A consumer has a Stone$-$Geary utility function

$U(x_1,x_2)=b_{1}ln(x_1-c_1)+b_{2}ln(x_2-c_2),(x_1>c_1,x_2>c_2)$

where $x_i$ denotes the consumption of the $i-$th commodity and $b_1,b_{2,}{c}_1$ and $c_2$ are positive

constants. The price of the $i-$th commodity is $p_i$ and the consumer's income, $m,$ is such

that

$m>p_1{c}_1+p_2{c}_2$

Show that each indifference curve is negatively sloped, convex and has the lines $x_1=c_1$ and

$x_2=c_2$ as asymptotes. Sketch the indifference curve pattern.

Express the consumer's problem as a constrained maximisation problem. Explain the

significance of the condition $(*).$

Explain with the aid of a diagram how the indifference curve pattern is modified when

$b_1,b_2$ and $c_2$ are positive but $c_1$ is negative.

In the case $c_1=-2,c_2=3,$ find the demand functions which are applicable whenever

$m>3p_2.$