Exercise 1 : Suppose a production function is given by f ( K , L)  K 1/ 2 L1/ 2 ,the
price of capital is $10 and the price of labor is $16.
1. Does the technology exhibits increasing, decreasing or constant return
to scale?
Answer: Constant return to scale.
2. What is the marginal product of capital? of labor? What is the Marginal
rate of technical substitution?
MPL
K
1 1/ 2 1/ 2
1
L K , MPK  K 1/ 2 L1/ 2 , MRTS  

2
2
MPK
L
3. The capital is fixed at the level K  4 . Is it long run or short run
Answer: MPL 
analysis?
(a) What is the quantity of labor that minimizes the cost of producing any
given input?
Answer: q  41/ 2 L1/ 2  2 L1/ 2  q  L 
q2
4
(b) What is the minimum cost of producing q units of output (long-run)?
Answer: cos t  10  4  16 
q2
 40  4q 2
4
4. What will be the long run optimal bundle to produce a given output? And
compare the long-run cost with the short-run cost?
Answer: MRTS  
w
K 10
  , then substitute this equation into the
r
L 16
production function, and you can solve out optimal input for labor and
capital. Put optimal labor and capital into the cost function, then we get the
long-run cost function.
Exercise 2 : Mary consumes only apples and chocolate. Fortunately she likes both
goods. The consumption bundle where Mary consumes x1 units of apples per week
and x2 units of chocolate per week is written as ( x1 , x2 ) . The preferences of
Mary are represented with the following utility function U ( x1 , x2 )  x12 x2 .
1. On a graph plot several points that lie on the indifference curve that
passes through the point (1, 4), and sketch this curve. What is the level of
utility that corresponds to this indifference curve? What is the equation of
the indifference curve? Do the same with the indifference curve that passes
through the point (2, 6). What is the level of utility? What is the equation
of the indifference curve?
2. Use pencil to shade in the set of commodity bundles that Mary prefers to
the bundle (1, 4).
Use red ink to shade in the set of all commodity bundles (x1, x2) such that Mary
prefers (2, 6) to these bundles.
3. Determine the marginal rate of substitution, MRS(x1, x2). Give the economic
definition of the marginal rate of substitution.
(a) What is the slope of Mary’s indifference curve at the point (1, 4)?
(b) What is the slope of her indifference curve at the point (2, 6)?
(c) What is the slope of his indifference curve at the point (2, 1)? and at the
point (1, 24)?
(d) Do the indifference curves you have drawn for Mary exhibit diminishing
marginal rate of substitution? Explain.
4. Suppose that the price of each good is 2 and the income of Mary is 20. Draw
her budget line.
5. What is the optimal bundle? At this optimal bundle, what is the level of
utility of Mary?
6. What happens to the optimal bundle if price of x1 increases? Decompose the
price effect into substitution effect and income effect graphically.
ANSWER: 1. The level of utility that corresponds to this indifference curve
that passes through the point (1, 4) is U(1, 4) = 1 ×4 = 4. The equation of the
indifference curve is x2  4 / x12 . The level of utility that corresponds to this
indifference curve that passes through the point (2, 6) is U(2, 6) = 2 2 × 6 = 24.
The equation of the indifference curve is x2  24 / x12 .
2. GRAPH. The set of commodity bundles that Mary prefers to the bundle (1, 4) is
on the right of the IC of equation x2  4 / x12 . The set of all commodity bundles
(x1, x2) such that Mary prefers (2, 6) to these bundles is on the left of the IC
of equation x2  24 / x12
.
3. MRS x2 forx1  
MU x1
MU x2

2 x1 x2
2x
 2
2
x1
x1
(a) The slope of Mary’s indifference curve at the point (1, 4) is MRS(1, 4) =-8.
(b) The slope of her indifference curve at the point (2, 6) is MRS(2, 6) = -6.
(c) The slope of his indifference curve at the point (2, 1) is MRS(2, 1) =-1
and at the point (1, 24) is MRS(1, 24) =-48.
(d) The indifference curves you have drawn for Mary exhibit diminishing
marginal rate of substitution as along the same IC of equation x2  4 / x12 , the
MRS is decreasing: at the point (1, 4) it is -8 and at (2, 1) it is -1. Same
along the IC x2  24 / x12 : the MRS is decreasing: at (1, 24) it is -48 and at
the point (2, 6) it is -6.
4. p1  p2  2 and m = 20. The budget line is 2 x1  2 x2  20 or equivalently
x2  10  x1 .
5. At the optimal bundle, the MRS x2 forx1  
p1
2x
2
. So  2   ,
p2
x1
2
i.e., x1  2 x2 .Plug that into budget constraint, we can solve out
x1 
20  10
, x2 
3
3
.
Thus the level of utility of Mary is U ( x1 , x2 )  148.15 .