Microeconomics 2
Tutorial 5: Production Possibility Frontiers
Simple economy:



Two firms (producing and respectively)
Two factors of production ( and )
Fixed amount of inputs available, n.b.
represents ‘quantity of input 1 used by firm 2’ and
so on:
o
o
Three examples:
1. Perfect substitutes technology
2. Perfect complements technology
3. Cobb-Douglas technology
Example 1 – perfect substitutes technology
Firm 1, 1:2 perfect substitutes:




“1 unit of input 1 can everywhere be substituted by 2 units of input 2”
Constant returns to scale
“1 unit of input 1 (and zero of input 2) produces 1 unit of output 1”
Firm 2, 2:1 perfect substitutes:



“2 units of input 1 can everywhere be substituted by 1 units of input 2”
Constant returns to scale
“1 unit of input 2 (and zero of input 1) produces 1 unit of output 2”

(n.b. represent firm 2 as starting in top right-hand corner)
Note: We represent the technologies by drawing isoquants for integer values of output. In principle,
output here is assumed continuous and so an infinite number of isoquants could have been drawn.
To contstruct the Edgeworth box we lay the two diagrams on top of one another, and form the
larger (landscape) diagram. Note that the height and width are both 10 units, this follows the setup
of the economy we are analysing, which has fixed amounts of input 1 and input 2 available.
We are asked to find ‘the locus of efficient points in the use of the two factors’, also known as the
contract curve.
Let’s start by considering when firm 1 produces 8 units of output:



The isoquant
gives possible combinations of inputs used by firm 1 that can produce 8
units of output. We are to take this as given, so the optimal point is where is maximised
subject to this.
Therefore firm 2 has to choose a point along this isoquant where the remaining inputs
available for firm 2 can be used to produce the highest possible output 2. As a result, firm 2
chooses point A – since this is the furthest isoquant (
), for firm 2 from its corner,
which intersects with the isoquant
.
Note that at this point, firm 1 is only using input 1, whereas firm 2 uses all of input 2 and
some of input 1.
Now we next want to consider if firm 1 had a higher output (
. Given this constraint, the
highest output firm 2 can achieve is 10 units, which is reached at point B. In moving from A to B, to
get firm 1 to produce 2 more units of output than in the earlier case, the most efficient way is to
take 2 units of input 1 away from firm 2 and give them to firm 1. Owing to firm 2’s technologies, its
output will only fall by 1 unit as a result of this re-allocation (whilst firm 1’s output increases by 2).
In order for firm 1 to reach an even higher level of output (
), we have to start giving firm 1
some of input 2, since it is already producing using all of the available input 1. Owing to firm 1’s
technologies, we need to give 2 units of input 2 to firm 1 in order to increase its output by 1. But,
because of firm 2’s technology this means will fall by 2, so
.
If we repeat this for all levels of and draw line through optimal points we obtain the contract
curve. In this case, the resulting contract curve runs along the bottom of the Edgeworth box until it
reaches the bottom right-hand corner and then goes up the right-hand side until it reaches the top
right-hand corner.
Points on the production possibility frontier represent the highest possible output for one firm, given
the output of the other. As such, they represent efficient combinations of inputs, and so are
resulting outputs from combinations of inputs on the contract curve.
The kink in the curve is where, to move to the right of the kink, we have to start giving firm 1 input 2
in order to increase its output beyond
. Since firm 1 requires 2 units of input 2 for each extra
unit of ouput, firm 2 will lose 2 units of output for every 1 gained by firm 1. To the left of this point,
giving a unit of input 1 to firm 1 increases its output by 1, and reduces that of firm 2 by only a half.
The optimal point depends upon the relative prices of the two goods, so giving outputs in terms of
:

, then optimal point:

, then optimal point:

, then optimal point:
(if
then optimal set of points lies on red line to the left of and including kink point at
; similarly if
kink point at
then optimal set of points lies on red line to the right of and including
)
Example 2 – perfect complements technology
Firm 1, 1 with 2 perfect complements:


“1 unit of input 1 combined with 2 units of input 2 produces 1 unit of ouput 1”
Constant returns to scale

Firm 2, 2 with 1 perfect complements:



“2 units of input 1 combined with 1 unit of input 2 produces 1 unit of ouput 2”
Constant returns to scale
In finding the locus of efficient points for this economy, with perfect complements for the firms’
technologies, we find that this is actually a contract ‘area’ and not a contract curve. This therefore
means that any allocation of inputs within the contract area will result in efficient output. This
results from the technology, suppose all inputs are given to firm 1:



the most output firm 1 can produce is
using 10 units of input 2 and 5 units of input 1.
Since firm 2 requires some of both inputs to produce anything, the remaining 5 units of
input 1 are wasted.
It therefore doesn’t matter if they are given to firm 1 or firm 2, no additional output can be
created from them.
Now suppose that we set
, in order to produce this firm 1 needs 2 units of input 1 and 4 units
of output 2. This implies that there are 8 units of input 1 remaining and 6 units of input 2. But firm 2,
like firm 1, produces output by using the inputs in a fixed ratio of 2 units of input 1 for every unit of
input 2. This means that firm 2 only needs 4 units of input 2 to produce an output of 4 (using the 8
units of input 1) and that 2 units of input 2 go to waste. The point O is where there exists a single
efficient use of inputs, and is where the inputs are used in their fixed ratios for both firms.
The PPF is constructed in the same way as in the previous example, by using the efficient allocation
of inputs for a given level of output for one of the firms to work out the highest possible output of
the other firm – and plotting these resulting outputs in
space.
Notice that the shape of the resulting PPF is similar to that in the earlier example. The kink point is
where there is no wasted input in the economy. To the left of the kink with increasing firm 2 is
limited by the quantity of input 1 available and wastes input 2, whereas to the right of the kink it is
the other way around.
Once again, the optimal point depends upon the relative prices of the two goods, so giving outputs
in terms of
:

, then optimal point:

, then optimal point:

, then optimal point:
(if
then optimal set of points lies on red line to the left of and including kink point at
; similarly if
kink point at
then optimal set of points lies on red line to the right of and including
)
Example 3 – Cobb-Douglas technology
Firm 1:
Firm 2:
In order to obtain contract curve need to find maximum output of firm 2 given a level of output for
firm 1 whilst taking account of the limited amounts of inputs available. To do this use Lagrangian
found in John Bone’s note (accessible here: http://wwwusers.york.ac.uk/~jdh1/micro%202/exercises/noteontutorial5fromJohnBone.pdf):
(since
and
)
Output for firm 1 when on contract curve, since
:
Output for firm 1 when on contract curve, since
:
Substitute into equation
When
,
:
:
With prices equal and owing to the symmetry of the problem, know that optimum is at
.
If prices weren’t equal to one another, such that the price of output 2 was lower than that of output
1, the optimal point would to be produce more of output 1 and less of output 2. As the price of
output 2 approaches 0 the optimal outputs would approach
and
.