# 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

are

where

is the agent’s endowment,

is her contribution to the public good,

is the marginal per capita return, and

is the group size.

$\alpha $

[Q6]Explainα and why in public goods game requires

$\frac{1}{n}<\alpha <1$1n<α<1 .

[your written answer here]

${e}_{i}=e=20$

[Q7]Supposeei=e=20 (i.e. everyone has 20),

$\alpha =0.4$α=0.4 and

$n=4$n=4 . Show that

${x}_{i}=0$xi=0 is a symmetric Nash equilibrium, but

${x}_{i}=20$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

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

punnoise

## 3.1 Description

*Use theme_classic() for all plots.*

[Q8]Recreate Table 1 and use`kable()`

to make a publication-quality table (in HTML).

*# your code here*

## 3.2 Inference

Consider the linear model

${x}_{1}$

[Q9]Write down the marginal effect ofx1 (in math).

[your written answer here]

Now suppose you have a non-linear model

where

is a “link function” that compresses the inputs so that the output

.

${x}_{1}$

[Q10]Write down the marginal effect ofx1 . How does this compare to the marginal effect in the linear model?

[your written answer here]

$\Phi $

[Q11]A probit model uses the Normal CDFΦ as the link function, where

${\Phi}^{\prime}=\varphi $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

$P\left(\text{contribute}\right)$

[Q13]Table 2 reports the average marginal effects (AMEs) of the variables onP(contribute) . Calculate the AME to the variable

`round`

as follows:

- Use
`predict()`

to create an object`predictions`

that contains the predicted z-scores. (i.e.Xβ^

$\hat{X\beta}$. Hint: use the option

`type="link"`

in`predict()`

.)

*# your code here*

- Use
`dnorm()`

to calculate the probabilities of the predicted z-scores and store the output in an object called`index`

.

*# your code here*

- 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*

- 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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