區域經濟學-Quality Selection and R&D Competition - 經濟

來源: 學校期中考古題

科目: 區域經濟學

問題:

In the market of "alcohol", the consumer preference for the
"degree of liquor" on the alcohol is during 8%-48%, and the
consumers preference with liquor degree during 8% and 48% is
uniform distribution, and assume that every consumer buys only one
unit of this alcohol, and its utility of the consumer associated
with the preference q with purchased the liquor from firm i is
specified by U(x) = a - p(xi) - c(q-qi)^2, where a denotes the
reservation price of consuming the alcohol for the consumer
associated with the preference q, in which the reservation price
is identical for all the consumers with the preference over q in
[8%,48%], the marginal cost of producing the alcohol by each firm
is identical and given by k, then(40%)

(i), if the production on the degree of the alcohol for both firms
is constrained with qi in [8%, 48%], and both firms compete the
choice of the degree of alcohol (First stage), then price
determination(Second stage), please find the equilibrium prices,
the degree of the alcohol, and the profits with Nash game for both
firms;
(ii), if there is not any constrain for the degree of the alcohol
to be produced by the both firms, that is, the firm can produce
the alcohol with any degree, then follow the previous two-stage
game, both firms can determine both the degree of the alcohol as
well as prices to maximize their profits, then please find the
Nash equilibrium degree of the alcohol, price, and profit for both
firms.

(iii), how to interpret the equilibrium result between assumption
with constrain and non-constrain?



我的想法:

在練習題目的時候，對於題目中的變數一直搞不清楚
q指的是消費者偏好，那 U(x) = a - p(xi) - c(q-qi)^2中x指的是什麼呢？
我是把它想做廠商所生產的degree
但是這樣子的話，式子中的q-qi又代表什麼意義呢？
我自己的想法是qi是某個消費者的偏好
而是式子中的q是廠商所生產的degree
所以我用xi來代換q，而變成U(x) = a - p(xi) - c(xi-qi)^2
亦即若廠商生產xi的與消費者偏好qi差越多則效用會減少
接著才開始求(i)兩廠商競爭的解


求解過程如下：

先假設 x1 >= x2 >= 0

U(x2) = a - p(x2) - c(x2-q2)^2

q2 = x2 +- (a/c - p2/c)^(1/2)

再利用

U(x1) = a - p(x1) - c(x1-q1)^2

= a - p(x2) - c(x2-q1)^2

q1 = [p(x1) - p(x2)] / c(x1 - x2) + (x1 + x2)


目前是做到這
因為不確定想法對不對
想來請教一下版上的高手
因為我是自己跑去外系修課，所以沒人能請教
希望大家可以幫忙一下，感謝！！！！




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