Question

Adrienne and Sarah consume pizza, Z, and cola, C. Adrienne’s utility function is UA = ZACA, and Sarah’s is Z0.5 D C0.5 D . Adrienne’s marginal utility of pizza is MUZA = CA. Similarly, MUAC = ZA, MUDZ = 1 2 Z-0.5 D C0.5 D , and MUDC = 1 2 Z0.5 D C-0.5 D . Their endowments are ZA = 10, CA = 20,ZD = 20,CD = 10. a. What are the marginal rates of substitution for each person? b. What is the formula for the contract curve? Draw an Edgeworth box and indicate the contract curve.

Answer 1

a.

$U_A=(Z.C)A, \space MRS_{ZC}=(\frac{\frac{\delta U}{\delta Z}}{\frac{\delta U}{\delta C}})^A=\frac{C}{Z}$

$U_B=(Z^{0.5}C^{0.5})^S,MRS_{ZC}=(\frac{\frac{\delta U}{\delta Z}}{\frac{\delta U}{\delta C}})^S=\frac{0.5C^{0.5}Z^{-0.5}}{0.5Z^{0.5}C6{-0.5}}=(\frac{C}{Z})$

b.

Contact curve is locus of all parieto optimal allocations, so at P.O.

$MRS_{ZC}^A=MRS_{ZC}^S$

$(\frac{C}{Z})^A=(\frac{C}{Z})^S$

As $W^A=(10,20),$ $W^S=(20,10)\space \&0$

$C^S=C-C^A=30-C^A\\Z^S=30-Z^A$

So at P.O

$\frac{CA}{ZA}=\frac{30-CA}{30-ZA}\implies30C_A-CAZA=30Z_A-CAZA\\C_A=Z_A..............contact \space curve.$

c)