Paying Not To Go to the Gym

Brooke Whitcomb 

Motivation is difficult, and it’s hard to stick to a plan. How should a person optimize a plan to make better decisions? In their paper, “Paying Not to Go to the Gym”, DellaVigna and Malmender find that gym members who pay $70 per month only visit the gym on average 4.3 times per month. This is startling and wasteful considering the pay-per-visit cost is 10 for non-members. The authors argue that consumers overestimate their future ability to stick to a plan. 

I seek to model why going to the gym is so difficult by modeling the utility function of going to the gym as a differential equation. Then I relate different utilities using a Lokta-Volterra Competition model. I present insights into the values of time used for competing tasks. 

A person signs up for a gym membership and pays a flat fee of $70 per month. Every time she goes to the gym, her utility cost decreases by 5%. It becomes less and less costly for her to go the more she goes. Likewise, she gets a present utility of 10 every time she goes to the gym. Under this structure how many times would she have to go to the gym to pay off her membership?

The person pays off the payment after just 9 trips to the gym a month– going to the gym about twice a week. The average involvement for gym members is 4.3 which under this model would yield a negative utility of So, the average gym member wastes almost 40 per month with a gym membership more than half of what she pays. The proof is below. Anyone who expects they won’t go to the gym at least twice a week should switch to a pay-per-visit scheme. So why is the decision to go to the gym so hard? 

I extend the previous model so that the rate of utility for going to the gym is also dependant on the utility of not going to the gym. These equations are modeled similarly to a competing species model as if they were competing for a person’s time. 

The return rate of not going to the gym is just slightly higher than going to the gym because it progressively gets easier to sit on the couch than go to the gym. I will show later that the growth rate of not going to the gym is non-trivial in the trade-off between going to the gym and not. Both actions are limited to occur 30 times i.e. a month. The effect of not going to the gym has twice the effect of going to the gym than the effect of going to the gym on not going to the gym. There is no present value for not going to the gym.

The two systems stabilize when both these derivatives are zero. This is a system of first-order ordinary differential equations. The system has two positive equilibrium points: (27.8452, 2.15477) and (2.15477, 27.8452). Figure 1 shows the implicit graph of these two equations. Notice that going to the gym has a nonlinear slope, this is because of the accumulative utility, as noted by the + 0.10 in the equation.

Figure 1: This is an Implicit Plot created using Wolfram

I determined the behavior of the equilibrium points using a Jacobian. The equilibrium point where the person goes to the gym about 27-28 and doesn’t go 2-3 times is a saddle point. The equilibrium point where the person goes to the gym about 27-28 and doesn’t go 2-3 times is stable and it’s a sink. Figure 2 shows the behavior around these two points. The proof is below.

Under these conditions, a person will go to the gym 2-3 times and not go to the gym 28-30 times. Thus, she would have an overall negative utility after a month and not break-even on the $70 flat fee.

Figure 2 a phase diagram of the two equilibrium points notice that the vectors converge at point 2.1, 27.8 but not 27.8 2.1.

The competitive factor reflects the effect of not going to the gym on going to the gym. Changing the model so that the competitive effects are reciprocals of each other without changing the growth rate. The system will converge when the competitive rates are 1.125 as figured below, such that the person will go to the gym 15 times and not go 15 times. This competitive equilibrium is regardless of the growth rate of not going to the gym. Any Rn between 0 and 1 and it would not affect the equilibrium competition factor. She will pay off the debt if the competitive rate is less than 1.195. The person will go to the gym every day if the competitive rate is less than 1.

Finally, for the average person who goes to the gym 4.3 times per month, her competitive factor is estimated to be 1.24. The competition factor measures interspecific competition relative to intraspecific competition. So, one trip to the gym is equivalent to 124% of a trip not going to the gym for that person. In other words, the time spent going to the gym is worth 80.6% less than not going to the gym. Accordingly, a person will go to the gym every day if their time spent going to the gym is worth more than not going to the gym. Under the conditions of this model and the pretense set up by DellaVigna and Malmendier, anyone who values going to the gym more than 83.7% of not going should get a monthly membership. Anyone below this threshold should pay by a visit. Consumers who are overconfident in their true trade-offs will lose utility.

REFERENCES

Competitive Lotka–Volterra Equations. (2020, April 19). In Wikipedia. Retrieved from https://en.wikipedia.org/w/index.phptitle=Competitive_Lotka%E2%80%93Volterra_equations&acti

DellaVigna, S., & Malmendier, U. (2006). Paying Not to Go to the Gym. American Economic Review, 96(3), 694-719.

Schreiber, S. J., Smith, K. J., & Getz, W. M. (2014). Calculus For the Life Sciences. John Wiley & Sons.

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