# Homothetic ```Question text
In the picture below there is are two isoquants one for producing 2q and the other for
producing q units of output. There are also 4 points labelled A,B,C,D
Suppose the coordinates of the point A are (a1,a2), B are (b1,b2), C are (c1,c2), and D are
(d1,d2).
Which of the following statements are correct.
Select one or more:
a.
If the production function is homothetic and 3(b1,b2)=(d1,d2), then 3(a1,a2)=(c1,c2).
True - treble the inputs produces the same amount of output
b.
If the production function is homogeneous degree 2 then then 2(b1,b2)=(d1,d2).
c.
If the production function is homogeneous degree 2 then then 21/2(b1,b2)=(d1,d2).
d.
If the production function is homogeneous degree 2 then then 4(b1,b2)=(d1,d2).
e.
If the production function f in the picture is homothetic, then so too is the production
function f2.
f.
If the production function f in the picture is homothetic, then so too is the production
function f(z1,z2)-z1.
We are told that f is homothetic so this means that if f(z1,z2) = f(z'1,z'2) so the bundles (z1,z2)
and (z'1,z'2) are on the same isoquant then so too are λ(z1,z2) and λ(z'1,z'2) that is f(λz1,λz2)
= f(λz'1,λz'2). Now let us try the same test with the amended production function and
suppose that
f(z1,z2) - z1 = f(z'1,z'2) - z'1
Are the outputs from the scaled up inputs equal?
Is f(λz1λ,z2) - λz1 (=?) f(λz'1,λz'2) - λz'1
Because we know that f is homothetic we know that f(λz1,λz2) = f(λz'1,λz'2). So the above
question is
Does - λz1 (=?)
- λz'1
This is clearly false because we never said z1 =z'1
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We are told that f is homothetic so this means that if f(z1,z2) = f(z'1,z'2) so the bundles (z1,z2)
and (z'1,z'2) are on the same isoquant then so too are λ(z1,z2) and λ(z'1,z'2) that is f(λz1,λz2)
= f(λz'1,λz'2). Now let us try the same test with the amended production function and
suppose that
f(z1,z2) - z1 = f(z'1,z'2) - z'1
Are the outputs from the scaled up inputs equal?
Is f(λz1λ,z2) - λz1 (=?) f(λz'1,λz'2) - λz'1
Because we know that f is homothetic we know that f(λz1,λz2) = f(λz'1,λz'2). So the above
question is
Does - λz1 (=?)
- λz'1
This is clearly false because we never said z1 =z'1
If the production function is homogenous degree 2, then
f(λz1,λz2) = λ2f(z1,z2) (*)
We are told that
2f(b1,b2) = f(d1,d2) and as B and D are on the same line we also know x(b1,b2) = (d1,d2), for
some number x&gt;1.
So combining these two we get
2f(b1,b2) = f(xb1,xb2)
Comparing this with the equation lablelled (*) above we can see that 2=λ2 so x has to be
equal to λ that is the square root of 2.
The correct answers are: If the production function is homothetic and 3(b1,b2)=(d1,d2), then
3(a1,a2)=(c1,c2)., If the production function is homogeneous degree 2 then
then 21/2(b1,b2)=(d1,d2). , If the production function f in the picture is homothetic, then so
too is the production function f2.
```