A consumer's utility function is given by: U(x,y) = 10xy

Currently, the prices of goods x and y are $3 and $5, respectively, and the consumer's income is $150.
a. Find the MRS for this consumer for any given bundle (x,y).
b. Find the optimal consumption bundle for this consumer.
c. Suppose the price of good x doubles. How much income is required so that the consumer is able to purchase the original consumption bundle (if you were unable to solve d., then take a guess at what the optimal bundle is before solving this)
d. Now that the price of good x has doubled, how much income is needed for the consumer to reach the original level of utility? Is this more or less that what you found in e.?

a. MRS = y/x
b. The MRS=MRT condition says y/x = 3/5. Plug into the budget constraint yields 3x + 3x = 150, or simply x = 25, y = 15.
c. Now the price of x is $6. To be able to purchase y=15 and x=25, this consumer will need an income of 225 which means we will need to increase her income by $75
d. The original bundle gave this person a utility of: U(25,15 )=3750. Now we wish to find how much income is needed to get this level of utility when the prices are now $6 and $5. So the first equation to consider is the indifference curve representing this utility level given by: 10xy = 3750

We also know that in order to be on this IC, the consumer will be at a point where the
MRS = MRT condition holds or: y/x = 6/5

And then plug this into the IC equation to get y = 21.2 and x = 17.7. To be able to purchase the bundle (17.7,21.2 ) the consumer will need an income of 212.2. Thus we need to give her $62.20.