$C(\delta_{min})-C(\delta_r)=\theta_C$

Assume consumer in control
$C'(\delta_r)(\delta_{min}-\delta_r)+C''(\delta_r)(\delta_{min}-\delta_r)^2 /2=\theta_L$

$F'(\delta_r)(\delta_{max}-\delta_r)+F''(\delta_r)(\delta_{max}-\delta_r)^2 /2=-\theta_H$

Define $\Delta_C = \delta_{min} - \delta_r$, $\Delta_F = \delta_{max}- \delta_r$

Then trade occurs when $\Delta_C \geq \Delta_F$

Assume firm in control

$C'(\delta_r)(\delta_{min}-\delta_r)+C''(\delta_r)(\delta_{min}-\delta_r)^2 /2=-\theta_H$

$F'(\delta_r)(\delta_{max}-\delta_r)+F''(\delta_r)(\delta_{max}-\delta_r)^2 /2=\theta_L$

Imagine prob of being high type is same, total surplus whenever weakly greater type is in control is the same no matter what

In the case where a strictly lower type has control of quality and there is no trade, total surplus is the same

It's possible that in one of these cases there is not trade and in one of these cases ther i

SOOOOOO all else equal, put the person more sensitive to price in contorl of quality

# Quality Control : Zoe & Benji

:::info
"Shifts in product or service quality affect inframarginal consumers. These people must be consulted if a correct evaluation of a quality change is to be made."

Spence (1975)
:::

## Screening problem problems

When thinking about the "mechanism design" approach, we got ourselves a bit tangled up trying to create a mechanism through which firms would accurately reveal their cost functions (for quality) and consumers would accurately reveal their valuation functions (for quality). Reading the literature has untangled things a bit, for me, and in some ways has shown me why we might want to braid things back together neatly... let me say what I mean.

We took as a starting point the quality distortion problem in Spence. And then we took the perspective of a regulator trying to regulate a monopolist.

It's valuable to pause and think about the "normal" regulator problem. In the normal (price) regulation of a monopoly, the firm's cost function is unknown to the regulator, as is the demand function. If the regulator knew the firms' cost and demand functions, the regulator could simply choose the surplus maximizing choice. Some models look at what happens when the regulator doesn't have info about firms' demand functions, others look at the absence of firms' cost functions(e.g. Baron and Myerson), and some look at what happens when the regulator knows neither firm costs nor demands (e.g. Lewis and Sappington 1988, Laffont Maskin and Rochet).

Now let's return to the quality issues. All of the problems in the normal regulator's problem remain---the regulator does not know firms' cost, or demands. But now these are functions of quality choices as well as price and quantity choices. Why does this matter? Well, let's think about the first best solution. If the regulator knew everything, the regulator would know (a) firms' cost functions, (b) demand functions and (c) whatever other info is needed to fix the Spence distortion. So, the way these problems usually go, we say "Oh, that's all information that the firm has, so we need to build an optimal (IC) policy that gets the firm to reveal that private information." But here's where there's a key departure from the "normal monopoly" case: firms are likely to know (a) and (b) but may not know ( c ).

Indeed, the fact that firms don't know ( c ) is its own screening problem! The fact that firms don't know much about consumer valuations of quality is what makes the product quality issue a multidimensional screening problem. So, firms are trying to solve this product design question---trying to figure out how to segment the market, etc., to extract the most consumer surplus. They can't achieve "first best" profit because they don't know consumers' exact types, only know distribution, so consumers typically get their information rent. The firm can't just ask consumers for their valuations---they'd always lie because they know the firm wants to maximize profit.

But here's an interesting way in which the pieces of the puzzle might fit together. The regulator wants to set the welfare maximizing price, quantity and quality. We can think of the regulator like an intermediary. And, if we think of the regulator like an intermediary, and we know she has overall welfare in mind, consumers might be willing to reveal their valuations. AND THEN! Maybe the firm would accurately reveal cost functions, because this information about quality could help them segment the market? This is sloppy thinking, but basically, I wonder if there's a neat way to put the pieces together---is there some mechanism that gets consumers to participate/reveal truthfully, and then the increased

Now, what I don't completely understand is how the pieces fit together. And I'm still not sure if there is some work that looks at the *regulator* problem with quality.

### Regulator problem

###

## Some takeaways:
1. BBM is very much about price discrimination and market segmentation. It is most relevant in that it suggests that there is a kind of information that could be given to producers that would allow them to segment the market and acehive the consumer surplus maximizing point (and this would yield the same producer surplus as the unfirom monopoly price). BUT they don't discuss strategic issues/how to get the consumer valuations.
2. Multidimensional screening is hard---if Gabe Carroll says so, then it must really be true. His approach is to look at a certain kind of robustness. Others' approaches are more about putting more structure on type distributions etc. to get something tractable.
3. Somethign I'm trying to puzzle through. I feel like we're not really interested in the "screening" problem... the screening problem is about how a firm can offer a menu of contracts to extract maximum surplus from consumers. It's cool how Mussa and Rosen and Carroll show us that there are some contracts that segment the market by quality pref and allow the producer to extract more surplus. BBM feels like it suggests (but doesn't pursue) a different path. In particular, it says: the consumer surplus maximizing point is achievable by a particular market segmentation. The producer surplus maximizing point is also achievable by some segmentation. BUt in order to do segmentation of either kind, the firm needs info in addition to its priors about the joint distribution. So BBM's results allow us to say: Suppose we want to achieve some combo of producer/consumer surplus. What information could we give to the firm, which would make them segment the market in order to achieve this surplus pair? This is nice... sort of. It's not very practical or helpful to a regulator, because there remains a problem about how to give extra information about consumer valuations to firms: what kind of intermediary would already have consumer valuations, and then face a question of how to optimally disclose this info on consumer valuations??
4. This last question helps me see, to some extent, why the problem we are interested in is not quite a screening problem. Screening problems that I looked at here take one of two perspectives it seems:
(a) a *firm* wants to extract more surplus from the consumer, offer a menu of contracts that sorts consumers, and thus pulls away some of the consumer's information rent. In the IO contexts the info rent is due to consumers' private informaiton about their valuations. In insurance contexts, the info rent is due to agents' private info about their risk type. In taxation context, the firm is the government trying to extract surplus from its taxable citizens---the info rent is due to agents' private ifno about their productivity---can "lie" about productivity by choosing a lower income and therefore lower tax bracket.

(b) a *regulator* wants to regulate the pricing of a monopolist but the firm has private info about its cost function (as in Baron and Myerson) and/or about quality "capacity" as in Blackorby and Szalay. So, regulator wants to choose regulatory policy that will get firms to truthfully reveal their costs.

*We are thinking about something slightly different*: we are looking at the problem from the perspective of a regulator, who is worried about regulating a quality-providing firm, where the firm is (potentially) segmenting the market to extract highest surplus. In other words, we've been thinking about a screening problem with another screening problem embedded in it.

## Spence 1977

## Mussa and Rosen 1978

Model. Let $q$ represent attributes that characterize a particular variety. $q$ is restricted to one dimension. Same price charged to all buyers.
- Equilibrium described by prices for each quality $P(q)$, number of each $q$ transacted $N(q)$ and observed breadth of product line $[\bar{q},bar{q}]$
- Consumer utility $U(x, q; \theta)$
- $U = x + \theta q$ is a fine approximation

Competitive solution easily described:
- comeptition and free entry imply profits 0 for all q
- So P(q)=C(1)
- Each consumer is self-assigned to the quality that maximizes utility , with $P'(q) = \theta$ for those who purchase.

:::info
Serving customers who place smaller valuations on quality creates negative externalities for the monopolist that limit possibilities for capturing consumer surplus from those who do value quality highly. In the nature of the case, these external effects all go in an “upstream” direction. The monopolist internalizes them by inducing less enthusiastic consumers to buy lower quality items, opening the possibility of charging higher prices to more adamant buyers of high quality units. Thus, the impersonal monopolist achieves some, but imperfect, discrimination by taking advantage of the natural tendency of consumers to sort themselves out along the quality spectrum.
:::

## Baron and Myerson 1982

Regulating a monopolist with unknown costs.

MOTIVATION. Dupuit and Hotelling assume...
:::info
While the
assumption of complete information may be too strong, the assumption that
information about demand is as available to the regulator as it is to the firm does
not seem unnatural. A second informational assumption is that the regulator has
complete information about the cost of the firm or at least has the same
information about cost as does the firm. This assumption is unlikely to be met in
reality, since the firm would be expected to have better information about costs
than would the regulator.
:::
The purpose of this paper is to develop an optimal regulatory policy for the case in which the
regulator does not know the costs of the firm.

:::info
This approach [delegate pricing decision to firm] has been proposed by Loeb and
Magat [6] but leaves the equity issue unresolved, since the firm receives all the
social surplus and consumers receive none. To resolve the equity issue, Loeb and
Magat propose that the right to the monopoly franchise be auctioned among
competing firms as a means of transferring surplus from producers to consumers. However, if there are no other producers capable of supplying the product
efficiently, an auction will not be effective
:::

## Matthews and Moore 1987

:::success
We consider in this paper a monopoly design problem in which each contract specifies a different quality product and the terms on which it will be sold. The intuition behind its solution starts with the observation that profit is potentially greatest on contracts designed for "high" type consumers, those with a high evaluation of quality. Because high type consumers cannot be prevented from choosing contracts meant for low types, this profit can only be realized by distorting the contracts meant for low types in a direction that makes them relatively unattractive to high types. All but the highest type of consumer should therefore receive products of inefficiently low quality
:::

One of their most striking results is that the allocation of
qualities is not necessarily monotonic with respect to types. As a consequence,
nonlocal incentive compatibility constraints may be binding at the optimum. - Rochet and Chone on Matthews and Moore

## Lewis and Sappington 1988

## Rochet and Chone 1998

MOTIVATION: We know lots of things about single dimension cases. For example...
:::success

(i) When the single-crossing condition is satisfied, only local (first and second
order) incentive compatibility constraints can be binding.
(ii) In most problems, the second order (local) incentive compatibility con-
straints can be ignored, provided that the distribution of types is not too
irregular.
(iii) If it is the case, bunching is ruled out and the optimal solution can be
found in two steps:
(a) First compute the expected rent of the agent as a function of the
allocation of (nonmonetary) goods.
(b) Second, find the allocation of goods that maximizes the surplus of the
principal, net of the expected rent computed above.

:::
... Do these extend to multiple dimensions??/

## Blackorby and Szalay 2008

COMMENT. I think this paper is the solution to somethign that Ben asked us to look at: what is the solution to the regulator problem when you turn off consumer heterogeneity entirely?

"The main difference to Matthews and Moore is that in their paper information is one-dimensional but there are two instruments to choose; in our paper both information and the set of instruments is two-dimensional."

. The firm produces a service of observable quantity and quality, but given an observed quality, the regulator does not know whether the firm would have been capable of delivering the good in still higher quality. In addition to that, the firm’s profit is unobservable to the regulator, so information about costs poses non-trivial problems as well.

## Bergemann Brooks and Morris 2015

We analyze the welfare consequences of a monopolist having addi- tional information about consumers’ tastes, beyond the prior distribution. The additional information can be used to charge different prices to different segments of the market, i.e., carry out "third degree price discrimination"

Motivation. THings we already know:
- no info beyond prior --> no segmentation
- lower bound on producer surplus
- consumers get positive surplus, standard information rent
- complete info --> perfect 1st degree px discrimination
- prodcuer captures all of gains from trade
- consumer surplus is zero
- what about all possible segmentations in between?? can put some bounds on possible outcoes because:
1. consumer surplus has to be nonnegative as a consequence of participation constraint
2. producer must get at least surplus that she could get if no segmentation and charged uniform monopoly price
3. sum of consumer and producer surplus cannot exceed total value that consumer receives from the good when that value exceeds marginal cost of production

MAIN RESULT. Every welfare outcome satisfying these constraints is attainable by some market segementation.

SOME INTUITION. Consider finite set of consumer valuations, cost of production is zero. A simple explanation of how to achieve max consumer surplus (i.e. point C). Suppose we can divide market into segments s.t.
- in each segment, consumer vals always greater than or equal to the price for that segment
- in each segment, producer indifferent between price for THAT segment and unfirom monopoly price
Together, these imply that producer indifferent between price on ALL segments --> surplus must be equal uniform monopoly profit. ALlocation is efficient so consumers must obtain rest of efficient surplus.

:::info
Suppose, however, that an Internet intermediary wanted to release its information about consumers to producers for free in order to maximize consumer welfare, say, because of regulatory pressure or as part of a broader business model. Our results describe how such a consumer-minded Internet company would choose to structure this information. An important subtlety of this story, however, is that this could only be done by randomly allocating consumers with the same valuation to different segments with different prices. Thus, it could be done by a benevolent intermediary who already knew consumers’ valuations, but not by one who needed consumers to truthfully report their values.
:::

BACKGROUND. Third degree price discrimination is a special case of the classic screening problem, in which a principal is designing a contract for an agent who has private info about the environment. Gross utility is: product of WTP for object and probability of receiving the good
- Riley and Zeckhauser -- when consumer qualiations linear in quantity, quality, prob of getting object, posted price is optimal mech
- no longer true if valuations are nonlinear in quantity (as in Maskin & Riley)
- no longer true if valuations are nonlinear in quality (as in Mussa & Rosen)

SITUATION/METHODOLOGY. "our results are a striking application of the methodologies of Bergemann and Morris and Kamenica and GEntzkow to the problem of price discrimination"

## Carroll WP - note discussion at bottom of page 10

:::info
Multidimensional screening stands out among current topics in economic theory as a
source of problems that are simple to write down, yet analytically intractable. Whereas
the canonical one-dimensional screening model is well understood, its multidimensional
analogue, as traditionally formulated, is much more complex and unruly.
:::

MOTIVATION. Even the simple two dimensional monopoly screening problem has been extremely hard for people.
- With one good, take it or leave it single posted price is optimal (Riley and Zeckhauser).
- In the two good case, we know the optimal mechanism when goods are independent unforim on [0,1]---prices for each good plus a price for the bundle (but proofs are complicated---Manelli and Vincent 2006).
- In other cases, optimum may involve probabilistic bundling, and menu of infinitely many bundles
- If values are correlated, restricting seller to any fixed number $k$ of bundles can lead to an arbitrarily small fraction of the optimal revenue (Hart and Nisan 2013), and finding optimal mech is computationally intractable
This paper proposes an alternative framework.

ALTERNATIVE FRAMEWORK. Agent has quasilienar preferences, several components of pvt info. For each component $g$ the agent is assigned an allocation $x^g$ from which he derives a value depends on corresponding type $\theta^g$. Total payoff is additively separable across compoenents.
- Unlike traditional model, where principal maximizes expected profit according to a prior distribution over full type $\theta$, assume that principal only has beliefs about marginal distribution of each component $\theta^g$---does not know how various components are correlated with each other
- Look for a guarantee on average profit that is robust to this uncertainty

MAIN RESULT. Optimal mechanism separates---for each component $g$, principal offers the optimal mechanism for that marginal distribution. In the monopolist example above, the optimal mechanism is to post a price for each good separately without any bundling.

:::success
Much of the literature on multidimensional mechanism design has emphasized the advantages of bundling (to use the monopolist terminology) — or, more generally, of creating interactions between the different dimensions of private information. In this
context, our result expresses the following simple counterpoint: If you don’t know enough to see how to bundle, then don’t.
:::

## AGENDA FOR 7/3
- Benji thoughts
- Zoe thoughts
- Revisiting Spence
- Farrell (1985) / Gans et al (2018)
- Goals for next week/talking to fac

Participation problems in mechs---mandated participation and trasnfers are natural

## Setup
There are two groups: $F$ and $C$. Group $F$ will refer to the firm which is a single entity that internalizes a single profit motive. $C$ refers to consumers, which consists of many heterogeneous agents with varying preferences. We will assume for now that they are interested in a particular value, $q$, which we refer to as quality.

From the firms perspective the change in quality will effect profits through the usual Spence (1975) type mechanism. Firms are expected to charge a single profit maximizing price conditional on the level of quality selected. Thus, we can define profits as being a function of the particular quality selected $\Pi = \Pi(q)$.

Similarly, for consumers, each consumer has heterogeneous preferences over quality. Each consumer has a certain willingness to pay for a particular level of quality given the profit maximizing price that will follow. Let $CS_i(q)$ be the level of consumer surplus for individual $i$. It may be useful to note that since a consumer always has the option not to purchase the good $CS_i(q)>0 \quad \forall \quad q \in Q$. Total consumer surplus is given by $CS(q) = \sum_i CS_i(q)$.

Define $q^{\pi}$ to be the profit maxmimizing market outcome level of quality. The logic of Spence (1975) is that there may be many qualities for which the inframarginal consumer would obtain greater value from the increase in quality than the marginal consumer. If welfare is taken to be $W(q) = \Pi(q) + CS(q)$, then the previous statement suggests $\frac{\partial W(q)}{\partial q}\Bigr|_{\substack{q=q^\pi}}>0$.

The goal of the mechanism designer is to maximize welfare by discovering $q^{opt}$ where $\frac{\partial W(q)}{\partial q}\Bigr|_{\substack{q=q^{opt}}}=0$. As a first pass imagine the mechanism where the designer simply elicits a menu of bids from both $F$ and $C$. For now, we can simply think of C as a disaggregated group of $N$ players. Therefore, the mechanism requests menu of valuations where each of $N+1$ provide their valuations. We should be able to show quite simply that because of the non-excludability of quality, it will not generally be the case that in a simple VCG style auction that members of C will put their true valuation. For instance, imagine you only enjoy quality up to a certain point and then after that you are indifferent. If you know that other people want very high quality, you have incentive to down weight your bid since the marginal value of your bid will be low if the others guarantee you are above your threshold value.

There is inherently a freerider problem. Unlike in a standard VCG setup where the item is being given to an individual, the decision of a particular quality is experienced by all of the bidders and participation is not enforced. Another way to consider this is to imagine a particular bidding function. Truthful elicitation of surplus results in the bidding schedule $\Pi_b(q) = \Pi(q)$ and $C_b(q) = C(q)$. The market clears at $q^{opt}$. In this case I think $F$ pays $C$: $C(q^c)-C(q^{opt})$ and $C$ pays $F$: $\Pi(q^\pi)-\Pi(q^{opt})$.

Now imagine an individual in $C$. They have relatively low valuation for quality at $q^{opt}$. Reducing their valuation for all $q$ will be welfare enhancing if their reduction in transfers is smaller than their loss in welfare associated with the relative decline in quality. I believe that this should always be the case for anyone with lower than average marginal utility from quality at this point.

## "Proof of concept"

A firm, currently producing a product at $\hat{p}$ and $\hat{q}$ wants to change prices and quality by $dp$ and $dq$ respectively. The new price quality pair is $(\hat{p}+dp, \hat{q} + dq)$. It knows that

$\pi_p dp + \pi_q dq >0.$

Current (inframarginal) consumers are asked to vote on whether they approve or disapprove of the change. How do consumers vote?

Recall from Spence...
- That surplus is maximized when $\frac{\partial W}{\partial q} = \int_0^x P_q dv - c_q = 0$.
- But profits are maximized when $\frac{\partial \pi}{\partial q} = xP_q - c_q = 0$
- How does welfare change with quality? $\frac{\partial W}{\partial q} = \frac{\partial S}{\partial q} + \frac{\partial \pi}{\partial q} = \int_0^x P_q(v,q)dv - xP_q(x,q) + \frac{\partial \pi(x,q)}{\partial q}.$
- When $\frac{\partial \pi}{\partial q} = 0$ then change in welfare depends on relative magnitude of $\frac{1}{x}\int_0^x P_q dv$ and $x P_q$.
- $P_q$ is marginal valuation of quality by marginal consumer
- $\frac{1}{x}\int_0^x P_q dv$ is average valuation of quality at the margin over all in market

If put to a vote, the consumers with the MEDIAN marginal value of quality detremines outcome.
- when median value is above mean
- when median value is below mean
- how to tell whether it's better or worse than profit max?

what do consumers have to know in order to vote {approve/disapprove}


## Some notes on Farrell (1985)
"Owner consumers and efficiency"--this paper investigates the implications of the fact that consumers are also shareholders of a firm.

**Key takeaway**: "If each household owns the same fraction of a firm as its share of consumption, shareholders unanimously want marginal-cost pricing. Otherwise, profits are overemphasised relative to consumer surplus"

**I know about this paper from a different one**: Gans, Leigh, Schmalz and Triggs--"Inequality and market concentration, when shareholding is more skewed than consumption"

Model:
- Single firm
- COntinuum of consumers $i \in [0,n]$
- Identicial utilities $u(x_i)$, non decreasing function of firms' output
- Per unit price of the good is $p$
- Consumers have equity share $\alpha_i$ in the firm, assuming $\alpha_i$non-decreasing in $i$ for simplicity
- Let $X + \int_0^n x_i di$
- Firm has costs $C(X)$
- Value of $i$'s shareholding is $\alpha_i \pi = \alpha_i (pX - C(X))$
- Consider owner-consumer preferences over $p$

**What do ppl optimize?**

Consumers solve $max_{x_i} u(x_i) - px_i$ which gives demand function $x_i(p) = u'^{-1}(p)$ and a surplus $v(p)$.

When voting for firm's pricing policy, shareholders maximize $max_p v(p) + \alpha_i(pnx(p) - C(nx(p)))nx'(p)$ (first equality comes from Roy's identity)

**Remarks:**

## Some notes on Spence (good to have)
"Price signals carry marginal information, while averages or totals are required in locating the optimum."

3 main takeaways of paper:
1) Under monopoly and monopolistic competition, product characteristics are not optimally set under pressure of market force
2) Regulation has many informational issues when price and quality are both decision variables
3) Rate of return regulation may have attractive properties

Consider firm contemplating a small increase in quality. Assume that each consumer buys one unit of the good.
- Increase in quality increases costs by $\Delta c$
- Increase in quality increases dollar benefits of product to marginal consumer by $\Delta p(x)$.
- Firm increases revenues by $x \Delta p(x)$ where $x$ is number of purchasers
- Increase is desirable for firm if $x\Delta p(x) > \Delta c$
- **BUT!!!!!**
- $x \Delta p(x)$ is not an accurate measure of social benefits of increase in quality
- Quality increase is desirable if average benefit $\frac{1}{x} \int_0^x \Delta p(v) dv$ is greater than average cost $\Delta c / x$.
- Or, equivalently total benefits greater than total cost
- So, increase corresponds to social benefit only if the marginal consumer is the average consumer, i.e. $\frac{1}{x}\int_0^x \Delta p(v) dv = \Delta p(x)$.

:::info
What distinguishes the regulatory duopoly in degree if not in kind, is the severe informational problem facing the regulatory authority. In addition to the familiar difficulty of knowing costs, there are two additional informational problems.

One is simply measuring quality. This problem has often been noted, to which I have nothing to add. But there is a second and, I think, equally serious problem relating to demand.
:::
On the demand problem....
- CS is defined by $\int_p^\infty D(v, q)dv$ = constant.
- But implied schedule of prices and qualities is difficult to compute
- Suppose that firm proposes changes in price and quality $dp$ and $dq$---going off info that profits will increase ($\pi_p dp + \pi_q dq > 0$)
- Regulator has to assess whether consumer surplus also rises, i.e. whether $dS = -D dp + \left(\int_p^\infty D_q(v,q)dv \right)dq >0$.
- Can rewrite as $\left(\left(\int_p^\infty D_q(v,q)dv \right)/D \right)dq >dp$ which says the average valuation of quality change has to exceed the price increase $dp$
- "this information is not conveyed by prices, or local experiments...The implication of this line of reasoning is that regulatory agencies facing firms with discretionary control over aspects of product quality require nonmarket info to evaluate changes in prices and quality." (!!!!!)
- Spence is pretty pessimistic, but suggests some ways to generate this type of info
- consumer surveys "suggests themselves as a starting point"
- "shifts in product or service quality affect inframarginal consumers. These people must be consulted if a correct evaluation of a quality change is to be made."
- Also considers some second-best alternatives: "rate of return regulation" may be a reasonable second-best
- Basic idea here---a rate of return constraint can cause the substitution of capital for other inputs may partially compensate for firm's tendency to undersupply quality
- But if quality is labor-using, rate of return regulation may exacerbate the quality problem

:::success
Ideally the regulatory authority would manage price-quality tradeoffs by confronting the firm, on behalf of consumers, with a reaction function that reflects rates of substitution between price and quality on the demand side of the market. But these rates of substitution are difficult to determine, because computing them requires knowledge of the value attached to quality by the full range of marginal consumers.
:::

## More mech design set up

Social planner doesn't know quality types and firm cost. $N+1$ agents. $N$ consumers with types $\theta_i \in [0,1]$ where $\theta_i$ is $i$'s value for quality. 1 firm.
- Consumers have $U(x, Q, \theta)$
- Firm gets profit $\Pi(x,Q)$

1. Turn on uncertainty about cost of quality
2. 2 classes of agents (firm and consumers)
3. Can't have individualized taxes and transfers

## Platform stuff

### conceptual stuff
people don't like uniform quality---not

mussa rosen

80s style price theory or nothing....haven't even seen real regulatory papers about this...laffont and tirole boook...

### Veiga and Weyl

- Read Sheshinski

In selection markets there's an additional sorting incentive when firms choose quality---"The per-customer sorting effect on profit of a marginal
increase in quality is the ratio of two terms. The denominator is
marginal consumers’ surplus. The numerator is the covariance,
among marginal consumers, between consumers’ marginal willingness to pay for quality and consumers’ cost to the firm.""

- the covariance term captures the extent to which those most attracted by quality are also those particularly costly
- sorting requires multidimensional types and endogenous quality
- monopoly's sorting incentive is socially optimal because "marginal consumers are indifferent to purchasingthe good, so the monopolist internalizes all relevant social welfare"
- but when firms compete, quality is inefficiently provided a la Rothschild Stiglitz

### Veiga Weyl White
A monopolistic platform chooses vector of platform characteristics $\rho$ and a special characteristic $p$. Variable $p$ is viewed as harmful by all consumers and always beneficial to platofrm.

This paper offers a model and some basic insights into the design and deployment of a monopoly platform in which (i) users make het- erogeneous contributions to platform success, and (ii) they are attracted by the characteris- tics of other platform users rather than just their number. We hope that future research provides a more complete picture of dynamic platform strategies in environments with these traits. We also hope that researchers will analyze platform competition in this setting. In analyzing analo- gous selection markets, Veiga and Weyl (2016) find that competition can give rise to socially harmful “cream-skimming,” which may be of policy concern for the regulators of platforms

## Notes 06/26

Why not just auction of control rights?

Moral of cap and trade models is that you can give them to anyone and marginal val of control rights can be allocated by Coase theorem

Different controls allocated to diff people?

Even in cap and trade, not all pollution is equal.

From Benji to Zoë from last time
:::warning
I think the confusion today was due to the fact that it was a pure exchange economy (quality 1 vs. 2) with linear indifference curves. So the price is only constrained by the relative valuations (not pinned down) and that the outcome is going to be a corner solution with one or the other players having all of one or the other good. So the planner would want to choose an initial allocation which is the optimal corner of the edgeworth box (this is assigning total control of one or the other qualities). Otherwise if the planner had limited information they would want to pick initial endowments so that the majority of aspects are under control by the efficient provider.
:::

- Hard to get a trade model without saying that more control plateaus (concavity)
- So maybe just think about this as an auction

Multiple good auction with heterogenous?

One seller and two buyers? (Gov + consumer and firm)

One seller (firm) and buyer (consumer)

Main information problem is that we don't know how people value different quality dimensions

Different consumers like different levels of quality

There's some tradeoff between how much someone iswililng to pay for quality, how much bargaining someone is willing to do, almost want to make an exact quality for each customer that

Spence distortion goes away if you can perfectly discriminate on quality---but we can't do that---so what are the products that allow me to maximally discriminate across qualities?

Could mandate that Facebook offer tranches of privacy, and that people pay for privacy?

Within each tranch you're still distorting, there would be some tradeoff....number of quality tiers... the more fine you make it the closer the average will be to the marginal to each group. But it's more expensive to have tranches.

Why do some platforms have lots of price tiers (spotify, chess.com) but not google or facebook for privacy?

Obvious solution: take firm qualities and force firms to sell control of a particular quality of their good to consumers. If it's worth it more to consumers vs. firms to have control over this quality, they should be able to find a price

Different Welfare weights for consumers vs. firm??

Propose regulation, propse a cash transfer. If everyone's on board, we do it.

Auction with some uncertainty about true valuation.
- If you broke up firms and they compete, you would eliminate Sdisortions by this much, change profits by this much, change welfare by this much....
- Or if you proposed an auction, you would get at least that much welfare.
- like electricity markets?
- if quality is a continuous value. firms have optimal values $q^f$ and consumers have $q^c$ optimal value. will end up with something in between those two points
- social planner asks for how much their WTP for their optimal point---but realy absolute values don't matter, only the change matters
- A differntial equation that says: at any point, if I were to increase how much I said I valued it, it makes it so that that point is more likely to happen---if I were at my exact WTP that should be negative deviation. If I were at a particular point and I decrease that value, it makes something on either side more likely---it would be worse if I preferred the point that I'm at by some positive amount
- social planner maximizes sum of CS and profits... (MAYBE WELFARE WEIGHTS)

## Quick Pass?

Consumers have utility over quality and quantity, where $x_1$ is quantity and $Q$ is a composite quality score.

$U(Qx_1) = (Qx_1)^{\frac{\epsilon-1}{\epsilon}}$
$Q(q_1,q_2)=f_1(q_1)+f_2(q_2)$

Note: quality dimensions are separable---no cross partials between dimensions.

Preferred quality level for consumer, for each dimension $i$ of $Q$.

$q_i^C = \max_{q_i} CS(q_i)$
$q_i^F = \max_{q_i} \Pi(q_i)$

Assign ownership of quality dimensions to firm or ocnsumer.
$\alpha = (\alpha_1, \alpha_2)$ with $\alpha_i \in \{c, f\}$

Consumer surplus: $\Delta CS_i = CS(q_i^C)-CS(q_i^F)$

Producer Cost: $\Delta \Pi_i = \Pi(q_i^C)-\Pi(q_i^F)$

From social planner perspective---wants all absolute trades to be executed. But if you get a trade where $\Delta A_i > 0$.

If $\Delta CS_i > -\Delta \Pi_i$, then consumers should make the decision on $q_i$

Consumer advantage: $\Delta A_i = \Delta CS_i -\Delta \Pi_i$

Without loss of generality assume: $\frac{\Delta CS_1}{\Delta \Pi_1} > \frac{\Delta CS_2}{\Delta \Pi_2}$

If you had full info, you could just rank the dimensions. You'd want anything with ratio >1 to be decided by consumer and anything with ratio <1 to be decided by the consumer. You would directly allocate, firm do these ones, consumers do these ones. But you want to find some way so that if you gave 2 out of 3 dimensions to consumers, you need a way that they can successfully trade to get the *right* dimensions under control.

In this case consumers want to control good 1 relatively more than good 2. In this discrete case

:::info
Note on absolute vs relative advantage

If consumers can only do 1 dimension, you would care more about the absolute difference.
:::

We just want to say there is some cost function for each $q_i$. Some cost to firm, some consumer surplus to be gotten. WTS we can rank good 1 and good 2 in terms of (consumer surplus)/(additional cost to firm). Or some other measure that compares the two.

What if $q_i = \sum_{\omega} q_i(\omega)$
Within each quality dimension, there are a bunch of equally weighted aspects. By choice of units, these are all equally valued as quality components within these categories.

Imagine
$\frac{\Delta CS_1(\omega)}{\Delta \Pi_1(\omega)} > 1> \frac{\Delta CS_2(\omega)}{\Delta \Pi_2(\omega)}$

Social planner can give out control rights over one aspect of a quality dimension. Consumer 1 is willing to trade of they have any control rights for quality 2, they should be able to successfully trade for quality 1. Because they relatively prefer it, and as long as there are enough goods blah blah they'll be able to make some exchanges.

Goal of social planner: Divvy up enough control so that after trading

::: info

Cap and trade mechanism... essentially puts a price on carbon. Could be informative to look at models of carbon trading schemes. The information from cap and trade gives a price, and therefore input for regulators.

:::

1. Say clearly how mechanism design framework doesn't work? Should be willing to put some weight on transfer to ocnsumers?
2. Insights from cap and trade schemes?

## Connection to HBR Porter article
Need more flexibility about how your
Shorter term holders vs. long term owner/investor long term holders, then you want to maximize how dynamic you can be, an agility thing

Longer-term strategy, you want more coordination within segments

Shorter term goals want more flexibility

Form of governance changes horizon Japan and Germany --> longer term goal vs. other places --> shorter term goal

:::info
Aside: relates sort of to tenure voting.

But this is weird! By revealed preference, the people who have been around for a while like what we're doing! I'm always getting the status quo.
:::

## High level - returning to DBS

Continuum of issues of quality. Going to arbitrarily allocate some fraction of control to one or other group. Then people can trade freely across control.

On a continuum, we can get all possible combinations and some effect of price.

Coase theorem would suggest that if we allow ex post bargaining, we'll always end up in the efficient allocation. But depending how we allocate, someone will always be better or worse off.

Could have a bidding thing, have to pay for the rights. Need a unit to price implicitly all of the qualities. Then could get some sense of surplus generated for consumer group who otherwise wouldn't have. Theoretically, everyone's better off, gains from trade, accurately ranked, partial information leads to full info.

In finite case you can only get specific couplings of prices.

## Thoughts on adapting DBS set up

Consumers have utility over $x_0$ and $x_1$. $x_1$ is the quantity of given good. Suppose there are two dimensions of quality $q_1$ and $q_2$. For example, if the firm is a technology platform, $q_1$ might be degree of privacy protection and $q_2$ might be graphic design.

$U(x_0, Qx_1) = x_0+\frac{\epsilon}{\epsilon-1}(Qx_1)^{\frac{\epsilon-1}{\epsilon}}$
$Q(q_1,q_2)=f_1(q_1)+f_2(q_2)$
$Q = \int_0^1f_i(q_i)di$

$Q$ is quality weight, so $Q x_1$ is a quality weighted quantity. $Q$ is a composite quality score, weighted sum over different dimensions of quality.

Marginal dollar val of increasing quality should be equal for q1 and q2.

Conjecture in discrete quality dimension case: consumers would be better at maximizing welfare for goods with "steep" $f_i$

In the continuum case, we can rank $f_i$ by the ratio of average benefit to cost of improvement. There may be some $i^*$ at which the cost to producer and benefit to consumer are equal.

Social planner would have to determine $i^*$ or some other primitive... would this information be revealed to them?

Could you cap the level that would allow you to bound $i^*$?

Something like this could limit the number of dimensions that you need to measure.... essentially creating a price for ownership of some dimension of quality

Could think of the sorting process like trade: could tell people arbitrarily 30% of decisions are made by consumers and 70% are made by board. Then the board is always going to prefer to give up some inexpensive quality dimension for something that it thinks is very expensive which the consumers might want.

Definition. WEIRD AUCTION. Both sides rank qualities they care about.

Could add a high cost in event of intractable disagreement.

More like trade---should get a comparative advantage between firms and consumers. The thing that's valued by both, really what matters is how its relatively more/less valued relative to other qualities