Answer to Question #247289 in Microeconomics for Chilu

Question

In a pure exchange economy with two goods, G and H, the two traders have Cobb-Douglas utility functions. Amos⠀ ™ utility is Ua = (Ga)Î ±(HÎ ±)1-Î ± and Elise⠀ ™s is U e = (G e Î ² ) (H e ) 1-Î ². What are their marginal rates of substitution? Between them, Amos and Elise own 100 units of G and 50 units of H. Thus, if Amos has Ga and Ha, Elise has Ge = 100 -Ga and He = 50 -Ha. Solve for their contract curve.

Answer 1

Rate of substitution (marginal) of Amos is: [MRSa​=[α/(1–α)] Ha​/Ga

Rate of substitution (marginal) of Elise is: MRSe​=[β/(1–β)] He​/Ge

The marginal rates of substitution are equal along the contract curves: MRSa​=MRSe​.

Equating the right-hand sides of the expressions for MRSa ​ and MRS_{e}MRSe

we then use information about the endowments and some algebra to write the quadratic formula for the contract curve with respect to the goods of Amos

MRSj​=∂Uj​/∂qj2​∂Uj​/∂qj1​​=∂Ud​/∂qd2​∂Ud​/∂qd1​​=MRSd​               

(β-α )G_{a}H_{a} +β (α – 1)50G_{a} +α (1 -β )100 H_{a} = 0(β−α) Ga​Ha​+β(α–1)50Ga​+α(1−β)100Ha​=0.

Substitute in α=β

If we set α=β 

the contract curve is (β^{2} – β)50G_{a} + (β – β^{2})100H_{a} = 0(β2–β)50Ga​+(β–β2)100Ha​=0

Dividing by (β^{2} – β)(β2–β) to obtain 50G_{a} – 100H_{a} = 050Ga​–100Ha​=0

Using algebra we can sum up the equations because the contract t curve is a straight line

Ga​=2Ha​.