Question

For Microeconomics: I get two utility functions for a baker and a farmer respectively. Ub=Xb*Yb, Uf=Xf*Yf^1/2. Both can produce up to 48 units per week. The price of bread Px is fixed at 1. How can I find the price Py of vegetables that clears the trade between both? And how much will each produce? Sorry but I didn't find a group for Economics somehow.

Answer 1

I took macroEcon; and the utility functions werent in the course .... what do the variables stand for?

Answer 2

Oh, well Macroeconomics is quite different. This is more about consumer behaviour. More neatly, the Utility functions are:
\[U _{b}=X _{b}Y _{b}\]
\[U _{f}=X _{f}\sqrt{Y _{f}}\]
The subscripts b and f stand for Baker and Farmer respectively.
X is units of bread and Y is units of vegetables.
Any ideas? Thanks!

Answer 3

are you using Makiew's book? it should literally be right in there.
Anyway tell your professor that economic theory is bs and that the world does not operate 'ceterus parebus.' :-)

Answer 4

You mean Mankiw? I'm using Snyder & Nicholson but it's not really in there...And yeah, I agree with your statement, but these models are still better than nothing...

Answer 5

So if I see it right, you want the utility of the farmer and the baker to equal each other given these formula?

Baker makes 48 loafs ; farmer picks 48 vegetables; and we divide them up so that the baker and farmer have equal utility and the balance of trade is at an equilibrium [Bu = Fu] ....

Bu = xy
Fu = x sqrt(y)

We can trial an error this to get some intuitive results ...

spose they split the loads evenly?

Bu = 24*24 = 576
Fu = 24*sqrt(24) = 117.58

What happens when we decrease they trade by 1?

Bu = 25*23 = 575
Fu = 23*sqrt(25) = 115 ; the farmers utility dropped more
than the bakers, bad direction

increase the trade by 1?

Bu = 23*25 = 575
Fu = 25*sqrt(23) = 119.9 ; that is definantly in the
right direction :)
..........................................................

But I have to wonder if im reading the equation right; are we trying to barter bread and veggies? Or determine a fair price for veggies based on bread being equal to 1 ? Or are these really the same concepts worded differently :)

Answer 6

Thanks for your reply. You are in the right direction but unfortunately I need a precise answer to this without approximations. I thought it would work by equalizing the MRS of both but somehow it doesnt...Thanks anyways...

Answer 7

Bu = 16 * 12 = 192
Fu = 32 * sqrt(36) = 192

Answer 8

The math i did went like this:

since bread is equal to each; then its value in the equation is moot;

How do we divvy up the veggies then?

y = sqrt(y); where Veggie Utilitiy compares
to baker(Vb) and farmer(Vf)

there is some relation "R" such that R = sqrt(y) - y; the rate at which the values change with respect to "y" is known as the derivative of R

R' = \(\frac{1}{2\sqrt{y}} -1\iff \frac{1 -2\sqrt{y}}{2\sqrt{y}}\) ; when this equation equals 0 we are at a point where the two reach an equilibrium.

a fraction = 0 when the top of it is 0 so: \(1-2\sqrt{y}=0\) when y = 1/4.

48 shares divided by 4 equals 12 and 36; the farmers equation suggests that has utility for veggies is lower than the bakers so.

Bu = x(12) : 12x
Fu = x(sqrt(36)) : 6x

Now we have to divvy up the bread to equal things out lol

Answer 9

the relationship between the utility obtained by baker and farmer for bread is:

12b = 6f ; we can rewrite this in terms of the bakers utility such that:

b = (1/2) f ; this tells us that after the veggies equal out, the baker will be content with 1/2 the amount of bread the farmer gets.

If we divvy the loaves up 6 ways we get:

o o o o o o ; which is good for our purpose since 4 + 2 = 6 and 2 is half of 4 :)

48/6 = 8 in each group; 4 groups = 32 loaves, and 2 groups = 16

Bu = 16 * 12 = 192
Fu = 32 * 6 = 192

Answer 10

As far as price goes now; the baker exchanges 32 loaves for 12 veggies

so $32 = 12 veggies; veggies = $(32/12) = 16/6 =abt. $2.66 ?

Answer 11

Without knowing what im talking about most of the time, thats the best I can do ;)

Answer 12

hey thanks a lot. yeah, your first part seems very reasonable. i'm sure this is about MRS though and my teacher would not accept an answer by intuition. sucks my book is not helpful in this case and i cant find much on google. anyways, lets just leave it like this and thanks for your help!

Answer 13

:) youre welcome, and good luck