{"id":1516,"date":"2023-06-06T19:13:28","date_gmt":"2023-06-06T10:13:28","guid":{"rendered":"https:\/\/saraheee.com\/?p=1516"},"modified":"2023-07-02T22:46:24","modified_gmt":"2023-07-02T13:46:24","slug":"game-theory-9-chap15-the-lemon-market","status":"publish","type":"post","link":"https:\/\/saraheee.com\/ko\/2023\/06\/game-theory-9-chap15-the-lemon-market\/","title":{"rendered":"Game Theory 9 \u2013 chap15. The lemon market"},"content":{"rendered":"

4. Bayesian Games
4.3 Applications: Lemons, Auctions, and Information Aggregation (Chapter 27)
Markets and Lemons<\/p>\n\n\n\n

Application 1: Lemon Market (Akerlof, 1970)<\/h3>\n\n\n\n

“Information asymmetry may bring about a market collapse<\/mark><\/strong>“<\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

George Akerlof, Nobel Iaureate in 2001<\/p>\n\n\n\n

Bilateral trade between a used-car seller and a buyer
Only the seller knows the exact quality of the car<\/p>\n\n\n\n

Quality<\/td>Buyer<\/td>Seller<\/td><\/tr>
H<\/td>10,000<\/td>8,000<\/td><\/tr>
L<\/td>5,000<\/td>3,000<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

As \\(v_{b} > v_{s}\\), the used car should be traded if the buyer knows the exact quality<\/p>\n\n\n\n

Suppose the buyer believes Pr(low quality) > 0.4 and makes a price offer to the seller<\/p>\n\n\n\n

10,000\u2219Pr(H) + 5,000\u2219Pr(L) < 8,000<\/p>\n\n\n\n

In this case, it is bad news to the buyer that his offer is accepted by the seller<\/p>\n\n\n\n

Consider a bilateral trade between a used-car seller (Freddie) and a buyer (Jerry)
Only Freddie knows the exact quality of the car<\/p>\n\n\n\n

Quality<\/td>Jerry<\/td>Freddie<\/td><\/tr>
Peach<\/td>3,000<\/td>2,000<\/td><\/tr>
Lemon<\/td>1,000<\/td>0<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Jerry knows only that the car is peach with probability q \u2208 (0, 1)
Both traders decide whether to trade (T) or not (N) at a given market price p<\/mark><\/strong><\/p>\n\n\n\n

J \\ F<\/td>\\(T^{P}\\)<\/td>\\(N^{P}\\)<\/td>J \\ F<\/td>\\(T^{L}\\)<\/td>\\(N^{L}\\)<\/td><\/tr>
T<\/td>3000 – p, p<\/td>0, 2000<\/td>T<\/td>1000 – p, p<\/td>0 ,0<\/td><\/tr>
N<\/td>0, 2000<\/td>0, 2000<\/td>N<\/td>0, 0<\/td>0, 0<\/td><\/tr>
<\/td>Peach (q)<\/td><\/td><\/td>Lemon (1-q)<\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

In Bayesian Normal Form, \\(S_{J}\\), \\(S_{F}\\) = {\\(T^{P}T^{L}, N^{P}T^{L}, T^{P}N^{L}, N^{P}N^{L}\\)}<\/p>\n\n\n\n

To investigate which type of the car is traded in equilibrium, we first convert the given game into Bayesian normal form:<\/p>\n\n\n\n

Jerry \\ Freddie<\/td>\\(T^{P}T^{L}\\)<\/td>\\(N^{P}T^{L}\\)<\/td>\\(T^{P}N^{L}\\)<\/td>\\(N^{P}N^{L}\\)<\/td><\/tr>
T<\/td>3000q + 1000(1-q) – p
p<\/td>		(1000-p)(1-q)
2000q+p(1-q)<\/td>		3000q-pq
pq<\/td>		0
2000q<\/td><\/tr>
N<\/td>0, 2000q<\/td>0, 2000q<\/td>0, 2000q<\/td>0, 2000q<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

(1) When is only the lemon traded in eqbm?<\/mark><\/p>\n\n\n\n

(T, \\(N^{P}T^{L}\\))
1) (1000-p)(1-q) \u2265 0 \u21d2 p \u2264 1000
2) 2000q+p(1-q) \u2265 p (\u2235 p \u2265 pq and 2000q \u2265 2000q + p(1-q)) \u21d2 p \u2264 2000
By 1) and 2), p \u2264 1000.<\/p>\n\n\n\n

(2) When are both cars traded in eqbm?<\/mark><\/p>\n\n\n\n

(T, \\(T^{P}T^{L}\\))
1) 3000q + 1000(1-q) – p \u2265 0
\u21d4 3000q + 1000(1-q) \u2265 p \u2265 2000 \u21d2 q \u2265 1\/2
2) p \u2265 2000q + p(1-q) (Always satisfied because p \u2265 2000)<\/p>\n\n\n\n

<\/p>\n\n\n\n