Mechanism Design

Jan Boone

jan.boone.uvt@gmail.com

Table of Contents

Introduction

What is mechanism design?

  • We will consider situations where a principal wants an agent to do something
  • Agent has private information that the principal cannot observe
  • in optimal tax problem: government cannot observe my productivity
    • in monopoly problem: monopolist cannot observe my valuation of the good
    • government regulating NS cannot observe costs of operating a railway track
    • ministry of health trying to limit health care expenditure cannot observe quality of hospitals, severity of patients
    • government selling telecom licenses cannot observe utility that consumers will enjoy from these or the costs of operating such networks
  • Agent’s private information gives him (bargaining/market) power and he can extract rents from the principal
  • Principal wants to design a way to get the desired action from the agent at the lowest cost
  • Instead of focusing on examples (like linear tax scheme, quadratic tax etc.), principal asks the agent directly what his type is
  • And makes sure that the agent will reveal his type to her truthfully
  • This is true for any mechanism that you can think of and –in fact– far easier to analyze
  • Mechanism design gives you the tools to analyze situations like this

Why is this useful?

  • It turns out to be analytically more convenient!
  • It separates the optimal outcome from the way you would like to implement this outcome
  • You get the optimal overall solution; not, say, the optimal linear solution
  • It forces you to think about the things that really matter:
    • what is the principal’s objective function?
    • how can agents deviate (incentive compatibility)
    • do agents want to participate (individual rationality)
    • how to trade off efficiency and rent extraction

Optimal taxation (2 types)

Without mechanism design

  • Consider utility function \(u(c)-n\) where \(c\) denotes consumption and \(n\) work effort
  • \(u'(c)>0\)
  • Decreasing marginal utility \(u''(c)<0\)
  • There are two types of workers: low productivity \(w^l\) (fraction \(\phi\) of the population) and high productivity \(w^h>w^l\) (\(1-\phi\))
  • Government cannot observe type \(w\) nor effort \(n\)
  • Government only observes gross income \(y = w n\)
  • At first sight, one would try to find a tax function \(T(y)\) to maximize (some) government objective
  • Worker then solves
\begin{equation} \label{eq:1} \max_n u(wn - T(wn)) - n \end{equation}
  • First order condition
\begin{equation} \label{eq:2} u'(wn-T(wn))(1-T'(wn))w-1 = 0 \end{equation}
  • Second order condition: oops?
  • That is why we need mechanism design

With mechanism design

  • Revelation principle: any mechanism that you can think of, can be described as “ask worker for his type \(n\) and then tell the worker to earn \(y^n\) and consume \(c^n\)”
  • We will create two options \((y^l,c^l),(y^h,c^h)\) such that each type chooses the option that is meant for this type
  • This we ensure by imposing incentive compatibility constraints:
\begin{align} \label{eq:3} \tag{$IC_l$} u(c^l)-y^l/w^l &\geq u(c^h)-y^h/w^l \\ \label{eq:4} \tag{$IC_h$} u(c^h)-y^h/w^h &\geq u(c^l)-y^l/w^h \end{align}
  • Adding these two equations yields:
\begin{equation*} y^h \left(\frac{1}{w^l} - \frac{1}{w^h} \right) \geq y^l \left(\frac{1}{w^l} - \frac{1}{w^h} \right) \end{equation*}
  • Hence in any mechanism we will find that the high type earns higher gross income than the low type
  • Moreover, high type always has higher utility than low type
\begin{equation} \label{eq:5} u^h = u(c^h)-y^h/w^h \geq u(c^l)-y^l/w^h \geq u(c^l)-y^l/w^l = u^l \end{equation}

where the last inequality is strict if \(y^l>0\)

  • We guess that (\ref{eq:4}) is binding: high productivity type tends to be “lazy” and wants to mimic low type
  • We ignore (\ref{eq:3}) and check at the end that indeed it is not violated
  • Rewrite (\ref{eq:4}) as
\begin{equation} \label{eq:6} u(c^h)-n^h = u(c^l)-(y^l/w^l)*(w^l/w^h) = u(c^l) - \omega n^l \end{equation}

with \(\omega = w^l/w^h < 1\)

  • Finally, we have the government budget constraint
\begin{equation} \label{eq:7} \tag{GBC} \phi c^l +(1-\phi) c^h + E = \phi w^l n^l + (1-\phi)w^h n^h \end{equation}
  • Government needs to raise tax revenue \(E\) to finance a public good
  • We write the government’s optimization problem as follows
\begin{split} \max_{c^l,c^h,n^l,n^h} & \phi (u(c^l)-n^l) + (1-\phi)(u(c^h)-n^h) \\ -&\lambda (\phi c^l +(1-\phi) c^h+E - \phi w^l n^l - (1-\phi)w^h n^h) \\ +&\mu (u(c^h)-n^h -u(c^l) + \omega n^l) \end{split}

where \(\lambda\) denotes the Lagrange multiplier on (\ref{eq:7}) and \(\mu\) on (\ref{eq:4})

  • First order conditions can be written as
\begin{align} \label{eq:FOCcl} \phi u'(c^l) - \phi \lambda -\mu u'(c^l) & =0 \\ \label{eq:FOCnl} -\phi + \phi \lambda w^l +\mu \omega & =0 \\ \label{eq:FOCch} (1-\phi) u'(c^h) - (1-\phi) \lambda +\mu u'(c^h) & =0 \\ \label{eq:FOCnh} -(1-\phi) + (1-\phi) \lambda w^h -\mu & =0 \end{align}
  • Routine to verify that
\begin{align} \label{eq:ch} u'(c^h) &= \frac{1}{w^h} \\ \label{eq:cl} u'(c^l) &= \frac{1}{w^l} \frac{\phi-\omega\mu}{\phi-\mu} > \frac{1}{w^l} \end{align}
  • High type faces a marginal tax rate equal to 0 (no distortion at the top)
  • Low type faces a positive marginal tax rate
  • Explain that to the SP!
  • Because \(u(c)\) is concave, government would like to redistribute from high to low type
  • But high type wants to mimic low type
  • What is best way to raise l-type’s utility such that h-type does not want to mimic it?
  • Reduce \(y^l\) below its first best value: positive marginal tax rate
  • We can solve first order conditions above for \(\lambda,\mu\):
\begin{align} \label{eq:8} \lambda &= \left( \frac{\phi}{w^l} + \frac{1-\phi}{w^{h}} \right) \\ \label{eq:9} \mu &= \phi(1-\phi) \left(\frac{1}{\omega} -1 \right) \end{align}
  • Marginal cost of public funds, \(\lambda\), is weighted average of effort costs \(1/w^l,1/w^h\):
    • asking each type to earn one unit more, gives government additional unit of budget and satisfies both IC constraints
  • Shadow price of (\ref{eq:4}) equals 0 if \(\omega = 1\): both types are the same; no informational rents; first best can be implemented with lump sum taxes
    • as \(\omega\) falls, \(\mu\) increases: government would like to redistribute more and (\ref{eq:4}) becomes “more binding”
  • Use (\ref{eq:4}) and (\ref{eq:7}) to solve for \(n^l,n^h\) and hence for \(y^l,y^h\):
\begin{align} \label{eq:10} y^l &= \phi c^{l}+(1-\phi)c^h+E - (1-\phi)w^h(u(c^h)-u(c^l)) \\ \label{eq:11} y^h &= \phi c^{l}+(1-\phi)c^h+E + \phi w^h(u(c^h)-u(c^l)) \end{align}

Implementation

  • consider an example with: \(w^l = 1, w^h = 2\) and \(\omega = 0.5\) \(\phi = 0.5\) \(E=1\) \(u(c) = 2 \sqrt{c}\)

taxfigure1.png

Optimal contracts for l-type and h-type

  • l-type works less and consumes less than h-type
  • h-type is indifferent between l and h contracts
  • if we would redistribute a bit more to l-type, h-type would mimic
  • both h-type and l-type pay taxes (contracts lie below the green line)
  • h-type pays (a lot) more tax than l-type
  • note that l-type does not want to mimic h-type: (\ref{eq:3}) is satisfied –which we needed to check
  • tax function \(T(y)\) should be such that: \(c(y) = y - T(y)\) lies everywhere below the two indifference curves \(c^l = y^l - T(y^l)\) \(c^h = y^h - T(y^h)\)
  • clearly there are a lot of functions \(T(y)\) that satisfy these criteria
  • around \(y =6\) black line seems to be paralel to green line: coincidence?

taxfigure2.png

Example of tax function \(T(y)\) for implementation

What have we learned?

  • Writing down an optimal taxation problem directly with a tax function \(T(y)\) is problematic
  • Analytically, it is more convenient to write down the options for each type directly and make sure that each type chooses “his own” option by imposing IC constraints
  • With optimal taxation we find that:
    • high productivity type pays high tax (high average tax rate)
    • but this does not imply that high type faces a (high) positive marginal tax rate
    • in fact, marginal tax rate for high type equals 0
    • no distortion at the top
    • low type faces a positive marginal tax rate
    • because high type wants to mimic low type, government needs to distort labor supply of low type (to make mimicking less attractive)
    • labor supply is reduced with a positive marginal tax rate
    • as high type works a lot, low type does not want to mimic high type

Monopolist (continuum of types)

Differences with optimal taxation

  • Instead of considering two types, we now consider a continuum of types
  • As people could not immigrate in the tax example, we only considered IC constraints
  • when buying from a monopolist, one can decide not to buy at all: we need IR constraints

Model

  • Monopolist can sell indivisible good to one consumer
  • Cost of production equals 0 for monopolist
  • If consumer buys the good from monopolist at price \(p\), his utility equals \(u(\theta) = \theta -p\) where \(\theta\) is distributed on \([0,1]\) with density [distribution] function \(f(\theta)[F(\theta)]\)
  • Below we focus on a uniform distribution: \(f(\theta)=1,F(\theta) = \theta\)
  • Monopolist cannot observe \(\theta\): this is private information for the consumer
  • Revelation principle: monopolist asks consumer for his type \(\theta\)
  • Given the consumer’s message \(\hat \theta\), he has to pay \(t(\hat \theta)\) and receives the good with probability \(x(\hat \theta)\)
  • Consumer and monopolist are risk neutral: it does not matter whether the price is always paid (\(t\)) or only conditional on actually giving the good to the consumer (\(p=t/x\))
  • Monopolist needs to choose \(x(.),t(.)\) in such a way that
    • IC: consumer is willing to truthfully reveal \(\theta\)
    • IR: consumer is willing to participate
  • Monopolist maximizes expected revenue
\begin{equation} \label{eq:12} \int_0^1 t(\theta)f(\theta) d\theta \end{equation}

Consumer’s optimization problem

  • Consumer chooses message \(\hat \theta\) to solve
\begin{equation} \label{eq:13} u(\theta) = \max_{\hat \theta} \{x(\hat \theta) \theta - t(\hat \theta) \} \end{equation}
  • Instead of looking at the first order condition for this problem, we use the envelope theorem:
\begin{equation} \label{eq:14} u'(\theta) = x(\theta) \end{equation}
  • Since \(x(.)\) denotes a probability, we find that \(u'(\theta) \geq 0\)
  • Hence the IR constraint \(u(\theta) \geq 0\) for all \(\theta \in [0,1]\) can be replaced by
\begin{equation} \label{eq:15} \tag{$IR$} u(0) = 0 \end{equation}

as there is no reason for the monopolist to “give away presents”

  • Consequently, we can write
\begin{equation} \label{eq:16} u(\theta) = \int_0^{\theta} u'(\tau) d\tau = \int_0^{\theta} x(\tau) d\tau \end{equation}

Monopolist’s optimization problem

  • Since \(t(\theta) = x(\theta)\theta - u(\theta)\), we can write the monopolist’s optimization problem as
\begin{equation} \label{eq:17} \max_{x(.)} \int_0^1 (x(\theta)\theta - u(\theta)) f(\theta) d\theta = \int_0^1 \left(x(\theta)\theta - \int_0^{\theta} x(\tau)d\tau \right) f(\theta) d(\theta) \end{equation}
  • Using partial integration, we can write this as
\begin{equation} \label{eq:18} \max_{x(.) \in [0,1]} \int_0^1 x(\theta) \left(\underset{MR(\theta)}{\underbrace{\theta - \frac{1-F(\theta)}{f(\theta)} }} \right) f(\theta)d\theta \end{equation}
  • By increasing \(x(\theta)\), social surplus increases by \(\theta f(\theta)\)
  • However, increasing \(x(\theta)\) makes it more attractive for types \(\theta'>\theta\) to mimic \(\theta\)
  • To stop them from doing that, the monopolist needs to give them an informational rent: there are \(1-F(\theta)\) of these types above \(\theta\)
  • This term is called marginal revenue (for reasons that become clear shortly)
  • As the monopolist’s problem is linear in \(x(.)\), the solution is quite simple:
\begin{equation} \label{eq:19} x(\theta) = \begin{cases} 1 & \text{if } MR(\theta) \geq 0 \\ 0 & \text{otherwise} \end{cases} \end{equation}
  • With a uniform distribution we have \(MR(\theta) = 2\theta -1\)
  • Hence the monopolist only sells to types with \(\theta \geq \frac{1}{2}\)
  • Why does the monopolist not sell to everyone with valuation \(\theta\) above production costs 0?

Marginal Revenue

  • To see where the term MR comes from in this context, we write \(q(\theta) = 1-\theta\)
  • In a standard demand context, the people with a high value of the good are on the left hand side, while above they are on the right hand side of the interval \([0,1]\)
  • Hence we “inverse things” with \(1-\theta\)
  • Also if you set a price \(p\), how many people would buy?
  • Consumers with \(\theta \geq p\) would buy and there are \(q = 1-p\) of such consumers
  • The demand curve (consumer valuation) can now be written as \(1-q(\theta)\)
  • Finally, \(MR(\theta) = [1-(1-\theta)] - \frac{1-\theta}{1} = 1 - 2q(\theta)\)

figure3.png

Demand curve and marginal revenue

Global optimum for consumer?

  • The envelop theorem is a local argument, is it clear that the optimum supposed to be chosen (\(\hat \theta = \theta\)) by the consumer is a global optimum?
  • What does the consumer optimization problem actually look like?
  • To see this, first note that
\begin{equation} \label{eq:20} u(\theta) = \int_0^{\theta} u'(\tau) d\tau = \int_0^{\theta} x(\tau)d\tau = \theta - 0.5 \end{equation}
  • Hence, we see that
\begin{equation} \label{eq:21} \theta-0.5 = \max_{\hat \theta} \{ x(\hat \theta) \theta - t(\hat \theta) \} \end{equation}
  • Clearly, people with \(\theta < 0.5\) will not participate: they do not buy (\(x(\theta)=0\)) and they do not pay (\(t(\theta)=0\))
  • For people with \(\theta \geq 0.5\), we see that \(x(\theta) =1\) and \(t(\theta)=0.5\)
  • Hence we can write the consumer’s maximization problem as
\begin{equation} \label{eq:22} \max_{\hat \theta} \{ x(\hat \theta) \theta - t(\hat \theta) \} = \max\{0,\theta-0.5\} \end{equation}
  • Either the consumer buys at a price equal to 0.5 or he does not buy at all
  • Consumer only needs to reveal whether \(\theta\) is above 0.5 (or not)
  • Put differently, the monopolist offers a menu with two options for \((x,t)\): \(\{(0,0),(1,0.5)\}\) and consumer chooses the best option from these two
  • Truthful revelation of type is indeed a global optimum
  • More generally, the reason why we can move from local to global here is the fact that \(x(\theta)\) is non-decreasing in \(\theta\)

What have we learned?

  • Mechanism design can also be applied when there is a continuum of types
    • in fact, this is often easier than the two type case
  • A monopolist who faces a consumer without knowing the consumer’s valuation for the good should do the following:
    • make a take-it-or-leave-it offer to the consumer
    • consumer then decides whether to accept or reject
    • this leads to social inefficiency (deadweight loss)
    • to reduce informational rents of high types, monopolist does not sell to consumers who value the good more than the cost of production
    • only consumers with a marginal revenue above the production cost can buy the good
    • if marginal revenue is non-decreasing in type, we do not need to worry about the second order condition of the consumer optimization problem

Conclusion

Who needs mechanism design?

  • If you want to analyze what the optimal policy is by, say, the government; instead of just analyzing the effects a specific policy (say a change in a linear tax rate), you need mechanism design
  • Why should taxes be linear (only)?
  • If you want to analyze a general tax function \(T(y)\), and you use a “standard approach”, the worker’s optimization problem may not be well defined
  • Hard to check whether the local optimum for the worker is a global optimum
  • Mechanism design forces you to think carefully about:
    • objective function for the planner
    • ways in which agents can deviate (incentive compatibility)
    • whether agents want to participate (individual rationality) and what their outside options are (may differ with type: easier to emigrate for high than low productivity type)
  • Mechanism design shows things generally:
    • high type (in the examples above) is always better off than low type, no matter what the mechanism is (i.e. this is also true for mechanisms that are not optimal)
    • high type can always mimic low type and then he is still better off
    • if you would focus on linear tax functions, you would not see that marginal tax rate for highest type should be zero
  • Important lesson from mechanism design: think in terms of informational rents
    • make mechanism as efficient as possible (the bigger the total pie, the more rents you can extract)
    • but make sure you do not leave too much rents to high types (who can mimic low types)

Created by jan.