Microeconomics I (Fall 2009) Problem Set 1 Sample Solutions

By dividing (1) by (2), we have x 2 /x 1 = p 1 , or x 2 = p 1 x 1 . By substituting this
into the budget constraint, the demand functions for goods 1 and 2 are given by
x 1 = Y
2p 1
,
x 2 = Y 2 .
Note that the demand function for good 1 is again hyperbolic.
2. u(x 1 , x 2 ) = min{2x 1 , x 2 }.
(a) (Omitted. Please refer to Figure 4.4 (b) on p. 82 in the textbook for the case of
u(x 1 , x 2 ) = min{x 1 , x 2 }.)
(b) Let us consider a point (x 0 1, x 0 2) on x 2 = 2x 1 . Then, x 0 2 = 2x 0 1 (or equivalently
x 0 1 = x 0 2/2) holds.
i. (x 1 , x 2 ) = (x 1 , x 0 2), x 1 > x 0 2/2: In this case, from 2x 1 > x 0 2, we have
u(x 1 , x 2 ) = min{2x 1 , x 0 2} = x 0 2. An increase in x 1 , therefore, does not raise
utility. Thus, MU 1 = 0. On the other hand, MU 2 = ∂u(x 1 , x 2 )/∂x 2 =
∂x 2 /∂x 2 = 1. Thus, MRS = −MU 1 /MU 2 = 0.
ii. (x 1 , x 2 ) = (x 0 1, x 2 ), x 2 > x 0 2: Since x 2 > x 0 2 = 2x 0 1, u(x 0 1, x 2 ) = min{2x 0 1, x 2 } =
2x 0 1. From ∂u(x 1 , x 2 )/∂x 2 = ∂(2x 1 )/∂x 2 = 0, MU 2 = 0. Besides, from
∂u(x 1 , x 2 )/∂x 1 = ∂(2x 1 )/∂x 1 = 2, MU 1 = 2. Therefore, MRS = −2/0 =
−∞.
iii. (x 1 , x 2 ) = (x 0 1, x 0 2): In this case, MRS is not defined.
(c) Rearranging the budget constraint, p 1 x 1 + x 2 = Y , gives x 2 = Y − p 1 x 1 . From the
shape of indifference curves, it is inferred that utility is maximized on x 2 = 2x 1 .
Solving the simultaneous equations,
x 1 (p 1 ) =
x 2 (p 1 ) =
Y
p 1 + 2 ,
2Y
p 1 + 2 .
You already know how to depict the graph of x 1 (p 1 ).
3. u(x 1 , x 2 ) = x 1 + x 2 .
(a)
MRS = − MU 1
MU 2
= −1.
(3)
2