ECN 201
Class Discussion # 10 (Ch. 18)
ANSWER KEY
True – False
1. If there are constant returns to scale, then doubling the amount of any input will exactly double the amount of
output. FALSE Tip: If the question had said “every input” instead of “any” input, it would have been correct.
2. If the production function is f ( x, y ) = min{2 x + y, x + 2 y} , then there are constant return to scale. TRUE
•
If you multiply both inputs by t, output multiplies by t.
3. If the production function is f ( x, y ) = min{12 x,3 y} , then there is convexity in production. TRUE
•
Leontief functions are convex.
4. If the production function is f ( x1 , x2 ) = x1 x2 , then there are constant returns to scale. FALSE
•
If you multiply both inputs by t, output multiplies by more than t, which means there are increasing
returns to scale.
5. The production function f ( x, y ) = x 2 / 3 + y 2 / 3 has increasing returns to scale. FALSE
•
If you multiply both inputs by t, output multiplies by less than t, which means there are decreasing
returns to scale.
Multiple Choices
1. A firm has the production function, f ( x, y ) = 50 x1/ 5 y 4 / 5 . The slope of the firm’s isoquant at the point (x,
y)=(50,30) is: (Pick the closest one.)
a)
b)
c)
d)
e)
–1.67
–0.25
–4
–0.15
–0.83
Solution : MRTS = - (MPx / MPy) = y / 4x = - (30 / 200) = - 0.15
2. If a firm moves from one point on a production isoquant to another point on the same isoquant, which of the
following will certainly NOT happen?
a)
b)
c)
d)
e)
A change in the level of output. The level of output is equal on every point on an isoquant.
A change in the ration in which the inputs are combined
A change in the marginal products of the inputs
A change in the rate of technical substitution
A change in profitability
3. A firm has the production function f ( x, y ) = x 2 / 3 y1/ 3 where x is the amount of factor x used and y is the amount
of factor y used. On a diagram we put x on the horizontal axis and y on the vertical axis. We draw some
isoquants. Now we draw a straight line on the graph and we notice that wherever this line meets an isoquant, the
isoquant has a slope of –6. The straight line we drew:
1
a)
b)
c)
d)
e)
is vertical
is horizontal
is a ray through the origin with slope 3
is a ray through the origin with slope 4
has a negative slope
MRTS = - 6 = - (2y / x) “3x = y” In order for
the slope of the isoquant to be -6, 3x must be equal
to y. The only way to achieve that is to have a ray
through the origin with a slope of +3 (because the
function 3x=y has a slope of +3 at all points).
4. A firm has the production function f ( x1 , x2 ) = ( x1b + x2b ) c where b>0 and c>0. This firm will have:
a)
b)
c)
d)
e)
Increasing returns to scale if and only if 2b+c>1.
Increasing returns to scale if and only if bc>1.
Increasing returns to scale if and only if b+c>1.
Increasing returns to scale if and only if c=1.
Increasing returns to scale if and only if b=c.
5. Suppose that the production function is f ( x1 , x2 ) = ( x1a + x2a ) b , where a and b are positive constants. For what
values of a and b is there diminishing technical rate of substitution?
a)
b)
c)
d)
e)
For any value of a if b<1.
For any value of a and b if ab<1.
For any value of a and b if a>b.
For any value of b if a<1.
None of the above
a-1
MRTS = (x1 / x2) In order for the derivative of MRTS with respect to x1 to be negative, “a” must be
smaller than 1. In that case we would have diminishing technical rate of substitution.
6. On separate axes draw typical production isoquants for each of the following production functions.
a)
b)
f ( x, y ) = min{2 x, x + y}
f ( x, y ) = xy
c)
f ( x, y ) = x + y 1 / 2
2