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Gaurav G.
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Calculus
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Question:

John is conducting an experiment in which he is growing bacteria. The population of a bacteria undergoes exponential growth. If there are 100 bacteria at noon on 15th June 2016. 200 bacteria at 2pm on the same date. John needs 1600 bacteria. At what time should be stop the experiment?

Gaurav G.

Approach 1. The bacteria population doubled from 100 to 200 in a span of 2 hours (noon to 2 pm). Thus the bacteria population is doubling every 2 hours. Therefore, after another 2 hours (i.e. at 4pm) the population will reach 400. Again after 2 hours (i.e. 6pm) the population will reach 800. Then after 2 hours (i.e. at 8pm), it will again double to reach 1600. Thus John should stop the experiment at 8pm Approach 2. The general formula for exponential growth is given by $$y=y_{0}(e^{kt})$$. where t is the time, $$y_{0}$$ is the initial value which is equal to 100. At noon, t=0 thus at 2pm, $$t=2$$ , $$y=200$$, $$y_{0}=100$$, putting these values into the equation, we get $$200=100(e^{2k})$$ which equals $$2=e^{2k}$$ Taking log on both sides, we get $$\ln 2 = 2k$$ or $$(\ln 2)/2 = k$$ Plug the value of k from above into the general equation $$y=y_{0}(e^{t(1/2)(\ln 2})$$ which equals $$y=y_{0}(e^{(\ln 2) t/2})$$ $$y=y_{0}(2^{t/2})$$ (because $$e^{\ln x}=x$$ )...........(1) Now we want y to be 1600, and using $$y_{0}=100$$ substituting these values into equation 1, we get $$1600=100(2^{t/2})$$ $$16=(2^{t/2})$$ ....(2) we know $$2^4=16$$ therefore substituting that into equation 2, we get $$2^4=(2^{t/2})$$ As terms of LHS and RHS are equal and base is equal, there power is also equal (power can only be positive as time can never be negative) $$4=t/2$$ $$8=t$$ There, John should stop the experiment at 8pm

Economics
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Question:

Why do markets fail?

Gaurav G.

Markets are not perfect and like any other institution in the world, are bound to fail. Major reasons for failing of a market are: 1. Market Power: When a producer or a supplier starts charging more than the level that can be sustained by the market (e.g. in a monopoly), inefficiency arises. This inefficiency can lead to failing of markets 2. Incomplete information: In cases where information available to all the stakeholders is not the same, the stakeholder with more information can use that information to his/her advantage which can lead to inefficient exchange and thus lead to failing of market. e.g. In the second hand car market, the seller has more information than the buyer. This is also the reason why insider trading is illegal 3. Externalities: Consumption or production can have indirect effect on a lot of things which are not reflected in the price. eg. Air is polluted by factories but the damage done is not reflected in the price. This is the reason carbon tax has come into picture these days 4. Public Goods: Firms are not willing to invent and are ready to copy technology developed by other firms. This is the reason patent law was brought into the economic scenario

Corporate Finance
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Question:

What is liquidity?

Gaurav G.

Liquidity in simple terms means "how easy is it for a person to get his hands on the cash". It effectively means how easy is it for me to convert something that I own into "cash in hand". In a way, it can be seen as how easy is it for me to sell an investment of mine so that I can get cash quickly. In a complex financial world, there are various investment instruments that are available to an investor- cash, gold, bank deposits, commodities, bonds, equity, antique art pieces etc. But not all instruments have the same liquidity. Cash, bank deposits are relatively more liquid and provide me easier access to my funds. On the other hand, antique art pieces is not as liquid as the others because of lesser demand in the market. Not every investor wants to buy antique art pieces hence there is not as much demand as there is for, lets say, shares of Microsoft. Now because, not a low of people are willing to buy that antique art piece, it would be difficult for me to find a buyer who is interested in buying it. Now, since finding a buyer is difficult, converting my antique art piece to cash is therefore more difficult. That makes it an asset with low liquidity.

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