Question

Consider a perfectly competitive economy with 200 individuals. Every individual is both a producer and a consumer. Each person has a production function as Y_i = 100 – X_i². There are 100 type-A individuals, whose preference is given by U_A= 5y_A (every person) and the 100 type-B individuals with U_B = 18lnx_B + 2y_B. Follow the steps below to check if p_x/p_y = 4 is an equilibrium price.

1. When p_x/p_y = 4, each producer produces __ units of good X and ___ units of good Y

2. When p_x/p_y = 4, the total (gross) quantity supplied of good X is __ units and that of good Y is __ units.

3. When p_x/p_y = 4, a type-A individual consumes __ units of good X and __ units of good Y

4. When p_x/p_y = 4, a type-B individual consumes __ units of good X  and __ units of good Y.

5. When p_x/p_y = 4, the total (gross) quantity demanded of good X is __ units and that of good Y is __ units.

Answer 1

1. When px/py = 4

MRT = dy/dx = - 2x

| MRT | =px/y

2X = 4

X = 2 and Y = 100 - 2² = 96

so, each producer produces 2 units of X and 96 units of Y.

2.

Total gross quantity of X = 2*200 = 400

Gross Quantity of Y = 96*200 = 19200

3. UA = 5Y

MRS = MUx/Muy = 0/5 =0 = px/py = 4 (not possible)

Hence, there will be a corner solutuon at which A will only consumer Y, thus X =0. X doesn't give any utility to A.

Budget Constraint of A :

x*px + y*py = M

where M = income from production = X*px + Y*py

x*px + y*py = X*px + Y*py

dividing the above equation by py

x*px/py + y = 2px/py + 96

4x + y = 8 + 96 = 104

if x = 0, y = 104

4. Type B

UB = 18lnX + 2y

MUx = 18/X and MUy = 2

MRS = 18/2X = 9/X = Px/Py = 4

X = 9/4

Budget Constraint(same as above) :

4x + y = 104

at x = 9/4, y = 104 - 9 = 95

5. Total quantity demanded of x = xA*100 +xB*100 = 0 + 9/4*100 = 225

Total quantity demanded of y = yA*100 + yB*100 = (104 + 95)*100 = 19900

Demand is not equal to supply, hence px/py = 4 is not an equilibrium price.

Hope the answer helped you :)