How could the economic cost of the world not speaking the same language be estimated? While languages are fun and add color and flair to cultures and make the world a more interesting place, they also have a cost. Beyond the monetary costs of translation/editing/proofreading, interpretation, multi-language signage and so on, there are well-documented instances when minor translation errors have caused serious international crises resulting in loss of life and/or of business.

What are some methods of estimating the (current or historical) global cost of language diversity? (I would like to limit the question to just the cost - language diversity also has benefits, such as tourism, or economic activity pertaining to language training.)
To measure the costs of different people speaking different languages, researchers use a "linguistic distance" metric, see for example this paper. However, measuring the cost of linguistic diversity appears to be challenging. Some effects are shown along the following lines.

Impact of linguistic distance on international trade

In this paper, they construct new series for common native language and common spoken language for 195 countries, which they use together with series for common official language and linguistic proximity in order to draw inferences about 


the aggregate impact of all linguistic factors on bilateral trade, 
the separate role of ease of communication as distinct from
ethnicity and trust, and 
the contribution of translation and interpreters to ease of communication.


Results show that the impact of linguistic factors, all together, is at least twice as great as the usual dummy variable for common language, resting on official language, would say.

Impact of linguistic distance on international migration

In this paper they examine the importance of language in international migration from multiple angles by studying the role of linguistic proximity, widely spoken languages, linguistic enclaves and language-based immigration policy requirements. To this aim they collect a unique data set on immigration flows and stocks in 30 OECD destinations from all world countries over the period 1980–2010 and construct a set of linguistic proximity measures. 

Results: Migration rates increase with linguistic proximity and with English at destination. Softer linguistic requirements for naturalisation and larger linguistic communities at destination encourage more migrants to move. Linguistic proximity matters less when local linguistic networks are larger.

This paper on the Costs of Babylon—Linguistic Distance in Applied Economics
may also be relevant.

Impact of linguistic distance on literary translations

Interestingly, one of the latest Freakonomics Radio episode is called "Why Don't We All Speak the Same Language? (Earth 2.0 Series)". There are 7,000 languages spoken on Earth. What are the costs — and benefits? They quote (with references)


  There's also the roughly \$40 billion a year we spend on global
  language services — primarily translation and interpretation. And
  another \$50-plus billion a year spent learning other languages.


Useful references


Victor Ginsburgh and Shlomo Weber, Culture, languages, and
economics, in Victor Ginsburgh and David Throsby, eds.,  Handbook
of the Economics of Art and Culture, vol. 2, Amsterdam: Elsevier,
2014.
Victor Ginsburgh and Shlomo Weber, How Many Languages Do We Need? The Economics of Linguistic Diversity, Princeton, NJ: Princeton University Press, 2011