Model Solutions
Econ 115b Midsem
Section I
1. Variable Cost (2 points):
A cost that varies with output, rising as more output is produced and falling as less output
is produced.
Note: Partial credit answers --- ‘Variable Cost = Total Cost – Fixed Cost’ with no
definition of fixed cost
2. Inferior Good (2 points):
A good for which quantity demanded falls as income rises. Its income elasticity is
negative.
3. Income elasticity of demand (2 points):
A measure of responsiveness of Quantity demanded to a change in income. Specifically,
% change in quantity demanded divided by % change in income
Note: Answers with ‘correct’ definition of price elasticity of demand gets partial credit of
one point.
4. Increasing returns to scale (2 points):
Output increases more than in proportion to inputs as the scale of production increases.
Equivalently, f(L, K) > f(L, K) where >1
Note: partial credit ---- When IRTS is ‘correctly’ explained in terms of Decreasing
Average Cost
5. Consumer Surplus:
The difference between the total value (willingness to pay) of all units consumed of a
good and the total paid for those units. Area between demand curve and price charged for
the good, out to total quantity exchanged.
Extra Credit:
Even though the answer is ‘yes’, no extra credit point will be given if no name, no TA
name, and indecipherable signature.
Section II
Question 1
(a)
Foobars
Widgets
(b)
The production possibilities frontier (PPF) needs to have a negative slope, given
fixed resources (inputs used in production) and the fact that the country of Whim is
producing efficiently (not wasting any resources). Production of one more unit of widgets
can only be achieved by cutting down the production of foobars and vice versa. This
implies a negative slope for the PPF.
The PPF also needs to be bowed out from the origin, to exhibit diminishing
marginal returns (as required by section a above). Diminishing marginal returns (DMR)
implies increasing opportunity cost in terms of the other good as more of this good is
produced. For example, as widget production increases for each additional widget the
amount of foobars that needs to be sacrificed increases. This shows up in the figure above
by an increasing absolute value of the slope of the PPF as we move right.
[Note: one point was deducted for not mentioning fixed resources, mentioning a tradeoff
in production is not enough. Similarly one point was deducted if DMR was not
mentioned or if DMR was explained in a sufficiently innovative (wrong) way. ]
(c)
As shown in the figure below if Whim was producing at A and nation 2 was
producing at B before trade, then Whim will have a comparative advantage in widgets
and nation 2 will have a comparative advantage in foobars. We can see this by the fact
that at A the slope of Whim’s PPF is lower in absolute value (or flatter) than the slope of
nation 2’s PPF at B. This implies that the opportunity cost of widgets or the number of
foobars that needs to be given up to produce one more widget is lower in Whim than in
nation 2. Therefore by definition Whim has a comparative advantage or lower
opportunity cost for widgets.
Foobars
Nation 2’s PPF
A
B
Whim’s PPF
Widgets
[Note : no points awarded if the slopes were reversed, i.e. if points A and B were chosen
such that the slope at A was actually steeper than at B (with widgets on the x-axis). ]
(d)
Starting from the points mentioned in part (c) above, one1 mutually beneficial
trade would be for Whim to produce one more unit of widget and nation 2 to produce one
less unit of widgets. Then the total amount of widgets stays the same but the total amount
of foobars available for consumption would have increased.
Say the slope of Whim’s PPF at A is 0.5, then to produce one more unit of
widgets it needs to give up half a unit of foobars. Similarly let us say that the slope of
nation 2’s PPF at B is 1.5 (it needs to be greater than 0.5 for Whim to have a comparative
advantage in widgets). Then by giving up one unit of widget nation 2 can produce 1.5
units of foobars. Therefore the overall impact of the trade is that total production of
widgets stays the same but the quantitiy of foobars increases by one unit. This extra unit
of foobars can be distributed between the two countries such that the consumption of
foobars increases in both countries.
All of the following are good reasons why this trade works : (i) Whim has a
comparative advantage in the production of widgets, (ii) the opportunity cost of
producing widgets is lower in the country of Whim, (iii) Whim is relatively more
efficient in the production of widgets. Conversely, one can say the same things about
nation 2 and foobars, because there are only two countries and two goods. (If Whim has a
comp. adv. in widgets then nation 2 has a comp. adv. in foobars and so on.)
[Note : two points for describing a trade, in addition one point for a reasonable
explanation of why the trade works, or full two points for mentioning any of the reasons
mentioned above. ]
1
There are more than one valid trades as long as the production of one or both goods increases and the
production of none decreases. Also remember that the countries should not overspecialize, it should not end
up at a point where the magnitudes of the slopes are reversed in the graph, they should stop when the slopes
are equalized as there are no more gains from trade from specializing.
Question 22.
(a)
$
S
P*
D
Q*
Quantity of Milk, Q
The equilibrium price (P*) and quantity (Q*) of milk are found where the market demand
(D) and market supply (S) curves intersect, as shown in the diagram. At the equilibrium
price, consumers receive the exact quantity of milk they are willing to purchase at that
price, and producers sell the exact quantity they are willing to sell at that price.
Therefore, at the equilibrium price and quantity, there is no incentive for either
consumers or producers to alter the price.
(b)
$
Excess
Supply
S
P’
P*
D
QD Q* Q S
Quantity of Milk, Q
A higher than equilibrium price (P’) creates an excess supply or quantity surplus3 equal to
Qs - QD. This is because at P’ producers are willing to supply Qs whereas consumers are
willing to purchase QD units of milk. As each producer discovers4 the surplus, she starts
lowering her price by a little bit so that she can undercut her competitors, and take
business away from them. However, other producers follow suit, since they do not want
to end up being the ones with the unsold surplus. At each new price above P* producers
continue to cut prices in an attempt to secure more of the market. Hence, the price of
milk is lowered continuously and consumers buy more at each lower price. This process
All diagrams refer to the “Market of Milk”
Do NOT confuse with Economic Surplus or Economic Welfare.
4
Recall that there is perfect information in a perfect competitive environment.
2
3
stops when the quantity supplied is equal to the quantity demanded. This happens at
exactly (Q*, P*), the equilibrium quantity and price.5
(c)
Excess
Supply
$
S
Pfloor
P*
D
Q’ Q*
Quantity of Milk, Q
As shown in the diagram, the result of the price floor on milk is a decrease in quantity
from Q* to Q’.
(d) The lost ‘economic surplus’ or dead-weight loss is the shaded triangle in the
diagram above.
(e)
$
S
Pfloor
P*
D
Q’’
Q’ Q*
D’’
Quantity of Milk, Q
The new, shaded triangle is the dead-weight loss with a more elastic demand curve (D’’).
The increase in the elasticity of demand led to an increase in the dead-weight loss as
expected.
5
Note that producers do not know the equilibrium price and/or quantity ahead and aim to achieve that by
cutting prices or quantity. They only observe an excess supply or demand and, therefore, understand that
they are not at equilibrium and act accordingly. They only know that they have reached the equilibrium
price and quantity when they eliminate the excess supply or demand.
Section III:
Question 1.
(a) At the optimum6 Bucky's consumption choice must satisfy the following condition
MU beer
p
MU beer MU bratwurst
.
 beer or equivalently

p beer
p bratwurst
MU bratwurst p bratwurst
This condition is fulfilled at the point of tangency between indifference curve and budget
constraint.
Why? There are many acceptable ways to explain it, you need only one.
1) The budget set (delimited upward by the budget constraint) is the set of bundles that
Bucky can afford. Each indifference curve represent a different level of utility, the further
out to the northeast, the higher the utility. Because of the shape of the indifference curves
(given by the assumption of diminishing MRS) and of the shape of the budget set, the
highest affordable indifference curve is the one just tangent to the budget constraint.
2) Bucky is equalizing per dollar marginal utility. By contradiction, imagine the situation
in which the per dollar marginal utility is not equalized. Suppose he has a higher per
dollar marginal utility in bratwurst, this means that he could increase his overall (beer +
bratwurst) utility by saving the last dollar spent in beer and buying one dollar more of
bratwurst. And so on, until the per dollar marginal utilities are equalized.
3) Same as explanation 2) but with symbols. Bucky is at the optimum when saving one
dollar on one good and spending it on the other leaves him with the same overall utility
(that is, it does not worth to change). If Bucky spends 1$ more in x1 he gets x1 units
more where x1 = 1/ p1. Multiply both sides by MU1 to find the impact that this change
has on his utility MU1* x1 = MU1/p1. In order to spend this 1$ more on x1 and still
fulfill his balance constraint Bucky has to save it from his consumption of x2. So he will
be able to buy x2 units less of x2 where x2 = 1/ p2. Multiply both sides by MU2 to find
the impact that this change has on his utility MU2* x2 = MU2/p2. Bucky is at an
optimum when the utility he gains from buying more of x1 is exactly equal to the utility
he looses by buying less of x2. That is when MU1* x1 = MU2* x2.
Putting the 3 equalities
MU1* x1 = MU1/p1
MU2* x2 = MU2/p2
MU1* x1 = MU2* x2
together we find the optimality condition MU1/p1 = MU2/p2
(b) Bucky's income has increased, the budget line moves outward with a parallel shift. A
good is normal if when the income increases, the quantity purchased of that good
increases too (and when the income decreases the quantity purchased decreases). Because
we are told that both goods are normal the consumption of both goods must increase too.
Let's denote by B and W the quantity of beer and bratwurst respectively. By pb and pw
their prices. And by Y and Y' the income before and after the increase. Intercepts on the
6
Provided that Bucky consumes a positive quantity of both goods.
vertical axis are Y/pw and Y'/pw and on the horizontal axis Y/pb and Y'/pb. The slope is pb/pw in both cases because the shift is parallel.
Bratwurst
W*'
new indifference
curve
W*
Beer
B*
B*'
(c and d)
Let's consider an increase in the price of beer from pb to p'b where p'b > pb
In real life we will have one total effect. The consumption choice shifts from A to C (the
exam did not require to show the total effect in a separate graph, you show the total effect
anyway in the income and substitution graph).
Bratwurst
new indifference curve
new budget
constraint
C
A
old indifference curve
Beer
Total effect
To drive the students crazy and for deeper theoretical reasons7, we want to split this
change from A to C (called total effect) into a substitution and an income effect. This
split is only fictional, you cannot see it on the market. When a price goes up it causes two
effects: 1) the relative price (= ratio of the two prices) changes, so one good now looks
more attractive than before, and 2) the old indifference curve is not achievable anymore,
from a practical point of view the person is "poorer".
The substitution effect singles out the effect on relative prices. To do so, it fictitionally
gives more8 income to the person, just enough income to make his old indifference curve
affordable and computes the fictional choice at the new price ratio.
The income effect considers the effect of the reduction9 in income from the fictional
income back to the actual income (of course, this is all fiction, the actual income never
changes). It is to this change in income (from fictional back to new/actual) that the
definitions of normal and inferior goods refer to10.
In the graph, the substitution effect is the movement from A to B and the income effect is
the movement from B to C
Old budget constraint: vertical intercept Y/pw, horizontal intercept Y/pb, slope -pb/pw
Fictional11 budget constraint: vertical intercept Y'/pw, horizontal intercept Y'/p'b, slope p'b/pw
Bratwurst
new indifference curve
fictional budget constraint
new budget
constraint
B
IE
C
A
old indifference curve
old budget constraint
Beer
IE
7
SE
e.g. the substitution effect is linked to the taxes' dead weight loss, while the income effect is not.
Recall that we are considering an increase in price, if it was a decrease it would be "less income".
9
Again, we are considering an increase in price, otherwise it would have been "an increase in income"
10
and NOT to the change from the old./actual to the fictional
11
More properly called compensated budget line
8
New budget constraint: vertical intercept Y/pw, horizontal intercept Y/p'b, slope -p'b/pw
Again, note that fictional budget constrain and new budget constraint are parallel and that
there is a change in income from the fictional to the new budget constraint.
In the picture both goods are normal because comparing B to C Bucky consumes less
beer and less bratwurst in C than in B.
Question 2.
(a) the present value of income Y is defined as the discounted sum of all future incomes.
It is customary to write the formula in the following way,
n
Yt
PV(Y) = 
t
t  0 (1  r )
but it created a lot of confusion.
Let's try not to make too much confusion with the t-index. When the formula is written is
this way, the t-index at the exponent of the denominator refers to how many years elapse
from the period in which we are computing the present value to the period in which you
receive the income. The t-index in the numerator instead is just notation.
In the exam there are two periods: period 1 and period 2. Because we are asked to
compute the present value in period 1, the t associated to period 1 is 0 (no period elapses
from period 1 to period 1) and the t associated with period 2 is 1 (one period elapses from
period 1 to period 2).
So PV(Y) = Y0/(1+r)0 + Y1/(1+r)1 = Y0 + Y1/(1+r) = Y/(1+r)
The above way is perfectly correct and you got full score if you did not get mixed up with
the indices. But I find the above way confusing because the income in period 1 is denoted
by Y0, while Y1 denotes the income in period 2.
So if it helps you, think in the following way12: the present value in t is
n
Yj
PV(Y) = 
j t
j t (1  r )
where the j represent the different periods in which you receive the incomes. So with this
notation, to compute the present value in period 1, we have Y1 as the income received in
1 discounted j-t = 1-1 = 0 times. And Y2 income received in period 2 discounted j-t = 2-1
= 1 times.
So PV(Y) = Y1/(1+r)1-1 + Y2/(1+r)2-1 = Y1 + Y2/(1+r) = Y/(1+r)
The same result, as it must be!!!!!!!!!!
Choose whatever notation you want as long as you do it correctly.
(b) The multi-period (here two-period) budget constraint is always PV(C) = PV(Y).
Where PV(C) means "present value of consumption".
We computed PV(Y) above.
12
If it does not help you, disregard it.
PV(C) = C1/(1+r)1-1 + C2/(1+r)2-1 = C1 + C2/(1+r) (sticking to my favorite way of writing
it)
So PV(C) = PV(Y) becomes C1 + C2/(1+r) = Y/(1+r) which is the two-period budget
constraint.
There are many acceptable, equivalent way of rearranging the above constraint. It is just
algebra (none of these were asked in the exam, but they have all been accepted).
- C1 + (Y- C2) /(1+r) = 0
C1 = (Y- C2) /(1+r)
In period 1, the price of C1 is 1 and the price of C2 is 1/(1+r).
How do you find them? It is way easier to see it algebraically than to understand why.
Let's start with the easy part. Algebraically, if we had two goods instead of two periods
the budget constraint would be p1*x1 + p2*x2 = y, which means (price)*(variable on one
axis) + (price)*(variable on the other axis) = (something which is not multiplied by
anything on the axis). So if we take the two-period budget constraint C1 + C2/(1+r) =
Y/(1+r) the variables on the axis are C1 and C2, so we find that the price of C1 is 1 and the
price of C2 is 1/(1+r)
Why? The price of C1 in period 1 is normalized to 1 by assumption. The price of C2 in
period 2 is also normalized to 1 by assumption. The price of C2 in period 1 is that amount
of money that once invested at interest rate r gives 1 in period 2. It is the amount x such
that (1+r)*x=1. So it is 1/(1+r).
Many people wrote the prices in period 2. In period 2, the price of C1 is (1+r) and price of
C2 is 1. I considered this answer correct because part b of the question did not specify in
which period the prices should have been quoted, although from part a you could have
understood that they were supposed to be quoted in period 1 units.
c) The optimal two-period consumption is at the tangency between the two-period budget
constrain and the indifference curve. Tangency occurs as usual when MRS = |ratio of
prices|. So MU1(C1)/MU2(C2) = 1/(1/(1+r)). That is, MU1(C1)/MU2(C2) = (1+r). The
optimal consumption points are C1* and C2*.
C2
pay back
what
borrowed
C2 *
Indifference curve
C1 *
borrowing
C1
(Vertical intercept Y, horizontal intercept Y(1+r).)
Because John has no income in period 1 he must borrow the entire amount C1*. If we
want to know how much that is, we solve the budget constraint for C1 and plug in the *.
So borrowing is C1* = (Y- C2*) /(1+r)
How much will he pay back in period 2? (the question did not ask to do it) In period 2 he
will pay back what he borrowed plus interest, that is (1+r)*C1* = (1+r)* (Y- C2*) /(1+r) =
Y- C2*
So in period 2 he consumes C2* and pays back Y- C2*, that is he uses up all his income
as it must be.
Section IV
Question 1.
(a) Because equipment is fixed in the short run, the firm’s demand for the variable input
(natural gas (G)) is its Marginal Revenue Product of Gas (MRPG). MRPG, you’ll recall,
is the monetary benefit to the firm from adding an extra unit of the input. MRPG is
determined by multiplying the marginal product of an additional unit of the input (MPG)
times the price of the firm’s output (P). Optimally, the firm should purchase a quantity of
gas such that the MRPG of the last unit of gas is equal to the price of gas (rG).
$
rG
MRPG
G*
Quantity of Gas
Points were deducted for 1) not identifying MRPG as the relevant concept and 2) not
indicating how rG determined the optimal amount of gas.
(b) If the firm can choose both inputs, the cost-minimizing way to produce Q units of
output is by finding the tangency between the Q isoquant and the lowest possible isocost
line. The coordinates of this tangency point are the optimal amount of gas (G) and
equipment (E).
Quantity of
Equipment (E)
E*
Q isoquant – slope=MPG/MPE
Isocost line – slope=PG/PE
G*
Quantity of Gas (G)
Points were deducted for 1) incorrect labeling of lines, intercepts, and axes, and 2) failure
to indicate the optimal combination of inputs.
(c) The prior graph provided the optimal quantities of gas and electricity (G* and E*) that
should be used to produced Q units. Combining this information with the prices of the
two inputs (rE and rG.), we calculate the total cost as follows: E*rE + G*rG.
(d) A firm should always produce the quantity of output where the last unit’s marginal
revenue is equal to its marginal cost. New Haven Power must sell electricity at $5 per
unit, so the marginal revenue is just the price, $5. The proper equation is: P=MC, or
$5=MC. The graph is:
$
MC
price=$5
Q*
Quantity of Output (Q)
Full credit was also given for a graph showing total revenue and total cost curves, with an
optimal output where the difference between the curves is greatest (where slopes are
equal):
$
TR
Q*
TC
Quantity of Output (Q)
Points were deducted for 1) failure to state the appropriate equation, 2) lack of a relevant
graph, and 3) failure to indicate optimal quantity on the graph.
(e) When there are multiple firms in the market, the supply curve is the horizontal
summation of the individual firms’ marginal cost curves. MCNH=New Haven Power
MCQ=Quinnipiac Power
$
MCNH
MCQ
Supply Curve
price=$5
QNH* QQ*
Qtotal= QNH*+QQ*
Points were deducted for failure to specify that the summation must be horizontal rather
than vertical.
(f) It is not possible for two firms purchasing inputs in the same market to lower costs by
trading inputs. Two different arguments received full credit:
--Argument from cost-minimization: Both companies are choosing their output level
where P=MC. They derived their marginal cost curves from their respective total cost
curves (MC is the slope of TC). The total cost curves themselves are a schedule of
lowest-cost ways to make each different level of output, derived from the tangency
between each and every isoquant and its particular cost-minimizing isocost line. The
tangency point results, in part, from the prices of inputs, which determine the slope of the
isocost line. Both firms buy inputs in the same market, so they face the same prices. So
both firms are already using the cost-minimizing combination of inputs; by definition, no
other combination can result in lower costs, so there is no reason to trade inputs.
--Argument from marginal product ratios: Trading inputs only lowers overall costs if the
trade allows at least one firm to make more output for the same cost. For instance, New
Haven Power might voluntarily trade a unit of gas to Quinnipiac Power if it received
exactly enough equipment in exchange to make up for the lost gas. The output that was
being produced at New Haven Power by that unit of gas was MPG-NH, and the firm would
use the following equation to determine how much equipment is needed in exchange:
(E)(MPE-NH)=MPG-NH. We can rewrite this as E=MPG-NH/ MPE-NH. Then overall costs
could only be lowered if Quinnipiac Power could make more output with the extra unit of
gas (MPG-Q) than with the equipment it is giving up in the trade (E). The equation
describing that situation is the following: MPG-Q > E(MPE-Q), which can be rewritten as:
MPG-Q/ MPE-Q > E. Since E is the same (the equipment Quinnipiac gives up is exactly
the equipment they trade to New Haven), the trade only lowers costs if MPG-Q/ MPE-Q >
MPG-NH/ MPE-NH. However, both firms are purchasing their inputs in the same markets,
so they face the same prices. Both are already choosing their inputs such that MPG/
MPE=rG/rE. Therefore rG/rE = MPG-Q/ MPE-Q = MPG-NH/ MPE-NH. The inequality that was
needed for a successful trade cannot exist.
Clarifications:
Section I: Minchul Kim
Section II
Q1: Mainak Sarkar
Q2: Paris Cleanthous
Section III: Renzo Comolli
Section IV: Shannon Bothwell