$(25)$ Now consider Cornelius and Wheaton as consumers who eat a balanced diet of both

foods. Denote Cornelius$$ consumption of corn by qc

c and wheat by qc

w$,$ and Wheaton$$s con$-$

sumption of corn by qw

c and wheat by qw

w $.($Note: That is$,$ the subscript denotes the good

consumed, and the superscript the consumer.$)$ Their utility functions are

uc$($qc

c $,$ qc

w$)=9$

$11$ log qc

c $+2$

$11$ log qc

w$,$ uw$($qw

c $,$ qw

w $)=2$

$11$ log qw

c $+9$

$11$ log qw

w $.$

$($a$)(5)$ Find the total income earned by Cornelius Ic and Wheaton Iw$.$

$($b$)(5)$ Set up each consumer$$s utility maximization problem, write the Lagrangians, and

find the first order conditions.

$($c$)(5)$ Find an equation that describes the line of Pareto e$$icient allocations by expressing

Cornelius$$ share of wheat consumption w $=$ qc

w

qc

w $+$qw

w as a function of his share of corn

consumption c $=$ qc

c

qc

c $+$qw

c $.$

$1$

$($d$)(5)$ Find each consumer$$s demand for corn, qc

c $($pc$,$ pw$)$ and qw

c $($pc$,$ pw$),$ and wheat, qc

w$($pc$,$ pw$)$

and qw

w $($pc$,$ pw$).$

$($e$)(5)$ Find the market demands for corn qd

c $($pc$,$ pw$)$ and wheat qd

w$($pc$,$ pw$).$