Answer to Question #254606 in Microeconomics for Sophie

Question

In a pure exchange economy with two goods, G and H, the two traders have Cobb-Douglas utility functions. Amos' utility is U_a= (G_a )^∝ (H_∝)^(1-∝) and Elise’s is U_e= (G_e)^β (H_e)^(1-β). 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 G_a and H_a, Elise has G_e =100-G_a and H_e =50-H_a. 

Solve for their contract curve. Show all your calculations clearly from your marginal utilities.

Answer 1

Amos: rate of marginal substitution [MRSa = [$\alpha$/ (1-$\alpha$) Ha/Ga

Elise: rate of marginal substitution [ MRSe = [$\beta$ / (1- $\beta$ ) He/Ge

the marginal rates of substitution are equal along the contract curves; MRSa= MRSe

equate the right-hand sides of the statements for MRSa and MRS_{e}MRSe

we finally make use of endowments information and some algebra to construct the contract curves quadratic formula respecting Amos goods to get the following equation;

($\beta$-$\alpha$)G_{a}H_{a}+$\beta$ ($\alpha$-1)50G_{a}+$\alpha$ (1-$\beta$)100H_{a}= 0($\beta$-$\alpha$) GaHa+$\beta$ (a-1)50Ga+$\alpha$(1-$\beta$)100Ha=0

solve this way by substituting $\alpha$ = $\beta$

therefore setting $\alpha$ = $\beta$

contract curve becomes ($\beta$ ^{2}-$\beta$ )50G_{a} + ($\beta$-$\beta$^{2})100H_{a}= 0($\beta$2-$\beta$)50Ga+($\beta$-$\beta$2)100Ha=0

Divide by ($\beta$^{2}-$\beta$ ) ($\beta$2-$\beta$ ) to get 50G_{a} -100H_{a} =50Ga-100Ha=0

we then use algebra to sum the equations. the straight line represents the contract curve t

Ga=2Ha