Exercise 1 : Suppose a production function is given by
1/ 2 1/ 2
( , )f K L K L
,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?
2. What is the marginal product of capital? of labor? What is the Marginal rate
of technical substitution?
3. The capital is fixed at the level
4K
. Is it long run or short run analysis?
(a) What is the quantity of labor that minimizes the cost of producing any
given input?
(b) What is the minimum cost of producing
q
units of output (long-run)?
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?
Exercise 2 : Mary consumes only apples and chocolate. Fortunately she likes
both goods. The consumption bundle where Mary consumes
1
x
units of apples
per week and
2
x
units of chocolate per week is written as
1 2
( , )x x
. The
preferences of Mary are represented with the following utility function
2
1 2 1 2
( , )U x x x x
.
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
1
x
increases? Decompose
the price effect into substitution effect and income effect graphically.
