training, and practice**. This is a good wage. But, while this amount seems appropriate given the level of effort and intelligence required to enter this occupation, it is rather disappointing given the public’s perception that doctors make a fortune. This surprisingly low result is due to the enormous amount of time and expense spent preparing to be a physician. The time and expense of this preparation is largely ignored by the public, who consider only what a doctor earns in his licensed practice. However, while this overall hourly wage is not quite as much as one might expect for a specialist physician, it is much greater than what is claimed in the article cited at the beginning of this blog post.

Now, what if the specialist physician retires before age 65? Below is a table giving \(q\phi_{q}\) (overall average hourly wage) for selected \(Y_{p}\) (the number of years of licensed practice). Note that the physician must perform licensed work for at least 0.55 years in order to break even (i.e., when \(q\phi_{q}=0\)).

\(Y_{p}\mbox{(years)}\) | \(q\phi_{q}\mbox{(\$/hour)}\) |
---|---|

0 | -5.28 |

0.55 | 0 |

5 | 32.87 |

10 | 56.24 |

15 | 72.77 |

20 | 85.64 |

25 | 96.36 |

30 | 105.75 |

34 | 112.62 |

And what if medical school tuition is free?^{7}“Cost of Attendance”, NYU Grossman School of Medicine, accessed 30 December 2019, https://med.nyu.edu/education/md-degree/md-affordability-financial-aid/cost-attendance/ In this case, \(m_{0}=0\), the career average only increases to **$115.27 per hour**.

#### APPENDIX: Symbol Definitions

\(V\) : present value of all expenses and income of a specialist physician over all phases (undergraduate education, medical school, residency training, and licensed practice) of his career.

\(W_{c}=P_{c}K_{c}Y_{c}=\left(50\right)\left(30\right)\left(4\right)=6000\) : total hours of study in the undergraduate education phase of a physician’s career.

\(W_{m}=P_{m}K_{m}Y_{m}=\left(70\right)\left(37\right)\left(4\right)=10360\) : total hours of study in the medical school phase of a physician’s career.

\(W_{r}=P_{r}K_{r}Y_{r}=\left(80\right)\left(50\right)\left(5\right)=20000\) : total hours of paid work in the residency (apprenticeship) phase of a physician’s career.

\(W_{p}=P_{p}K_{p}Y_{p}=\left(60\right)\left(50\right)\left(34\right)=102000\) : total hours of paid work in the licensed practice phase of a physician’s career.

\(i=0.05\) : assumed rate of interest (annually compounded) on investments having risk similar to the cash flows associated with all phases of a medical career.

\(\delta=\ln(1+i)=0.04879\) : assumed force of interest (continuous) on investments having risk similar to the cash flows associated with all phases of a medical career.

\(w\) : time (in hours) spent in required study or paid work. In the formulas above we only integrate over these study or work hours, not over calendar hours.

\(c_{0}=-\$26593\) : average annual cost in today’s dollars of undergraduate education. A growth factor will be applied to this initial amount to reflect the increasing nominal cost of college education.^{8}“Tuition costs of colleges and universities” *Digest of Education Statistics 2017* (NCES 2018-070), U.S. Department of Education, National Center for Education Statistics, https://nces.ed.gov/fastfacts/display.asp?id=76

\(m_{0}=-\$50000\) : average annual cost in today’s dollars of medical school. This cost varies widely by school, but here we will use a figure of $50,000. A growth factor will be applied to this initial amount to reflect the increasing nominal cost of medical school education.^{9}“What’s the real cost of medical school?” American Medical Student Association, 10 November 2018, https://www.amsa.org/2018/11/10/real-cost-of-medical-school/

\(r_{0}=\$61200\) : average annual salary in today’s dollars of residents (medical apprentices). A growth factor will be applied to this initial amount to reflect the increasing nominal salary paid to medical residents (trainees).^{10}“What residents are getting paid in 2019” *The DO*, American Osteopathic Association, Updated July 2019, https://thedo.osteopathic.org/2018/07/what-residents-are-getting-paid-in-2018/

\(p_{0}=\$329000\) : average annual starting salary in today’s dollars of new specialist physicians. A growth factor will be applied to this initial amount to reflect the increasing nominal salary paid to licensed specialist medical doctors.^{11}O’Connell, Brian. “How Much Do Doctors Make in 2018?” *TheStreet*, TheStreet, Inc., 13 November 2018, https://www.thestreet.com/personal-finance/how-much-do-doctors-make-14779617

\(q\) : equivalent annual income in today’s dollars over all four phases of a specialist physician’s career calculated by amortizing \(V\) over this entire period.

\(G_{c}=0.0508\) : average annual rate of growth of the cost of undergraduate education (assumed to be a constant rate).^{12}“Tuition costs of colleges and universities” *Digest of Education Statistics 2017* (NCES 2018-070), U.S. Department of Education, National Center for Education Statistics, https://nces.ed.gov/fastfacts/display.asp?id=76

\(g_{c}=ln(1+G_{c})=0.04955\) : continuously compounded average annual rate of growth of the cost of undergraduate education (assumed to be a constant rate).

\(G_{m}=0.0702\) : average annual rate of growth of the cost of medical school (assumed to be a constant rate).^{13}“Medical School Tuition Frequently Asked Questions” American Medical Student Association, accessed 16 November 2019, https://www.amsa.org/advocacy/action-committees/twp/tuition-faq/

\(g_{m}=ln(1+G_{m})=0.06785\) : continuously compounded average annual rate of growth of the cost of medical school (assumed to be a constant rate).

\(G_{r}=0.0226\) : average annual rate of growth of the salary of medical residents (assumed to be a constant rate).^{14}Levy, Sandra. “Medscape Residents Salary & Debt Report 2018” *Medscape*, WebMD LLC, 18 July 2018, https://www.medscape.com/slideshow/2018-residents-salary-debt-report-6010044

\(g_{r}=ln(1+G_{r})=0.02235\) : continuously compounded average annual rate of growth of the salary of medical residents (assumed to be a constant rate).

\(G_{p}=0.045\) : average annual rate of growth of the salary of licensed specialist medical doctors (assumed to be a constant rate).^{15}Linderfelt, Heather. “Physician salary 2019: Salaries rise again, but so does paperwork” Weatherby Healthcare, 5 June 2019, https://weatherbyhealthcare.com/blog/physician-salary-2019

\(g_{p}=ln(1+G_{p})=0.04402\) : continuously compounded average annual rate of growth of the salary of licensed specialist medical doctors (assumed to be a constant rate).

\(g_{e}\) : continuously compounded average annual rate of growth of the salary of licensed specialist medical doctors based on their years of experience (assumed equal to \(g_{p}\) due to lack of good data).

\(\phi_{c}=\frac{1}{P_{c}K_{c}}=\frac{1}{\left(50\right)\left(30\right)}=0.000667\) : scaling factor; calendar years per hour spent in undergraduate study.

\(\phi_{m}=\frac{1}{P_{m}K_{m}}=\frac{1}{\left(70\right)\left(37\right)}=0.000386\) : scaling factor; calendar years per hour spent in medical school study.

\(\phi_{r}=\frac{1}{P_{r}K_{r}}=\frac{1}{\left(80\right)\left(50\right)}=0.00025\) : scaling factor; calendar years per hour spent in paid medical training (residency).

\(\phi_{p}=\frac{1}{P_{p}K_{p}}=\frac{1}{\left(60\right)\left(50\right)}=0.000333\) : scaling factor; calendar years per hour spent in paid licensed medical practice.

\(\phi_{q}=\frac{Y_{c}+Y_{m}+Y_{r}+Y_{p}}{P_{c}K_{c}Y_{c}+P_{m}K_{m}Y_{m}+P_{r}K_{r}Y_{r}+P_{p}K_{p}Y_{p}}=\frac{4+4+5+34}{(50)(30)(4)+(70)(37)(4)+(80)(50)(5)+(60)(50)(34)}=\frac{47}{138360}=0.00033969\) : scaling factor; calendar years per hour spent at work and required study over all phases of a specialist physician’s career.

\(\tau=0.30\) : annual effective income tax rate (assumed to be a constant rate).

\(\pi=0.0322\) : the rate of inflation, or the average annual rate of increase of prices for all goods and services. Correspondingly, inflation is the rate of loss of the purchasing power of money (i.e., the rate of decrease of the amount of real goods and services that a unit of money can buy). Assumed here to be a constant rate.^{16}“Long Term U.S. Inflation”, Capital Professional Services LLC, 1 April 2014, https://inflationdata.com/Inflation/Inflation_Rate/Long_Term_Inflation.asp

\(P_{c}=50\) : hours of study per week in the undergraduate education phase of a physician’s career.

\(P_{m}=70\) : hours of study per week in the medical school phase of a physician’s career.

\(P_{r}=80\) : hours of paid training per week in the residency (apprenticeship) phase of a physician’s career.^{17}“Residency (medicine)” Wikipedia, accessed 16 November 2019, https://en.wikipedia.org/wiki/Residency_(medicine)

\(P_{p}=60\) : hours of paid work per week in the licensed practice phase of a physician’s career.^{18}Kane, Leslie. “Medscape Physician Compensation Report 2019” *Medscape*, WebMD LLC, 10 April 2019, https://www.medscape.com/slideshow/2019-compensation-overview-6011286

\(K_{c}=30\) : weeks per year spent in the undergraduate education phase of a physician’s career.

\(K_{m}=37\) : weeks per year spent in the medical school phase of a physician’s career.^{19}Institute of Medicine (US) Division of Health Sciences Policy. Medical Education and Societal Needs: A Planning Report for the Health Professions. Washington (DC): National Academies Press (US); 1983. Appendix F, TYPICAL PATTERN OF MEDICAL EDUCATION IN THE UNITED STATES. Available from: https://www.ncbi.nlm.nih.gov/books/NBK217675/

\(K_{r}=50\) : weeks per year spent in the residency (apprenticeship) phase of a physician’s career.

\(K_{p}=50\) : weeks per year spent in the licensed practice phase of a physician’s career.

\(Y_{c}=4\) : duration in years of the undergraduate education phase of a physician’s career.

\(Y_{m}=4\) : duration in years of the medical school phase of a physician’s career.

\(Y_{r}=5\) : average duration in years of the residency (apprenticeship) phase of a physician’s career.^{20}“Medical education in the United States” Wikipedia, accessed 16 Nov 2019, https://en.wikipedia.org/wiki/Medical_education_in_the_United_States

\(Y_{p}=34\) : average duration in years of the licensed practice phase of a physician’s career.^{21}Hedden, Lindsay. “Patterns of physician retirement and pre-retirement activity: a population- based cohort study” Canadian Medical Association Journal, 11 December 2017, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5718891/pdf/189e1517.pdf