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Principles of General Equilibrium Theory

Question 1: Consider a simple economy with two goods, apples (A) and bananas (B), and two individuals, Alice and Bob. Alice's utility function is given by U_A = A^(1/2)B^(1/2), and Bob's utility function is given by U_B = AB. The initial endowments of Alice and Bob are (4, 8) and (6, 2) units of apples and bananas, respectively. Find the initial equilibrium allocation of goods.

Solution: To find the initial equilibrium allocation, we need to determine the allocation of goods that maximizes the total utility of both individuals subject to their initial endowments.

Let x_A and x_B represent the quantities of apples consumed by Alice and Bob, respectively, and y_A and y_B represent the quantities of bananas consumed by Alice and Bob, respectively.

The optimization problem can be stated as follows:

Maximize U_A + U_B = (x_A^(1/2)y_A^(1/2)) + (x_By_B) Subject to the following constraints: x_A + x_B = 4 + 6 (total apples) y_A + y_B = 8 + 2 (total bananas)

Taking the first-order conditions, we have:

.

where is the Lagrange multiplier. λ

From equation (1), we get: y_A^(1/2) = 2 (5) λ

From equation (2), we get: x_A^(1/2)y_A^(-1/2) = 2 (6) λ

Combining equations (5) and (6), we get: x_A^(1/2)y_A^(1/2) = 4^2 (7) λ

From equation (3), we get: y_B = (8) λ

From equation (4), we get: x_B = (9) λ

Now, substituting equations (8) and (9) into equation (7), we have: x_A^(1/2)y_A^(1/2) = 4^2 => x_A^(1/2) = 4^2 => x_A^(1/2) = 4 λλλλ

Squaring both sides, we get: x_A = 16^2 (10) λ

Substituting equation (10) into the total apples constraint, we have: x_A + x_B = 4 + 6 16^2 + = 10 16^2 + - 10 = 0 λλλλ

Solving the quadratic equation, we find: = 0.5 or = -0.625 λλ

Since utility functions are strictly concave, we disregard the negative value for .λ

Using = 0.5, we can calculate the equilibrium allocation as follows: λ

From equation (8), we have: y_B = = 0.5 λ

From equation (9), we have: x_B = = 0.5 λ

From equation (5), we have: y_A^(1/2) = 2 = 1 => y_A = 1 λ

From equation (10), we have: x_A = 16^2 = 16(0.5)^2 = 4 λ

Therefore, the initial equilibrium allocation of goods is: Alice: (x_A, y_A) = (4, 1) Bob: (x_B, y_B) = (0.5, 0.5)

Question 2: Consider an economy with two goods, X and Y, and two individuals, Adam and Eve. Adam's utility function is given by U_A = X^(1/2)Y^(1/2), and Eve's utility function is given by U_E = XY. The initial endowments of Adam and Eve are (8, 6) and (4, 10) units of X and Y, respectively. Find the initial equilibrium allocation of goods. Solution: To find the initial equilibrium allocation, we need to determine the allocation of goods that maximizes the total utility of both individuals subject to their initial endowments.

Let x_A and x_E represent the quantities of good X consumed by Adam and Eve, respectively, and y_A and y_E represent the quantities of good Y consumed by Adam and Eve, respectively.

The optimization problem can be stated as follows:

Maximize

the following constraints:

Taking the first-order conditions, we have:

From equation (1), we get:

From equation (2), we get:

From equation (4), we get: x_E = (9) λ

Now, substituting equations (8) and (9) into equation (7), we have: x_A^(1/2)y_A^(1/2) = 4^2 => x_A^(1/2) = 4^2 => x_A^(1/2) = 4 λλλλ

Squaring both sides, we get: x_A = 16^2 (10) λ

Substituting equation (10) into the total X constraint, we have: x_A + x_E = 8 + 4

16^2 + = 12 16^2 + - 12 = 0 λλλλ

Solving the quadratic equation, we find: = 0.75 or = -0.5 λλ

Since utility functions are strictly concave, we disregard the negative value for .λ

Using = 0.75, we can calculate the equilibrium allocation as follows: λ

From equation (8), we have: y_E = = 0.75 λ

From equation (9), we have: x_E = = 0.75 λ

From equation (5), we have: y_A^(1/2) = 2 = 1.5 => y_A = 2.25 λ

From equation (10), we have: x_A = 16^2 = 16(0.75)^2 = 9 λ

Therefore, the initial equilibrium allocation of goods is: Adam: (x_A, y_A) = (9, 2.25)

Eve: (x_E, y_E) = (0.75, 0.75)

Question 3: Consider a two-person economy with two goods, X and Y, and two individuals, Amy and Ben. Amy's utility function is given by U_A = X^(1/3)Y^(2/3), and Ben's utility function is given by U_B = XY^(1/2). The initial endowments of Amy and Ben are (9, 4) and (5, 7) units of X and Y, respectively. Find the initial equilibrium allocation of goods.

Solution: To find the initial equilibrium allocation, we need to determine the allocation of goods that maximizes the total utility of both individuals subject to their initial endowments.

Let x_A and x_B represent the quantities of good X consumed by Amy and Ben, respectively, and y_A and y_B represent the quantities of good Y consumed by Amy and Ben, respectively.

The optimization problem can be stated as follows:

Taking the first-order conditions, we have:

Cubing both sides, we get: y_A^2x_A = 81^4/16 (10) λ

Substituting equation (10) into the total X constraint, we have: x_A + x_B = 9 + 5 => x_A + x_B = 14

Substituting equation (8) into equation (9), we get: y_B^(1/2) = => y_B = ^2 (11) λλ

Substituting equations (10) and (11) into the total Y constraint, we have: y_A^2x_A + y_B^2x_B = 4 + 7 => 81^4/16 + (^2)(x_B) = 11 λλ

From equation (10), we can express x_A in terms of as: x_A = (81^4/16)/y_A^2λλ

Substituting this into the total X constraint, we have: (81^4/16)/y_A^2 + x_B = 14 λ

Simplifying, we get: 81^4 + 16x_By_A^2 = 224y_A^2 λ

Solving this equation for x_B, we have: x_B = (224y_A^2 - 81^4)/(16y_A^2) (12) λ

Substituting equation (11) into equation (12), we have: x_B = (224 - 81^2)/(16) λ (13)

From equation (8), we have: y_B = λ

From equation (9), we have: x_B^(1/2)y_B^(-1/2) = => x_B^(1/2)^(-1/2) = 1 λλ => x_B = ^(2)λ

Therefore, the initial equilibrium allocation of goods is: Amy: (x_A, y_A) = (9, 4) Ben: (x_B, y_B) = (^(2), )λλ

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