Question: In the economy...........  Edit

Answer: For consumer 1, dU1/dx11 = 2(x11x124)/(x112x124)= 2/x11
dU1/dx12 = 4(x112x123)/(x112x124)= 3/x12
dx11/dx12 = 2x12/3x11
For consumer 2, dU2/dx21 = (1/3)(x22/x21)2/3
dU2/dx22 = (2/3)(x21/x22)1/3
Dx21/dx22 = (1/2)x22/x21

The consumers will allocate resources until dx11/dx12 = dx21/dx22 to achieve Pareto Optimality.

B) The allocation is Pareto Optimal as there is no other feasible distribution which can satisfy the consumers more than the Pareto Optimal allocation of good.

C) The equilibrium condition says that the MRS of consumer 1 and consumer 2 must be equal at the optimum quantity as the tangency condition of the indifference curves is satisfied at that point to achieve Pareto Optimum.

D)
Feasibility conditions
x11 + x21 = 4
x12 + x22 = 32



Setting Lagrange for consumer 1,
L= x112x124 - n(x211/3x222/3 -4) - q(x11 + x21-4) -r (x12 + x22-32)

FOC
dL/dx11 = 2/x11 -q =0 (1)
dL/dx12= 3/x12 - r =0 (2)
dL/dx21 =(1/3)(x22/x21)2/3 -q (3)
dL/dx22 =(2/3)(x21/x22)1/3-r (4)


2x12/3x11 = q/r (Dividing 1 by 2)
(1/2)x22/x21 =q/r (Dividing 2 by 3)

2x12/3x11 = x22/2x21 (or, MRS 1 = MRS 2)
2(32- x22)/ 3( 4 - x21)= x22/2x21 (Replacing with feasibility conditions)
Or, 256 x21 - 4x22x21 = 12x22 - 3 x22x21
Or, 256 x21 -x22x21 = 12x22
Or, x21 = 12x22 /(256- x22)
x22 = 256 x21 /(12 + x21 )
Given utility 2 equals 4,
x211/3x222/3 -4 = 0
x211/3[256 x21 /(12 + x21 )]2/3 = 4
2562 x21 3= 64(x21 2 + 24x21 + 144)
x21 = 0.5 (using cubic equation calculator)
x11 = 4-0.5= 3.5
Again,
x211/3x222/3 -4 = 0
Or, x21x222= 64
X22 = 11.3 (Using x21 = 0.5)
X12= 32 - 11.3 = 20.7





















E) If x21 = 4 and x22 = 32, which are the maximum values they can take (consumer 1 gets zero), U= 41/3*322/3= 16
If consumer 2 gets no allocation (x21 = 0 and x22 = 0), his utility is 0
Hence, value of consumer’s 2 utility ranges from 0 to 16, ie, the minimum is where he gets no goods and maximum is when he gets the entire allocation.

F) x211/3x222/3 -1.5 = 0
x211/3[256 x21 /(12 + x21 )]2/3 = 1.5
2562 x21 3= 3.375(x21 2 + 24x21 + 144)
x21 = 0.19 (using cubic equation calculator)
x11 = 4-0.19= 3.81 (using feasibility equation)
Again,
x211/3x222/3 -1.5 = 0
Or, x21x222= 3.375
X22 = 1.9
X12= 32 - 1.9 = 30.1



X12= 32 - 11.3 = 20.7
No, consumer 2 is worse off as his utility curve is lower

G) x21 = 0.5
x11 = 3.5
X22 = 11.3
X12 = 20.7

P1x11 + p2X12 = 3.5p1 + 20.7p2


p* = p1/p2

P*x11 + x12/p* = 3.5p* + 20.7/p* (5)
x21p* + x22/p*= 0.5p* + 11.3/p*

2x12/3x11 = p* (MRS of consumer 1)

2x12 = 3x11 p* (6)

(1/2)x22/x21= p* (MRS of consumer 2)

We use the equation 5
P*x11 + 3/2x11 = 3.5p* + 20.7/p* (Replacing with 6)
Or, 3.5p* + 5.25 = 3.5p* + 20.7/p*
Or, 3.5p*2 + 5.25p* = 3.5p*2 + 20.7
Or, p*= 3.942

H) A Walrasian equilibrium is a Pareto efficient allocation as both are achieved when MRSs of utility functions are equal. Hence, any Walrasian equilibrium is Pareto optimal.
  Edit