Section #: 01

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Many aspects of the environment are public goods. Public goods are nonrivalrous and nonexcludable. Nonrivalrous means that you deriving benefit from the good does not diminish the amount of the good available for others. Nonexcludable means that there is no way to prevent people from deriving benefits from the good. Examples of environmental public goods include obvious things like the ozone layer, but can also be more subtle like the absence of litter in a park.

In experiment 4 we saw that the health of a society is also public good, where individuals can take costly actions to contribute. Experiment 4 put you in a situation where you faced a trade-off between a private good (having fun) and a public good (the health of your society). Your task was to choose a costly effort level that would impact the transmission of a virus. Effort involves things like wearing a mask, washing your hands and staying socially isolated. Obviously these things are un-fun: we assume that effort reduces fun 1 for 1. Of course the flip-side is that these actions are what keeps the virus under control. We assume that public health is 1.5 times the average effort in your group, and all members of your group benefited equally from public health. Thus, individual 1’s payoff was:

\(\pi_1=10-e_1+1.5\times\frac{e_1+...+e_n}{n}\)

where 10 is the maximum effort you can exert: you can think of this as your endowment.

The treatments determined how subjects were rematched between rounds. In the not treatment subjects played all 20 rounds with the same group. In the randomly treatment subjects were randomly rematched for each round. Finally in the assortatively treatment subjects played each round with players who had made similar contributions in the previous round. Higher payoffs can be sustained in a public goods game if behaviour can be coordinated (each member of the group contributing a similar amount.) These treatments were designed to influence how easy it would be to coordinate behaviour.

1 (10 marks)

If I increased your endowment from 10 to 11, but you kept your effort level constant, by how much would your payoff increase? If instead you responded by increasing your effort by one unit, by how much would your payoff increase (assume 3 people in your group)? If a player is maximizing their own payoff (as given by the equation above) how much effort should they exert? Does your answer depend on group size? Does your answer depend on round number? Does your answer depend on the effort level of others?

  1. Effort remains unchanged, endowment rises by 1, payoff rises by 1
  2. n=3, if effort1 rises by 1, payoff changes -0.5
  3. Maximize Π: -1+0.5 = -0.5<0 e1 rises and Π1 falls. e1 = 0, maxΠ
  4. The result of optimal effort has nothing to do with group size n, because marginal payoff of fun=1 is always greater than marginal payoff of public goods=1.5/n
  5. The optimal effort result has nothing to do with the round number
  6. The result of optimal effort has nothing to do with the effort level of others

2 (10 marks)

Suppose that each group had a “government”, which could mandate effort levels: (all members have to exert the same effort) Would this government intervention increase player payoffs relative to the equilibrium prediction? What would be the optimal mandated effort level?

If the government enforces public interest, personal benefits will increase. There is no government law enforcement, e1=0, Π1=10. If there is a government forced e1>0. Assuming e1=2 Π1=11>10, better Charity 1.5e* Optimal mandatory effort = 10. The social return of marginal pleasure =1 <the marginal social return of public goods = 1.5, then the socially efficient effort level = endowment e*=10

3 (5 marks)

Use dataframe mydf and functions ggplot(), geom_boxplot(), and geom_jitter() to visualize how effort choice differs by treatment rematching. Use labs() to give the plot an informative title.

4 (10 marks)

Use dataframe mydf and functions ggplot(), geom_boxplot(), and geom_jitter() to visualize how effort choice varies across round. Use function facet_wrap(~rematching) to create a different facet for each treatment. Use labs() to give the plot an informative title.

5 (10 marks)

What differences do you notice in the first round choices across the three treatments?

The not group is the most concentrated group

6 (10 marks)

In your .R file, create a new dataframe called within_group_variability by taking dataframe mydf THEN group_by variables whichgroup, round, rematching THEN summarise by calculating variability_contributions=sd(choice). Include a copy of your code in the chunk below (note eval=FALSE) 

within_group_variability <- mydf%>%
  group_by(whichgroup,round,rematching)%>%
  summarise(variability_contributions=sd(choice))

7 (10 marks)

Use dataframe within_group_variability and functions ggplot(), geom_boxplot(), and geom_jitter() to visualize how variability_contributions differs by treatment rematching. Use labs() to give the plot an informative title. Make use of function fix_gg_labs() to fix the labels of third_plot.

8 (10 marks)

Randomization of which treatment each subject receives is meant to ensure that subjects are comparable. Above we saw some evidence that in the first round, prior to being re-matched the subjects in the randomly treatment made much larger choices than the other two treatments. This suggests that the subjects were not in fact comparable prior to receiving their treatments. One way we can address this is look at choices relative to the choice made in the first round. To this end, create a new dataframe called relative_to_r1 by taking dataframe mydf THEN group_by() variables oneid,rematching THEN nest() THEN mutate(new_df=map(data,calc_relative)) THEN select(-data) THEN unnest(cols = new_df) THEN filter(round!=1). Include a copy of your code in the chunk below (note eval=FALSE)

relative_to_r1<-mydf%>%
  group_by(oneid,rematching)%>%
  nest()%>%
  mutate(new_df=map(data,calc_relative))%>%
  select(-data)%>%
  unnest(cols=new_df)%>%
  filter(round!=1)

9 (5 marks)

Explain what function calc_relative() does (it can be found in the .R file)

calc_relative is used to calculate choice relative to first round = choice- fisrt round choice

10 (10 marks)

Use dataframe relative_to_r1 and functions ggplot(), geom_boxplot(), and geom_jitter() to visualize how choice_relative_to_first_round differs by treatment rematching. Use labs() to give the plot an informative title. Make use of function fix_gg_labs() to fix the labels of fourth_plot

11 (0 marks)

Use dataframe mydf and functions ggplot(), stat_density_ridges(), scale_fill_viridis_c() and facet_wrap() to create a density ridge plot for choice across rematching and round. Use labs() to give the plot an informative title. Use function fix_gg_labs() to fix the labels for this fifth_plot.

12 (10 marks)

Use dataframe relative_to_r1 and functions ggplot(), stat_density_ridges(), scale_fill_viridis_c() and facet_wrap() to create a density ridge plot for choice_relative_to_first_round across rematching and round. Add a geom_vline() with arguments xintercept = 0, col="white",alpha=.5,lwd=2 as a guide. Use labs() to give the plot an informative title. Use function fix_gg_labs() to fix the labels for this sixth_plot.