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## Stuff not covered in lectures

Hi sir

I was wondering whether we have to do question 7 of tutorial 2? As we did not go over this in the lecture.

Kind regards

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As I explained, I do not attempt to cover everything in the lectures: the reason you have a textbook is so that you can study things I have not covered in the lectures. I did mention the Laspeyres price index in the lecture. So in particular, it will be part of the syllabus

In any case, there is no sanction if you do not do some exercises: if you want them marked you need to hand them in the week before. If you do not want to do them, fine by my (and less work for the tutors :).

## Bertrand competition

Dear Professor

Question 1 (b) from section A

1. Two Firms, A and B, can produce each unit of a good at a respective cost of a and b. Market
demand equals 1 minus the price. The two Firms compete by simultaneously announcing their
prices.
(a) Which prices would the Firms announce if a = b < 1? [50%]
(b) Which price would consumers pay if a < b < 1? [50%]

The answer suggested is that ‘the consumer could pay any price between a and the minimum of b and (1+ a)/2. Firm B could charge less than b if its price does not undercut Firm A’s price, provided that A then sells to the entire market…’

I think the price paid should be (1+a)/2 if it is lower than b, and if it is greater than b, a can charge a price slightly lower than b and can gain all consumers. I don’t quite understand why the answer is an interval. Could you explain it in detail?

Best wishes

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I did not set this question, so a student’s ability to answer it may well depend on what the lecturer had concentrated on in their lectures at the time.

For part (a) I would say that both firms set a.

For part (b): I would say that the price is the lower of (1-a)/2 (minus not plus) and b. Unless, when consumers choose, they ALL choose to buy from firm A if the firms charge the same price. In which case, indeed, any price between (1-a)/2 and b would be chosen by both firms: firm B cannot increase its profit, which is zero, and would become negative if it lowered its price (and remain zero if it increased it).

## The core

Dear Gianni,

Why does the core of an exchange economy converge to the set of competitive equilibria as the number of consumers increases?

I can’t find anything on the core in the PowerPoint or in the Varian book.

Do we need to know this? If we do, could you provide a longer explanation of the answer?

Kind regards,

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If you scroll down this blog, you will find that the question was set in a year where the syllabus was different. There is no harm in studying this, but it will not be a topic asked in the exam.

## Still on coordinates of a point

Dear Gianni,

I saw the previous posts on how to draw indifference curves for U(x, y) = min[3x+y, 5y} but I still have questions.

you said that 4k/5 is the intersection of the lines y=k/5 and y=3x/4

but how is k/5 found?as well as y=k-3x?

Also, for the kink points why are they (x, 3x/4)?

Kind Regards,

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“you said that 4k/5 is the intersection of the lines y=k/5 and y=3x/4 ” I didn’t: I said that 4k/15 is the intersection of the lines y=k/5 and y=3x/4: which it is.

k/5 is the consumption of good y which is necessary to consume in the relevant part of the consumption space to have utility level k. You should remember that the indifference curve at level k is the combination of points that give utility  level k. Change k and you get a different indifference curve.

The co-ordinates of the points on the line y = 3x/4 have co-ordinates ( x, 3x/4 ). This is something that I know was covered in your first year maths courses.

## Cournot games with three or more firms

Hi Gianni,

Hope you are well. I need some help with cornout questions, especially in the case where n>2 firms. I really have no idea where to begin. Please find below examples found in microeconomics past papers.

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There is no conceptual difference with three firms. You have a three player game, apply the definition of Nash equilibrium. In the case of continuous Cournot games, simply write three (or N) first order conditions, and solve.

Sometimes the foc are symmetric and so you can assume that the firms will choose the same quantity levels; other times it may be necessary to think of other ways for solving the problem: think about adding up the foc of the firs, and then solving for the TOTAL quantity.

## Engel curves

Dear Dr Gianni

I have a question from the above paper

1. (a) Define precisely an individual’s Engel curve for a good [30%]

(b) Derive the Engel curves (for good 1 and good 2) for a consumer with the following utility functions [35%]
U (x1;x2) = lnx1 + 5lnx2

U (x1;x2) = min(x1,x2)

For part (b)

the solution for the both graphs has income on the horizontal axis and consumption on the vertical axis. Shouldn’t income be on the vertical axis to draw engel curves?

Also, why the lnx1 + lnx2 has to have 2 straight lines coming through the origin and min(x1,x2) one?

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As long as you indicate what is on the axes, either is fine: there are conventions (such as the price on the vertical axis for the demand curve, or x1 on the horizontal axis in a x1,x2 consumption good cartesian diagram), but they are convention.

I don’t understand what you mean by ” lnx1 + lnx2 as to have 2 straight lines coming through the origin”. Did I draw both goods on the same diagram?

## Dumbledore and Snape

Hi,

Question 6e on the 2016/17 paper is:

Would Snape benefit from knowing what payment scheme Dumbledore has chosen before choosing whether or not to train? Would Dumbledore benefit from knowing Snape’s choice before deciding on the payment scheme?

The mark scheme said neither would benefit as the Nash equilibrium (training, revenue sharing) is unique. If Snape knew Dumbledore has selected  ‘fixed wage’ instead, wouldn’t he prefer to play ‘no training’ as this provides a higher payoff to Snape than still playing ‘training’, regardless of whether Snape believes Dumbledore is rational or not, as he knows for certain Dumbledore has selected ‘fixed wage’ in this game?

I have attached a screenshot of the matrix and mark scheme

Thanks

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Indeed, this is precisely what the suggested answer (I presume this is what you mean by “marking scheme”) suggests: If Snape observes that Dumbledore has choosen “fixed wage”, then Dumbledore is not rational.