{"id":11562,"date":"2024-10-04T15:16:46","date_gmt":"2024-10-04T15:16:46","guid":{"rendered":"https:\/\/mycryptomania.com\/?p=11562"},"modified":"2024-10-04T15:16:46","modified_gmt":"2024-10-04T15:16:46","slug":"what-is-the-differentiated-bertrand-model","status":"publish","type":"post","link":"https:\/\/mycryptomania.com\/?p=11562","title":{"rendered":"What is the Differentiated Bertrand Model?"},"content":{"rendered":"

Photo by Campaign Creators<\/a> on\u00a0Unsplash<\/a><\/p>\n

The Differentiated Bertrand Model<\/strong> is a theoretical model used in economics that explores competition and pricing behavior among firms producing, as inferred, differentiated products. The Bertrand model <\/strong>alone, on the other hand, focuses on homogeneous products.<\/p>\n

In the differentiated Bertrand model, firms compete for imperfect substitutes. So, consumers have preferences for certain product characteristics, and as a result firms have some degree of market power, letting them charge prices above marginal cost. The more differentiated the products are, the higher the markup over marginal\u00a0costs.<\/p>\n

The model is often used to explore the entry and exit of firms, as well as deriving profit-maximizing prices, quantity produced, and revenue. In this article, we\u2019ll focus on those applications, namely how to derive price, quantity, and revenue given the marginal costs and demand functions of two firms. There\u2019s more to the model than such, but this both provides the best groundwork and is the most common application of the differentiated Bertrand model among for students.<\/p>\n

As an example, let\u2019s look at a market with two firms called Firm 1<\/strong> and Firm 2<\/strong>. Both firms have a marginal cost of 0 and respectively have demand functions of q1 = 72\u20133p1 + 2p2 and q2 = 72\u20133p2 + 2p1. This means that the pricing behavior of Firm 1 affects Firm 2, and vice\u00a0versa.<\/p>\n

To calculate the optimal price, let\u2019s first find marginal revenue, since marginal revenue = marginal cost at the competitive equilibrium:<\/p>\n

For firm\u00a01:<\/p>\n

TR1 = p1*q1 = p1(72\u20133p1 + 2p2) = 72p1\u20133p1\u00b2 +\u00a02p1p2<\/p>\n

*to calculate marginal revenue, we\u2019ll take the partial derivative of TR (dTr1\/dP1).<\/em><\/p>\n

MR1 = 72 \u2013 6p1 +\u00a02p2*now, solve for Firm 1\u2019s\u00a0price.<\/em>MR1 =\u00a0MC72 \u2013 6p1 + 2p2 =\u00a00-6p1 = -2p2 \u2013\u00a072p1 = 12 +\u00a01\/3p2<\/strong><\/p>\n

*reaction function; given any P2 price, we\u2019ll get P1\u2019s best response, e.g., it\u2019s profit maximizing price.<\/em><\/p>\n

For firm\u00a02:<\/p>\n

TR2 = p2*q2 = p1 (72\u20133p2 + 2p1) = 72p2\u20133p2\u00b2 +\u00a02p2p1MR1 (dtr2\/dp2) <\/em>= 72 \u2013 6p2 +\u00a02p172\u20136p2 + 2p1 =\u00a00-6p2 =\u00a0-2p1\u201372p2 = 12 +\u00a01\/3p1<\/strong><\/p>\n

*now, let\u2019s substitute Firm 1\u2019s reaction function into Firm 2\u2019s reaction function:<\/em><\/p>\n

p1 = 12 + 1\/3(12 +\u00a01\/3p1)p1 = 12 + 4 + 1\/9p1 = 16 +\u00a01\/9p18\/9p1 =\u00a0168p1 =\u00a0144p1 =\u00a018<\/strong><\/p>\n

Therefore, the profit-maximizing price for Firm 1 is 18. If we do the math for Firm 2, we\u2019ll get the same result given the symmetric demand curves and marginal costs. We can furthermore solve for quantity by plugging the profit maximizing price into the original demand functions. For Firm\u00a01:<\/p>\n

q1 = 72\u20133p1 +\u00a02p2q1 = 72\u20133(18) +\u00a02(18)q1 = 72- 54 +\u00a036q1 =\u00a054<\/strong>TR =\u00a0PQTR = (18)(54) =\u00a0972<\/p>\n

So, in sum, both firms will sell 54 goods for a price of $18, resulting gross revenue of\u00a0$972.<\/p>\n

That\u2019s all for our introduction to the differentiated Bertrand model! Let me know in the comments if you would like to see any additions to this article, or further articles on similar subjects.<\/p>\n

What is the Differentiated Bertrand Model?<\/a> was originally published in Coinmonks<\/a> on Medium, where people are continuing the conversation by highlighting and responding to this story.<\/p>","protected":false},"excerpt":{"rendered":"

Photo by Campaign Creators on\u00a0Unsplash The Differentiated Bertrand Model is a theoretical model used in economics that explores competition and pricing behavior among firms producing, as inferred, differentiated products. The Bertrand model alone, on the other hand, focuses on homogeneous products. In the differentiated Bertrand model, firms compete for imperfect substitutes. So, consumers have preferences […]<\/p>\n","protected":false},"author":0,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-11562","post","type-post","status-publish","format-standard","hentry","category-interesting"],"_links":{"self":[{"href":"https:\/\/mycryptomania.com\/index.php?rest_route=\/wp\/v2\/posts\/11562"}],"collection":[{"href":"https:\/\/mycryptomania.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mycryptomania.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"replies":[{"embeddable":true,"href":"https:\/\/mycryptomania.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11562"}],"version-history":[{"count":0,"href":"https:\/\/mycryptomania.com\/index.php?rest_route=\/wp\/v2\/posts\/11562\/revisions"}],"wp:attachment":[{"href":"https:\/\/mycryptomania.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11562"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mycryptomania.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11562"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mycryptomania.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11562"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}