Problem set

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In this problem set you will summarize the paper “Imperfect Public Monitoring with Costly Punishment: An Experimental Study” (Ambrus and Greiner, AER 2012) and recreate some of its findings.

1 Big Picture

[Q1] What is the main question asked in this paper?

[your written answer here]

[Q2] Summarize the experiment design.

[your written answer here]

[Q3] Summarize the main results of the experiment.

[your written answer here]

[Q4] Why are these results valuable? What have we learned? Motivate your discussion with a real-world example. In particular discuss the tradeoffs to transparency in groups and how these tradeoffs might be navigated in a firm, or more broadly, a society.

[your written answer here]

[Q5] If punishment is ineffective under imperfect monitoring, what else can you lean on to ensure people cooperate (at least a little) in a public goods problem?

[your written answer here]

2 Theory

Payoffs to agent i

i

are

πi=(eixi)+αi=1nxi

πi=(eixi)+αi=1nxi

where ei

ei

is the agent’s endowment, xi

xi

is her contribution to the public good, α

α

is the marginal per capita return, and n

n

is the group size.

[Q6] Explain α

α

and why in public goods game requires 1n<α<1

1n<α<1

.

[your written answer here]

[Q7] Suppose ei=e=20

ei=e=20

(i.e. everyone has 20), α=0.4

α=0.4

and n=4

n=4

. Show that xi=0

xi=0

is a symmetric Nash equilibrium, but xi=20

xi=20

is the social optimum. (Recall that in a Nash equilibrium i

i

cannot increase her payoff by changing her contribution.) Hint: you can use code to answer this problem by calcuting payoffs to a representative agent and plotting them. You might the curve() function useful.

[your written answer here]

3 Replication

punnoise = read_csv("../data/punnoise_data.csv")

3.1 Description

Use theme_classic() for all plots.

[Q8] Recreate Table 1 and usekable() to make a publication-quality table (in HTML).

# your code here

3.2 Inference

Consider the linear model

y=α+β1x1+β2x2+ε

y=α+β1x1+β2x2+ε

[Q9] Write down the marginal effect of x1

x1

(in math).

[your written answer here]

Now suppose you have a non-linear model

y=F(α+β1x1+β2x2+ε)

y=F(α+β1x1+β2x2+ε)

where F()

F()

is a “link function” that compresses the inputs so that the output y^[0,1]

y^[0,1]

.

[Q10] Write down the marginal effect of x1

x1

. How does this compare to the marginal effect in the linear model?

[your written answer here]

[Q11] A probit model uses the Normal CDF Φ

Φ

as the link function, where Φ=ϕ

Φ=ϕ

is the Normal PDF. Use glm() to estimate Model 1 in Table 2. Assign the model to the object m1. Cluster the standard errors at the group level.

# your code here

[Q12] Interpret the coefficients. (For more on the probit model, see the appendix.)

[your written answer here]

3.2.1 Average marginal effects

[Q13] Table 2 reports the average marginal effects (AMEs) of the variables on P(contribute)

P(contribute)

. Calculate the AME to the variable round as follows:

  1. Use predict()to create an object predictions that contains the predicted z-scores. (i.e. Xβ^
    Xβ^

    . Hint: use the option type="link" in predict().)

# your code here
  1. Use dnorm() to calculate the probabilities of the predicted z-scores and store the output in an object called index.
# your code here
  1. Now calculate the marginal effects by multiplying the predicted probabilities times the estimated coefficient for round and store the output in dydxround.
# your code here
  1. Use mean() to calculate the AME.
# your code here

[Q14] Verify your calculations with margins(), the plot the AMEs. (Note: these will not be exactly the same as those in the paper, since the paper uses an outdated method in Stata.

# your code here
# your code here

[Q15] Interpret the AMEs.

[your written answer here]

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