Question

1. Ryan likes to read and watch premium CSPAN. His utility for a given month is represented by U(x1,x2) = x1,x2, where x₁ equals the number of books he reads and x₂ is the number of hours of CSPAN he watches. He has a budget of $160 per month for entertainment expenses, and these are the only two entertainment activities he spends his money on. Reading is $8 per book and each CSPAN rental costs him $16.
   1. Graph Ryan’s budget constraint.
   2. In June, Ryan read 12 books and rented 4 CSPANs. Would his utility increased, decreased, or remained the same if he bowled 8 books and watched 6 CSPANs in July?
   3. Did Ryan behave rationally in June? Why or why not?

Answer 1

Ryan likes to read and watch premium CSPAN. His utility for a given month is represented by

U(x1, x2) = x1.x2

x1 = the number of books he reads

x2 = the number of hours of CSPAN he watches

He has a budget of $160 per month for entertainment expenses, and these are the only two entertainment activities he spends his money on. Hence, his income is

M = $160

Reading is $8 per book (x1). Hence, price per book is

P1 = $8

And, each CSPAN (x2) rental costs him $16. Hence, price of CSPAN is

P2 = $16

Now, Ryan buys Book (x1) and watches CSPAN (x2) with budget M = $160. Hence, Ryan's budget constraint is

P1.x1 + P2.x2 = M

Here, P1 = $8, P2 = $16 and M =$160

Hence,

  8.x1 + 16.x2 = 160

or, x1 + 2.x2 = 20..........(1)

This is the budget constraint of Ryan. This is drawn below in the following diagram.

✓ Ryan's utility function is

U(x1, x2) = x1.x2

In June, Ryan read 12 books and rented 4 CSPANs.

Hence, x1=12 and x2=4.

His utility in June is

U(12, 4) = 12×4 = 48

or, U(June) = 48

In July, if Ryan bowled 8 books and watched 6 CSPANs, Hence, x1=8 and x2=6. Hence,

His utility in July is

U(8, 6) = 8×6 = 48

or, U(July) = 48

We can see that,

U(12, 4) = U(8, 6)

or, U(June) = U(July)

Hence, Ryan's utility remained the same if he bowled 8 books and watched 6 CSPANs in July.

✓ Let us check Ryan's optimal consumption bundle in June.

From the utility function, let's calculate the MRS or Marginal Rate of Substitution.

Hence, U(x1, x2) = x1.x2

Marginal Utility of x1 is

MU1 = dU/dx1 = x2

And, Marginal Utility of x2 is

MU2 = dU/dx2 = x1

Hence, MRS = MU1/MU2

or, MRS = x2/x1.........(2)

Now, at optimal consumption, MRS equals the price ratio of x1 and x2 i.e. (P1/P2)

Hence, at equilibrum,

MRS = P1/P2

or, x2/x1 = 8/16

or, x1 = 2.x2.........(3)

From the budget line, we put x1=2.x2 and get

x1 + 2.x2 = 20

or, 2.x2 + 2.x2 = 20

or, 4.x2 = 20

or, x2* = 5

And, from (3) we get

​​​​​​ x1* = 2.x2* = 2×5

or, x1* = 10

Ryan's Rational Choice is 10 books and 5 CSPANs.

If Ryan was bevaving rationally in June, then he must have read 10 books and watched 5 CSPANs in June.

But, in June, Ryan read 12 books and warched 4 CSPANs. This is not Rational.

Hence, Ryan did not behave rationally in June.

Hope the solutions are clear to you my friend.