math and economics

Question

1. Two persons have the utility functions, 𝑈𝐴 = 𝑥𝐴𝑦𝐴 and 𝑈𝐵 = 𝑥𝐵𝑦𝐵. Let the initial endowments be 𝑥𝐴̅ = 90, 𝑦̅𝐴 = 35, 𝑥̅𝐵 = 30, 𝑦̅𝐵 = 25, so the total quantities available are 𝑥̅= 90 + 30 = 120 and 𝑦̅ = 35 + 25 = 60. Normalizing prices such that 𝑝𝑦 = 1, persons A and B have incomes of 𝑚𝐴 = 𝑝𝑥90 + 35 and 𝑚𝐵 = 𝑝𝑥30 + 25, respectively.

A. Solve the constrained optimization problem for person A. That is, maximize 𝑈𝐴 = 𝑥𝐴𝑦𝐴 subject to the income constraint 𝑝𝑥90 + 35 = 𝑝𝑥𝑥𝐴 + 𝑦𝐴, which gives us the Marshallian demand functions 𝑥𝐴 ∗ = 𝑓(𝑝𝑥) and 𝑦𝐴 ∗ = 𝑔(𝑝𝑥) for person A.

B. The corresponding demand functions for person B are 𝑥𝐵 ∗ = 15 + 12.5(1⁄𝑝𝑥) and 𝑦𝐵 ∗ = 15𝑝𝑥 + 12.5. Plug this expression for 𝑦𝐵 ∗ and the expression for 𝑦𝐴 ∗ from part A into the market equilibrium condition, 𝑦𝐴 ∗ + 𝑦𝐵 ∗ = 𝑦̅, and solve for 𝑝𝑥 ∗ , the market equilibrium price. [By Walras’ law, when the market for good 𝑦 is in equilibrium, the market for good 𝑥 will be in equilibrium.]

C. Substitute 𝑝𝑥 ∗ into all the demand functions for persons A and B to find their quantities demanded for all the goods. Check whether the total quantity demanded for each good equals the total endowment of that good.

D. Is this market equilibrium allocation Pareto efficient. That is, are the marginal rates of substitution, 𝑀𝑅𝑆𝐴 = (𝜕𝑈𝐴⁄𝜕𝑥𝐴)⁄(𝜕𝑈𝐴⁄𝜕𝑦𝐴) and 𝑀𝑅𝑆𝐵, for the two persons equal when evaluated at the market equilibrium allocation?

Answer 1

For A , MU(x) =y and MU(y)=x so MRS = y/x py = 1 (y is the numeraire)

so MRS=px = y/x so y = pxX . and X =Y/px The budget constraint is M=pxX + pyY = pxX +Y.

Substitute pxX= Y into the budget constraint and get M =pxX +pxX = 2PxX solving for X we get A's demand curve = X= M/2Px. we also get Y=M/2 as A's demand for Y.

Now substitute the initial endowment M=pxXA +YA = px90 +35 and Get

X_A^D= (px90 +35)/2px = 45+35/2px

and Y_A^D=px90/2 +35/2

We can look for solving via the market for Y or the market for X.

for B (let's call him Brown and Call A Adams.) we see Adams and Brown have the same preferences but differ in endowments so we solve for Browns demands by substituting his endowment into the demand equations. M_B= px30 +25 and get

X_B^D=M/2px= px30/2px +25/2px =15 +(25/2px)

and Y_B^D =M/2 = px30/2 +25/2 = 12.5 + 15px.

We know that total amounts of X and Y in the market are

X_A+X_B=120 and Y_A+Y_B=60.

Let's Use Y_A+Y_B=60 =

We can solve for px by using excess demand curves or via offer curves ( If you are a bit confused here I would be more than happy to help, I coan also lay out the edgeworth box and graph out the equilibrium) I can't do that in this note).

Lets use Y_A+Y_B=60 = 12.5+15px+35/2 +45 px or 60= 30 +60px or

60px=30 or px=1/2.

When px = 1/2 X_A =45 +35/2Px = 45 + 35 = 80

and Y_A = 90/4 +35/2 = 160/4 = 40.

so Adam's consumes X_A=80 and Y_A=40 and Poor Brown has X_B=40 and Y_B=20

Recall MRS^A=MRS^B Has Y_A/X_A = Y_B/X_B = 80/40 = 40/20 so yes this is a Pareto Optimal reallocation!!!

This is a great problem and if you would like me to go over all of the graphs on my whiteboard, give me a shout!!!!