Please show all steps and derivations. Make sure that your graphs and all their axes are correctly and neatly labelled.

Q$.$ Lucy works in her family farm and gets paid $M$.$ The only goods that she consumes are rice $()$ and wheat $($w$).$ Her utility function for these two goods is given as:

Ur$,$ w$)=$ In$($r $-1)+$ w

a$.$ What is Lucy's MRS for rice and wheat?

b$.$ If the price of rice is denoted as P$,$ and the price of wheat is denoted as Pw$,$ derive Lucy's demand functions for rice and wheat $($Hint: consider all possible cases of prices and income$)$

c$.$ IfM$=$ $$5,$ P$,=1,$ and Pw $=3,$ what is the quantity of rice and wheat that will maximise her utility? Graphically illustrate Lucy's budget line, indifference curve and the optimal bundle in this case.

d$.$ Considering prices to be the same as part $($c$)$ above, graphically illustrate Lucy's Engel curve for rice. Make sure to mark its slope.