Question

Question 1 Chepa’s utility function is given by U(x,y) = lnx + 4lny. Assume that Chepa has endowments (10,10) and that P_y = 10 throughout the problem.

1. (Note: this part of the question is intended to reduce your workload. If you prefer not to work with a general demand function but calculate the demand separately for each of the cases, you can do this part after part (f).) Given P_y = 10, solve for the optimal bundle for Chepa as a function of P_x and money income M.
2. Express Chepa’s endowment income as a function of P_x. Using this expression and your answer to (a), find the range of values of P_x such that Chepa will be a net seller of good x.

Answer 1

According to the given question, Chepa’s utility function is given by U(x,y) = lnx + 4lny. We have to assume that Chepa has endowments (10,10) and that Py = 10 throughout the problem. Now we will try to answer the follwoing questions with the help of the information.

(a) We have to find the optimal bundle i.e. (x*,y*) as functions of Px and M.

First we will find Chepa's budget constraint and will maximize the utility function subject to the budget constraint.

Hence, Price of x is Px and Price of y is given as Py. Money income is M.

Hence, Budget Line: x.Px+10Py=M........(1)

Hence our problem is,

Max U(x,y)=lnx+4lny subject to x.Px+10Py=M

According to the Utility Maximization Theory, at the optimum, the Marginal Rate of Substitution i.e. MRS equals the Price Ratio i.e. Px/Py. We will do the same here first. The calculations are shown below.

From the above calculations we get 10y=4x.Px.......(2)

Now we will put the value in the budget line to solve for x* and y*.

From the budget line

x.Px+10y=M

or, x.Px+4x.Px=M (from (2))

or, 5x.Px=M

or, x* = M/5Px

and, from (2)

10y=4x.Px

or, 10y=4.(M/5Px).Px

or, y* = 2M/25

Hence the optimal bundle is

(x*,y*)=(M/5Px , M/25).

(b) Chepa's endowment is (10,10). Now her endowment income is the value of her endowment.

Hence, thr price of x is Px and price of y is Py=10.

Hence, Endowment Income = Px.10+10×10 =10Px+100

Hence, Chepa's endowment income as a function of Px is M = 10Px+100.

Hence, if we put this value of M in part (a), we will get x* and y* as functions of Px only.

Hence, x* = M/5Px = {10Px+100}/5Px........(3)

and, y* = M/25 = {10Px+100}/25.............(4)

fNow we will have to find the range of values of Px such that Chepa will be a net seller of good x.

Chepa's endowment of x is 10. If she would be a net seller of x, then her endowment of x must be higher than her consumption of x, thus she will have some of x left to sell. Here will will set the consumption of x i.e. x* less than 10. The calculations are shown below.

From the above calculations, we can see the range of Px should be greater than 5/2 or 2.5.

Hence, Chepa will be a net seller of good x if range of Px is Px>2.5.

Hope the solution is clear to you my friend.