Question

6. Sarah and Jane are two representative individuals living in an economy that produces two goods, X and Y. Sarah's and Jane's utility functions are given as: Sarah: US -100X0.75y0.2.5 Jane: UJ 50X0.6Y0.4 The market determined prices of X and Y are $10 and $20, respectively. Sarah's endowment 80 is units of X and no Y, while Jane's endowment is 10 units of X and 20 units of Y. (Sarah's current income is $800 per time period, while Jane's income is $500 per time period.) a. Write expressions for Sarah and Jane's marginal rates of substitution b. Determine the quantities of X and Y that Sarah and Jane should consume in equilibrium. c. Do the values calculated in part (b) satisfy the conditions for equilibrium in exchange? Explain using numbers. d. Examine your answers in parts (b) and (c). If equilibrium has not been achieved what would be necessary to reach equilibrium? If equilibrium has been achieved comment on the process by which equilibrium was reached

Answer 1

U^S = 100 (X)^.75 Y^.25 a) MRS_S = partial differential u/partial differential x/partial differential u/partial differential y = 100 (.75) (x)^-.25 y^.25/100 (.25) (x)^.75 (y)^-.75 = 3y/x MRS_J = (partial differential u/partial differential x)^J/(partial differential u/partial differential y)^J, u^J = 50 times x^.6 y^.4 = 50(.6) (x)^-.4 y^.4/50 (.4) (x)^.6 (y)^-.6 = 3/2 y/x b) Since cobb Douglas utility function, so X_S * = 3 M_s/4 P_x, M_s rightarrow endowment income of = 80 P_x + 0P_y = 800. P_x = 10, P_y = 20. M_J = 10P_x + 20P_y = > 100 + 400 = 500. So, X_S * = 3/4 times 800/10 = 60, X_J *= (500)_.6/P_X = 500 times 6/10 times 10 = 30. Y_S * = 1/4 middot 800/20 = 10, y_J * = (500) times 64/200 = 10 (c) Yes two values of P_x & P_y satisfy eq^n in exchange Market because at eq^n X_S * + X_J * = omega_X = 80 + 10 = 90 Y_S * + Y_S