21.12 Product Quality, Durability, Warranties
Horizontal Differentiation
\[ U = V -p - t (\theta - a)^2 \]
Vertical Differentiation
\[ U_B = \theta s_B - p_B \\ U_A = \theta s_A - p_A \]
Assume that product B has a higher quality
\(\theta\) is the position of any consumer on the vertical differentiation line.
When \(U_A < 0\) then customers would not buy
Point of indifference along the vertical quality line
\[ \theta s_B - p_B = \theta s_A - p_A \\ \theta(s_B - s_A) = p_B - p_A \\ \bar{\theta} = \frac{p_B - p_A}{s_B - s_A} \]
If \(p_B = p_A\) for every \(\theta\), \(s_B\) is preferred to \(s_A\)
- K. S. Moorthy (1988)
\[ \pi_A = (p_A - c s_A^2) (Mktshare_A) \\ \pi_B = (p_B - cs_B^2) (Mktshare_B) \\ U_A = \theta s_A - p_A = 0 \\ \bar{\theta}_2 = \frac{p_A}{s_A} \]
- Wauthy (1996)
\(\frac{b}{a}\) = such that market is covered, then
\[ 2 \le \frac{b}{a} \le \frac{2s_2 + s_1}{s_2 - s_1} \]
for the market to be covered
In vertical differentiation model, you can’t have both \(\theta \in [0,1]\) and full market coverage.
Alternatively, you can also specify \(\theta \in [1,2]; [1,4]\)
\[ \theta \in \begin{cases} [1,4] & \frac{b}{a} = 4 \\ [1,2] & \frac{b}{a} = 2 \end{cases} \]
Under Asymmetric Information
Adverse Selection: Before contract: Information is uncertain
Moral Hazard: After contract, intentions are unknown to at least one of the parties.
Alternative setup of Akerlof’s (1970) paper
Used cars quality \(\theta \in [0,1]\)
Seller - car of type \(\theta\)
Buyer = WTP = \(\frac{3}{2} \theta\)
Both of them can be better if the transaction occurs because buyer’s WTP for the car is greater than utility received by seller.
- Assume quality is observable (both sellers and buyers do know the quality of the cars):
Price as a function of quality \(p(\theta)\) where \(p(\theta) \in [\theta, 3/2 \theta]\) both parties can be better off
- Assume quality is unobservable (since \(\theta\) is uniformly distributed) (sellers and buyers do not know the quality of the used cars):
\[ E(\theta) = \frac{1}{2} \]
then \(E(\theta)\) for sellers is \(1/2\)
\(E(\theta)\) for buyer = \(3/2 \times 1/2\) = 3/4
then market happens when \(p \in [1/2,3/4]\)
- Asymmetric info (if only the sellers know the quality)
Seller knows \(\theta\)
Buyer knows \(\theta \sim [0,1]\)
From seller perspective, he must sell at price \(p \ge \theta\) and
From buyer perspective, quality of cars on sale is between \([0, p]\). Then, you will have a smaller distribution than \([0,1]\)
If \(E[(\theta) | \theta \le p] = 0.5 p\)
Buyers’ utility is \(3/4 p\) but the price he has to pay is \(p\) (then market would not happen)
21.12.1 Akerlof (1970)
- This paper is on adverse selection
- The relationship between quality and uncertainty (in automobiles market)
- 2 x 2 (used vs. new, good vs. bad)
\(q\) = probability of getting a good car = probability of good cars produced
and \((1-q)\) is the probability of getting a lemon
Used car sellers have knowledge about the probability of the car being bad, but buyers don’t. And buyers pay the same price for a lemon as for a good car (info asymmetry).
Gresham’s law for good and bad money is not transferable (because the reason why bad money drives out goo d money because of even exchange rate, while buyers of a car cannot tell if it is good or bad).
21.12.1.1 Asymmetrical Info
Demand for used automobiles depends on price quality:
\[ Q^d = D(p, \mu) \]
Supply for used cars depends on price
\[ S = S(p) \]
and average quality depends on price
\[ \mu = \mu(p) \]
In equilibrium
\[ S(p) = D(p, \mu(p)) \]
At no price will any trade happen
Example:
Assume 2 groups of graders:
First group: \(U_1 = M = \sum_{i=1}^n x_i\) where
\(M\) is the consumption of goods other than cars
\(x_i\) is the quality of the i-th car
n is the number of cars
Second group: \(U_2 = M + \sum_{i=1}^n \frac{3}{2} x_i\)
Group 1’s income is \(Y_1\)
Group 2’s income is \(Y_2\)
Demand for first group is
\[ \begin{cases} D_1 = \frac{Y_1}{p} & \frac{\mu}{p}>1 \\ D_1 = 0 & \frac{\mu}{p}<1 \end{cases} \]
Assume we have uniform distribution of automobile quality.
Supply offered by first group is
\[ S_2 = \frac{pN}{2} ; p \le 2 \]
with average quality \(\mu = p/2\)
Demand for second group is
\[ \begin{cases} D_2 = \frac{Y_2}{p} & \frac{3 \mu}{2} >p \\ D_2 = 0 & \frac{3 \mu}{2} < p \end{cases} \]
and supply by second group is \(S_2 = 0\)
Thus, total demand \(D(p, \mu)\) is
\[ \begin{cases} D(p, \mu) = (Y_2 + Y_1) / p & \text{ if } p < \mu \\ D(p, \mu) = (Y_2)/p & \text{ if } \mu < p < 3\mu /2 \\ D(p, \mu) = 0 & \text{ if } p > 3 \mu/2 \end{cases} \]
With price \(p\), average quality is \(p/2\), and thus at no price will any trade happen
21.12.1.2 Symmetric Info
Car quality is uniformly distributed \(0 \le x \le 2\)
Supply
\[ \begin{cases} S(p) = N & p >1 \\ S(p) = 0 \end{cases} \]
Demand
\[ \begin{cases} D(p) = (Y_2 + Y_1) / p & p < 1 \\ D(p) = Y_2/p & 1 < p < 3/2 \\ D(p) = 0 & p > 3/2 \end{cases} \]
In equilibrium
\[ \begin{cases} p = 1 & \text{ if } Y_2< N \\ p = Y_2/N & \text{ if } 2Y_2/3 < N < Y_2 \\ p = 3/2 & \text{ if } N < 2 Y_2 <3 \end{cases} \]
This model also applies to (1) insurance case for elders (over 65), (2) the employment of minorities, (3) the costs of dishonesty, (4) credit markets in underdeveloped countries
To counteract the effects of quality uncertainty, we can have
- Guarantees
- Brand-name good
- Chains
- Licensing practices
21.12.2 Spence (1973)
Built on (Akerlof 1970) model
Consider 2 employees:
Employee 1: produces 1 unit of production
Employee 2: produces 2 units of production
We have \(\alpha\) people of type 1, and \(1-\alpha\) people of type 2
Average productivity
\[ E(P) = \alpha + 2( 1- \alpha) = 2- \alpha \]
You can signal via education.
To model cost of education,
Let E to be the cost of education for type 1
E/2 to be the cost education for type 2
If type 1 signals they are high-quality worker, then they have to go through the education and cost is E, and net utility of type 1 worker
\[ 2 - E < 1 \\ E >1 \]
If type 2 signals they are high-quality worker, then they also have to go through the education and cost is E/2 and net utility of type 2 worker is
\[ 2 - E/2 > 1 \\ E< 2 \]
If we keep \(1 < E < 2\), then we have separating equilibrium (to have signal credible enough of education )
21.12.3 S. Moorthy and Srinivasan (1995)
Money-back guarantee signals quality
Transaction cost are those the seller or buyer has to pay when redeeming a money-back guarantee
Money-back guarantee does not include product return (buyers have to incur expense), but guarantee a full refund of the purchase price.
If signals are costless, there is no difference between money-back guarantees and price
But signal are costly,
Under homogeneous buyers, low-quality sellers cannot mimic high-quality sellers’ strategy (i.e., money-back guarantee)
Under heterogeneous buyers,
when transaction costs are too high, the seller chooses either not to use money-back guarantee strategy or signal through price.
When transaction costs are moderate, there is a critical value of seller transaction costs where
below this point, the high-quality sellers’ profits increase with transaction costs
above this point, the high-quality sellers’ profits decrease with transaction costs
Uninformative advertising (“money-burning”) is defined as expenditures that do not affect demand directly. is never needed
Moral hazard:
- Consumers might exhaust consumption within the money-back guarantee period
Model setup
| High-quality sellers (\(h\)) | Low-quality sellers (\(l\)) | |
|---|---|---|
Cost \(c_h > c_l\) |
\(c_h\) | \(c_l\) |