Utility Function and MRS

Beau

I'm in a bit of a bind, have a Microeconomics exam tomorrow and there's a question on an assignment that I have no idea how to do. Could you help? I'd be so grateful!

Q. Two individuals, Dave and Bob, consume two goods, X and Y. The utility functions for the
two individuals are given as:
Bob’s utility function:
UB = 30X0.25 Y0.75
Dave’s utility function:
UD = 50X0.5 Y0.5
Bob is currently consuming 5 units of X and 10 units of Y. Dave is currently consuming 12
units of X and 8 units of Y. The current prices of X and Y are $10 and $15, respectively.

a. Determine the marginal rate of substitution for each individual.
b. In light of the information given above, have the two individuals achieved exchange
equilibrium? Would it be possible to make one individual better off without harming the
other?

Économiste Monétaire

Why is there a space in your utility function? Is it UB=30X^0.25.Y^0.75, or 30X^0.25 + Y^0.75, i.e. substitutes. MRS is simply - MUx/MUy... Just a little calculus.

Also, and I'm not a big micro guy, but question (B) seems to be inferring have they achieved a pareto optimal solution? Maybe it's really asking, is the indifference curve a tangent to the budget constraint. Any of the micro-nerds might be able to say whether I've interpreted that question wrongly.

Beau

Hey,

Yeh thanks, there shouldn't be a space there. I figured out part a its 1/3(y/x) for Bob and y/x for Dave. That gives 2/3 for Dave and .22 for Bob. That makes Bob wrong, in the solution it says 2/3 as well but I don't know how that is possible. How does the market price effect things?

Well the answer to b could be that they are Pareto efficient (assuming the solution I have is right! Both MRS equal each other). I don't know how it could be related to a budget constraint if we don't know the pair's income.

Économiste Monétaire

[1/3(10/5)] is 2/3, not .22, for Bob.

Beau

ok thats embarrassing! Don't know where I got that from. That clears up part b. The prices must be just thrown in there to confuse. Thanks a million.

Économiste Monétaire

You understand the method, you just used Dave's X and Y consumption values, 12 & 8, as opposed to Bob's, so it's not that bad

Good luck with the exam

Bah. Micro! Brings me out in a rash!

Time Magazine

I'll try explain these.

UBob = 30.X^(1/4).Y^(3/4)

So for Bob, MUx/MUy = (1/4)X^(-3/4).Y^(3/4) / (3/4)X^(1/4).Y^(-3/4)
(the 30s cancel)
This simplifies to Y/3X
So Bob will give up 3Xs to get one Y in a pure exchange economy type of way.

A quick way to calculate these, if you're dealing with a Cobb-Douglas, is to look at the relative powers of the goods. For Bob, y goes to 3/4 while x goes to 1/4. So he likes y three times as much as x, all other things equal. Thus he'll give up three x's to get one y. Because Dave's powers equal each other, he likes them equally, so he's indifferent between them if the prices are the same. Thus the MRS when there're no prices is x/y.

Given that there are prices, Bob and Dave will adjust things accordingly. Dave likes x and y equally so say x was twice the price of y, then he'd buy twice as much y as x. In equilibrium the MRS = price ratio. Does this hold? Price of x ("Px") = 10 and price of y ("Py") = 15. So Py = 1.5(Px). This means you'd expect the quantity of x to be 1.5 the quantity of y. Is that true? Yep, he buys 12 and 8.

For Bob, he likes y three times as much as x, but y costs 1.5 times as much. So you'd expect him to buy (3/1.5) = 2 times as much y. Which he does.

To check whether either person could be made better off you check to see if their indifference curves are tangental. If they're not, it's not Pareto efficient.

Edit: Damn forgetting to submit things!

Économiste Monétaire

Deleted User wrote: »

Bah. Micro! Brings me out in a rash!

+1

Seriously, vanilla Micro blows. Honestly, the only thing worse than liking Micro, is tutoring in Micro... *cough* :pac::D:pac: *cough*

Time Magazine

Économiste Monétaire wrote: »

+1

Seriously, vanilla Micro blows.

Beats exchange rate "determination", Purchasing Power "Parity", Uncovered Interest "Parity", arbitrary use of Central Bank functions, an atheoretic empirical "rule" that's an embarrassment to the discipline, pretending there's a consensus on Keynesian stimuli/Rational Expectations....

Optimal patent length and abstract theorems FTW.

Honestly, the only thing worse than liking Micro, is tutoring in Micro... *cough* :pac::D:pac: *cough*

Second semester cough cough.

Microeconomist: Firms can't make profits!

Microeconomst: Reads FT - firms appear to be making profits

Microeconomist: ......



Macroeconomist: I assume a Blanchard Model!

RA whispers something in his ear..

Macroeconomist: You mean you weren't born fully trained with the instant ability to work? And we don't all have a constant probability of death?

Macroeconomist: .......



Both economists get published in AER. No one cares that the models don't work!

Time Magazine

Deleted User wrote: »

Microeconomist: Firms can't make profits!

Fail.

Firms shouldn't make rents.

But that's subject to our entrepreneurial budget constraint.

The Economist

The Economist wrote: »

Fail.

Firms shouldn't make rents.

But that's subject to our entrepreneurial budget constraint.

Not according to Mas Collell et al. Zero profit condition.

But this micro, 'tis crazy stuff! Also, UIP now appears to be holding. Seems the less people in the Forex market, the more efficient it gets.

nesf

The Economist wrote: »

Beats exchange rate "determination", Purchasing Power "Parity", Uncovered Interest "Parity", arbitrary use of Central Bank functions, an atheoretic empirical "rule" that's an embarrassment to the discipline, pretending there's a consensus on Keynesian stimuli/Rational Expectations....

Pfft, PPP? Just give them Big Macs!

Time Magazine

Deleted User wrote: »

Not according to Mas Collell et al. Zero profit condition.

Yes, if you theoretically impose the condition of zero profits firms won't earn profits. Similarly if you assume monopolistic profits hold... well then firms earn monopolistic profits :P

Theoretically in that game situation you can derive whatever output levels you like given a certain concentration of competition. Since not everyone is an entrepreneur, of course, you're simply not going to have anything approaching perfect competition in every industry. And even PC is damaging to TFP dynamics.

I look forward to the rise of the "innovation-monopoly trade-off patent-length optimisation condition" (IMTOPLOC) parlance myself.

nesf wrote: »

Pfft, PPP? Just give them Big Macs!

The similarity to "Let them eat cake" is striking

Économiste Monétaire

How dare you question the Macro... DSGE is making some progress for CBs... Check back in a few years... Kinda like behavioural economics, eh? Psychologists posing as economists :pac:

I'm sure the commerce students will be a joy to teach

nesf

Deleted User wrote: »

Also, UIP now appears to be holding. Seems the less people in the Forex market, the more efficient it gets.

Well, there is across most financial markets way more trading volume than is "needed" (can't remember the paper off hand, but they found something like 5 or 6 times the volume needed) which would make things far more inefficient normally.

nesf

The Economist wrote: »

The similarity to "Let them eat cake" is striking

Very much intended I assure you.

Économiste Monétaire wrote: »

Kinda like behavioural economics, eh? Psychologists posing as economists :pac:

But some of it is so interesting! Plus instead of optimising equations you get to make experiment on people. Win-win tbh.

Time Magazine

Économiste Monétaire wrote: »

I'm sure the commerce students will be a joy to teach

Oh they are.

"So to get the optimising point here you need to use the chain rule. I know it was two years ago for you but can anyone remember the chain rule from Leaving Cert maths?

No?

What?

You're serious?

You can't differentiate?

"

You just wouldn't get that crap at the home of Edgeworth and Longfield.

Économiste Monétaire

Économiste Monétaire wrote: »

How dare you question the Macro... DSGE is making some progress for CBs... Check back in a few years... Kinda like behavioural economics, eh? Psychologists posing as economists :pac:

I'm sure the commerce students will be a joy to teach

Have you read any of the Svensson papers on that?

They are really interesting, and they also pretty up front about what doesn't work and steps taken to fix it.

Économiste Monétaire

nesf wrote: »

But some of it is so interesting! Plus instead of optimising equations you get to make experiment on people. Win-win tbh.

Indeed, it is. UCD is actually giving a module on behavioural economics for the first time this year, the lecturer is making the notes and such available on a website. It's here if you're interest in reading some of his stuff.

The Economist wrote: »

Oh they are.

"So to get the optimising point here you need to use the chain rule. I know it was two years ago for you but can anyone remember the chain rule from Leaving Cert maths?

No?

What?

You're serious?

You can't differentiate?

"

You just wouldn't get that crap at the home of Edgeworth and Longfield.

Oh, the real invasion will begin this semester. I know someone in Intl. Commerce, doing that module you're referring to this semester, and most of the commerce people leave it until semester two. That level of math retardation is shocking... It could be worse for you, i.e. it could be 1st Quants

nesf

Économiste Monétaire wrote: »

Indeed, it is. UCD is actually giving a module on behavioural economics for the first time this year, the lecturer is making the notes and such available on a website. It's here if you're interest in reading some of his stuff.

Cheers, PM me if you want a list of books on the topic + Neuroeconomics I got from someone deeply involved in the field.

The Economist

The Economist wrote: »

Oh they are.

"So to get the optimising point here you need to use the chain rule. I know it was two years ago for you but can anyone remember the chain rule from Leaving Cert maths?

No?

What?

You're serious?

You can't differentiate?

"

You just wouldn't get that crap at the home of Edgeworth and Longfield.

Being a pass maths student (and a former [but temporary] business student) I have kind of sympathy for them.
No excuse for them not to go off and learn the methods though.

nesf

nesf wrote: »

Cheers, PM me if you want a list of books on the topic + Neuroeconomics I got from someone deeply involved in the field.

We have that behavioral economics option over here in Warwick, however it was either that or General Equilibrium. Might as well get the hard stuff over and done with first! Can look into the interesting stuff later! Plus I'm doing a GE dissertation and my supervisor teaches the course so I didn't have much choice in the matter!

nesf

Deleted User wrote: »

Being a pass maths student (and a former [but temporary] business student) I have kind of sympathy for them.
No excuse for them not to go off and learn the methods though.

Pfft, you people should see first year Honours Physics in UCC. The class used to drop from 100 to 20 within 2 weeks every year before they changed the course around a lot. Vector fields tend to make people cry.

nesf

Deleted User wrote: »

We have that behavioral economics option over here in Warwick, however it was either that or General Equilibrium. Might as well get the hard stuff over and done with first! Can look into the interesting stuff later! Plus I'm doing a GE dissertation and my supervisor teaches the course so I didn't have much choice in the matter!

Run, run from the neo-classical dear boy! It'll make you forget that Economics was ever meant to resemble reality.

Économiste Monétaire

Deleted User wrote: »

Have you read any of the Svensson papers on that?

They are really interesting, and they also pretty up front about what doesn't work and steps taken to fix it.

Unfortunately, I can't say I've read his most recent work. Chronic procrastination His work on liquidity traps might be worth linking to Bernanke et al

Its a pretty macro orientated GE Model and I am relaxing a lot of the restrictions on the original paper.

Fight 'em from within I say!

Économiste Monétaire

Économiste Monétaire wrote: »

Unfortunately, I can't say I've read his most recent one. Chronic procrastination His work on liquidity traps might be worth linking to Bernanke et al

The Blanchard 'State of Macro' paper sums up that topic quite well. I liked the Svensson papers, can't remember if they were his most recent or not but I found them pretty good. Bernanke gave a speech somewhere recently which gave a lot of support to the concept of DSGE. Really good, non technical overview of how they are used and the advantages to them.

nesf

nesf wrote: »

Well, there is across most financial markets way more trading volume than is "needed" (can't remember the paper off hand, but they found something like 5 or 6 times the volume needed) which would make things far more inefficient normally.

there goes my PhD idea.

If you have a link, that would be cool!

nesf

Deleted User wrote: »

there goes my PhD idea.

If you have a link, that would be cool!

Don't unfortunately, it came up in conversation at a conference I was at. I'll try and get it from the guy I was chatting to about it, I need to email him about some other stuff anyway.

Économiste Monétaire

As far as I understand, Svensson has published quite a bit over the last year, or so, on the topic. I'm trying to load the NBER site to link to the paper, but it seems down... I'll go and find Ben B's speech, thanks.

Its here:
http://www.federalreserve.gov/boarddocs/speeches/2004/20041202/default.htm