Entrepreneurship

Ada has started a new software company that is really taking off.  Her income in period 0 is $100,000 and she expects it to be $500,000 in period 1.  She has Cobb-Douglas preferences about her consumption in periods 0 and 1 and her utility function is:

U = (C0)^0.4 * (C1)^0.6. 

Ada can borrow or lend at an interest rate of 20%.  Her demand equations can be shown to be the following, where PVI indicates the present value of her income:

C0 = 0.4*PVI

C1 = 0.6*PVI/(1/(1+r)) = 0.6*(1+r)*PVI

Please answer the following questions:

1. How much does Ada consume in each period?  What is her utility?
   
2. Does she borrow or lend in period 0?  How much?
   

The remaining questions focus on how much Ada benefits from being able to borrow and lend. They extend the use of the expenditure function to intertemporal problems and apply it in a slightly different way.

3. Suppose Ada had not been able to borrow or lend.  What would her utility have been?
4. Following the usual series of steps, it is possible to derive an intertemporal version of the expenditure function.  The main difference is that PVI plays the role of M. Please derive Ada's intertemporal expenditure function and show that it can be written:
   
   PVI = U/( 0.4^0.4 * (0.6*(1+r))^0.6 )
   
5. Please use the expenditure function to determine what level of PVI would have been necessary for Ada to have the utility you calculated in part 3.
6. The difference between the value from part 5 and Ada's actual PVI is formally known as an "equivalent variation" because it is the change in her income that would be equivalent (in terms of utility) to removing her ability to borrow or lend. How large is it?