Download Problem Set #3

Problem Set #3
Suggested Solution
1.
(a) Marginal Revenue Product –
The change in revenue as a result of a change in production input such as labor.
(b) Iso -Cost Curve –
A curve illustrating all input combinations that can be purchased for a given total cost,
TC.
(c) Cost-Minimizing Input Choice –
The least expensive input combination that yields a certain desired output level Q.
(d) Marginal Cost –
The change in total cost as a result of a small increase in units produced.
2.
(a) Large Substitution effect – for a given change in relative price, the change in the
optimal consumption bundle is large.
The substitution effect will be larger as the indifference curve is “flatter”. A relatively flat
indifference curve represents a consumer who views the two goods we consider as good
substitutes and therefore is more responsive to a given change in relative price.
The extreme case is a straight line indifference curve representing perfect substitutes. In
such a case, as the relative price change the consumer can move from one extreme case
of purchasing only one good to purchasing only the other good (corner solutions). A
relatively “flat” indifference curve will occur when the two goods we consider are
similar.
(b) Large Income effect – as the purchasing power change, the change in the optimal
consumption bundle is large.
A large income effect will happen when the percentage spending on the good we
consider, out of our total income, is large. Related to this, income effect for a luxury good
is large relatively to the income effect for a necessity.
(c) Demand more Elastic –
Recall that the relationship between price and quantity demanded is a result of both the
substitution and income effect, i.e. the total effect. As the total effect is greater, the
responsiveness to a change in price is greater, meaning a flatter and more elastic demand
curve.
Assuming a downwards sloping demand curve, the larger the substitution effect the larger
is the total effect, leading to a flatter and more elastic demand curve. A large substitution
effect is a result of a relatively flat indifference curve, as explained in (a). In addition,
because the total effect is composed also from the income effect, then a large normal
income effect, will lead to a more elastic demand curve. A large income effect is a result
of the factors mentioned in (b). Notice that an income effect for an inferior good is in the
opposite direction to the substitution effect and thus decreases the total effect generating
a less elastic demand curve, all else equal.
(d) Upwards sloping demand curve - the relationship between price and quantity
demanded is a result of both the substitution and income effect, i.e. the total effect.
Substitution effect is always in the direction of a downwards sloping demand curve.
Thus, to have an upwards sloping demand curve, we need to have an income effect that is
stronger than, and in the opposite direction of, the substitution effect. An income effect,
in the opposite direction of substitution effect, is the case of inferior good. Therefore, if
we have an inferior good, where the income effect dominates the substitution effect then
theoretically we will have an upwards sloping demand curve. Such a good is a special
case and is called a “Giffen” good.
(e) Complements in Demand – Consider two goods x and y. These two are
Px  Q y 
complements in demand if, for example,
.
Px  Q y 
Suppose Px  : substitution effect leads to buying more of good y and purchasing power
decreases. We are given that the total effect of Px  on consumption of y is negative
(purchasing less of y). This implies that as purchasing power decreases we buy less of y.
Moreover, the income effect needs to be stronger than the substitution effect to receive a
final decrease in the quantity of y. The conclusion is that y must be a normal good
(purchasing power decreases – we decrease the amount of y) and to have a small
substitution effect (such that the income effect will dominate.)
One can do the same exercise for a change in Py and will get similar conclusions
regarding good x.
To summarize, for complements in demand, the two goods need to be normal and to have
a small substitution effect.
3.
(a)
In the graph below we illustrate the Marginal Productivity of labor ( MPL ) and Marginal
Revenue Product of Labor ( MRPL ). These two are downwards sloping curves, since they
reflect diminishing marginal productivity of labor – as we have more workers, the
marginal productivity of an additional worker declines. You can see that also by
analyzing the formula: MPL 
K
2 L
denominator) and MPL decreases.
. As more workers are employed, L increases (in the
MPL, MRPL, W
W
MRPL
MPL
L*
L
An equivalent way to see this is:
MPL, W/P
W/P
MPL
L*
L
(b) The optimal labor in the short run is determined by: MRPL  W .
MRPL is defined as P  MPL  P 
1 K
2 L
. Thus, L* is determined by the following
2
 P 
equation: P 
 W and is equal to L  
 K.
 2W 
2 L
(c)
1 K
*
MPL, MRPL, W
W2
Wage
Increase
Wage
Decrease
W0
W1
MRPL
MPL
L2* L0* L1*
L
Increase
in Labor
Decrease in
Labor
Or, alternatively:
MPL, W/P
W 2/P
W 0/P
W 1/P
MPL
L2* L0* L1*
Increase
in Labor
Decrease in
Labor
L
4.(a), (b) The optimal output of the firm, Q*, is determined by equating marginal cost
(mc) to the price (p), as you can see in the graph below:
mc,
p
mc
p<mc
p
p>mc
Q*
Q
The reason is as follows: on all units where p>mc the firm makes positive profit;
therefore it is optimal for the firm to continue and produce them. On all units such that
p<mc, the firm makes negative profit; therefore the optimal decision is not to produce
these units. As a result, the profit maximizing output level (optimal production) is at the
point where p=mc. At this output level, the firm is producing all units and only units that
contribute positively to its profits.
In this example, the optimal output will be:
pa
p  mc  p  a  bQ  2wQ  Q * 
b  2w
mc1
mc,
p
Wage
increase
p
Q1 Q0
Q2
mc0
Wage
decrease
mc2
Q