2. In the same model , how much consumers' surplus would be crated by randomly assigning buyers

2. In the same model , how much consumers' surplus would be...

Key Concepts

Comparative Efficiency of Assignment Methods

Comparative efficiency involves evaluating and contrasting different assignment methods—for instance, random assignment versus an optimal matching rule (such as one that might yield a surplus of x???). By comparing these methods, one can determine which assignment mechanism results in a higher consumer surplus and overall greater market efficiency.

Optimal Matching

Optimal matching is an assignment method that pairs buyers with sellers in a way that maximizes the total surplus, often by matching high-valuation buyers with low-cost sellers. This approach seeks to enhance overall market efficiency and maximize consumer surplus compared to non?optimized methods.

Consumer Surplus

Consumer surplus is an economic measure representing the difference between what consumers are willing to pay for a good or service and what they actually pay. It reflects the net benefit to consumers from participating in the market and is a key indicator of market welfare and efficiency.

Random Assignment

Random assignment refers to a matching process where buyers are paired with sellers without considering differences in buyer valuations or seller costs. This method does not take into account optimal matching criteria and typically results in a distribution of surplus that is not maximized across all transactions.

Transcript

00:03 So let's on x1, x2, up to xn, b the random variable is denoting n independent bits for an item that is for sale.

00:21 So we are going to suppose xi is uniformly distributed.

00:27 Sorry, is uniformly distributed on this interval.

00:39 So the probability density function of xi is given by this.

00:48 It is uniformly distributed.

00:51 We are going to get f of xi here x to be 1 over b minus a.

01:00 Why we know this is b and this is a.

01:03 So we have 200 minus 100.

01:07 So we have 1 over 100.

01:16 So the cumulative distribution function, which is the cdf, we have the cdf x i giving us big f of x i of x to be equal to x minus a over b minus a so we are going to get x minus hundred over 200 minus hundred so we have hundred over here where x belong to this interval we have hundred from a 200 and so we are going to let let y be the highest bit so we are going to get y to be equal to the max x i sorry x1 up to x n so x1 up to xn over here so our main name is to find for the expected value of x so you want to find for this over here so now the cumulative distribution function of y can be obtained as so the cdf of y so cdf of y can be obtained as f of y to be equal to p of y less than or equal to small y so we have p max x i where x i where i is from one up to n less than or equal to y so from here we are going to get an p of x i where x i is from one up to n less than or equal to y so from here we are going to get a p of x one less than or equal to y to p of x1 less than or equal to y um so we are doing this because um earlier we know um x i are independent so we know the x i are independent yeah so if they are independent it means we are going to get the product so we are going to use this instead so so i from 1 to n, the product of this.

04:12 So we have y minus 100, all over 100.

04:25 So f of y, which is the cumulative, will be equal to y minus 100, or divided by 100, or to the power n.

04:39 Where y belongs to 100, 200.

04:46 So i'm just enter one.

04:48 So now we can copy the probability density function of y by differentiating the cdf with respect to y...

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