CHAPTER 19: Technology
TRUE/FALSE
1. The production set of a firm is the set of all products the firm can produce.
2. A production isoquant is a locus of combinations of inputs that are equally profitable.
3. If there are constant returns to scale, then doubling the amount of any input will exactly double the
amount of output.
4. The economist’s distinction between the long run and the short run captures the idea that quantities of
some factor inputs can be varied in the short run but not in the long run.
5. If the production function is f(x, y) = min2x + y, x + 2y, then there are constant returns to scale.
6. If the production function is f(x, y) = x + minx, y, then there are constant returns to scale.
7. If the production function is f(x, y) = min12x, 3y, then there is convexity in production.
8. If the production function is f(x1, x2) = x1x2, then there are constant returns to scale.
9. It is possible to have decreasing marginal products for all inputs, and yet have increasing returns to
scale.
10. A production function has well-defined marginal products at every input combination. If factor x is
shown on the horizontal axis and factor y is shown on the vertical axis, the slope of the isoquant
through a point (x*, y*) is the negative of the ratio of the marginal product of x to the marginal product
of y.
11. The production function f(x, y) = x2/3+ y2/3 has increasing returns to scale.
12. The production function f(x, y) = x + y has constant returns to scale.
13. If there is one input used in production and if there are decreasing returns to scale, then the marginal
product for the input will be diminishing.
14. A firm’s production function is f(x1, x2) = x1 + 2x2. This means that x2 is twice as expensive as x1.
15. A firm has two variable factors and a production function f(x1, x2) = (2x1 + 4x2)1/2. The technical rate of
substitution between x1 and x2 is constant.
16. If the marginal product of each factor decreases as the amount of that factor used increases, then there
must be decreasing returns to scale.
MULTIPLE CHOICE
1. In any production process, the marginal product of labor equals
a.
the value of total output minus the cost of the fixed capital stock.
b.
the change in output per unit change in labor input for “small” changes in the amount of
input.
c.
total output divided by total labor inputs.
d.
total output produced with the given labor inputs.
e.
the average output of the least-skilled workers employed by the firm.
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.
A change in the level of output
b.
A change in the ratio in which the inputs are combined
c.
A change in the marginal products of the inputs
d.
A change in the rate of technical substitution
e.
A change in profitability
3. A firm has the production function f(x, y) = x.5 + y, where x is the amount of factor x it uses and y is the
amount of factor y. 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 the slopes of all the
isoquants that it meets have the same slope at the point where they meet this line. The straight line we
drew was
a.
vertical.
b.
horizontal.
c.
diagonal through the origin with slope 0.5.
d.
diagonal with slope 2.
e.
diagonal with slope greater than 2.
4. Which of the following production functions exhibit constant returns to scale? In each case y is output
and K and L are inputs. (1) y = K1/2 L1/3. (2) y = 3K1/2 L1/2. (3) y = K1/2 + L1/2. (4) y = 2K + 3L.
a.
1, 2, and 4
b.
2, 3, and 4
c.
1, 3, and 4
d.
2 and 3
e.
2 and 4
5. A firm has the production function f(x, y) = 60x 4/5 y1/5. The slope of the firm’s isoquant at the point (x,
y) = (40, 80) is (pick the closest one)
a.
−0.50.
b.
−4.
c.
−0.25.
d.
−8.
e.
−0.25.
6. A firm has the production function f(x, y) = 20x3/5 y2/5. The slope of the firm’s isoquant at the point (x,
y) = (20, 40) is (pick the closest one)
a.
−3.
b.
−0.67.
c.
−1.50.
d.
−0.50.
e.
−0.25.
7. A firm has the production function f(x, y) = 20x3/5 y2/5. The slope of the firm’s isoquant at the point (x,
y) = (50, 70) is (pick the closest one)
a.
−1.50.
b.
−0.67.
c.
−0.71.
d.
−2.10.
e.
−0.36.
8. A firm uses only two inputs to produce its output. These inputs are perfect substitutes. This firm
a.
must have increasing returns to scale.
b.
must have constant returns to scale.
c.
could have increasing returns to scale, constant returns to scale, or decreasing returns to
scale.
d.
must have decreasing returns to scale.
e.
must have decreasing returns to scale in the short run and constant returns to scale in the
long run.
9. A firm has the production function f(X, Y) = X 1/2 Y 1/2, 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 23. The straight line we drew
a.
is vertical.
b.
is horizontal.
c.
is a ray through the origin with slope 3.
d.
is a ray through the origin with slope 4.
e.
has a negative slope.
10. A firm has the production function f(X, Y) = X3/4 Y1/4, 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 −9. The straight line we drew
a.
is horizontal.
b.
is a ray through the origin with slope 3.
c.
is vertical.
d.
is a ray through the origin with slope 4.
e.
has a negative slope.
11. A firm has the production function f(X, Y) = X3/4 Y1/4, 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 −9. The straight line we drew
a.
is a ray through the origin with slope 3.
b.
is a ray through the origin with slope 4.
c.
is vertical.
d.
is horizontal.
e.
has a negative slope.
12. If output is produced with two factors of production and with increasing returns to scale,
a.
there cannot be diminishing marginal rate of substitution.
b.
all inputs must have increasing marginal products.
c.
on a graph of production isoquants, moving along a ray from the origin, output more than
doubles as the distance from the origin doubles.
d.
the marginal product of at least one input must be increasing.
e.
all inputs must have decreasing marginal products.
13. A firm has the production function f(x1, x2) = (xb1 + xb2)c, where b 0 and c 0. This firm will have
a.
increasing returns to scale if and only if 2b + c 1.
b.
increasing returns to scale if and only if bc 1.
c.
increasing returns to scale if and only if b + c 1.
d.
constant returns to scale if and only if c = 1.
e.
constant returns to scale if and only if b = c.
14. A firm has the production function f(x, y) = x + minx, y. The isoquants for the firm
a.
are L-shaped with the corners of the L’s on the line y = x.
b.
are L-shaped with the corners of the L’s on the line y = x + 1.
c.
consist of two line segments, one vertical and the other with a slope of −1.
d.
consist of two line segments, one horizontal and the other with a slope of −1.
e.
are upside down L-shaped.
15. Suppose that the production function is f(x1, x2) = (xa1 + xa2)b, where a and b are positive constants. For
what values of a and b is there a diminishing technical rate of substitution?
a.
For any value of a if b 1
b.
For any values of a and b if ab 1
c.
For any values of a and b if a b
d.
For any value of b if a 1
e.
None of the above.
16. A firm has the production function f(x1, x2) = x0.601x0.302. The isoquant on which output is 803/10 has the
equation
a.
x2 = 80x-21.
b.
x2 = 80x3.331.
c.
x1/x2 = 2.
d.
x2 = 80x-0.301.
e.
x1 = 0.30x-0.702.
17. A firm has the production function f(x1, x2) = x11x0.502. The isoquant on which output is 305/10 has the
equation
a.
x2 = 30x-21.
b.
x2 = 30x21.
c.
x2 = 30x-0.501.
d.
x1/x2 = 2.
e.
x1 = 0.50x-0.502.
18. A firm has the production function f(x1, x2) = x0.801x0.202. The isoquant on which output is 702/10 has the
equation
a.
x2 = 70x51.
b.
x1/x2 = 4.
c.
x2 = 70x-41.
d.
x2 = 70x-0.201.
e.
x1 = 0.20x-0.802.
19. A firm has the production function f(x, y) = x1.40y1. This firm has
a.
decreasing returns to scale and diminishing marginal products for factor x.
b.
increasing returns to scale and decreasing marginal product of factor x.
c.
decreasing returns to scale and increasing marginal product for factor x.
d.
constant returns to scale.
e.
None of the above.
20. A firm has the production function f(x, y) = x1.40y0.90. This firm has
a.
decreasing returns to scale and increasing marginal product for factor x.
b.
constant returns to scale.
c.
increasing returns to scale and decreasing marginal product of factor x.
d.
decreasing returns to scale and diminishing marginal products for factor x.
e.
None of the above.
21. A firm has the production function f(x, y) = x0.90y0.80. This firm has
a.
constant returns to scale.
b.
decreasing returns to scale and diminishing marginal products for factor x.
c.
decreasing returns to scale and increasing marginal product for factor x.
d.
increasing returns to scale and decreasing marginal product of factor x.
e.
None of the above.
22. A firm uses 3 factors to produce its output. Its production function is f(x, y, z) = minx3/y, y2,
(z4 − x4)/y2. If the amount of each input is multiplied by 3, its output will be multiplied by
a.
27.
b.
9.
c.
3.
d.
0.30.
e.
The answer depends on the original choice of x, y, and z.
23. A firm uses 3 factors to produce its output. Its production function is f(x, y, z) = minx3/y, y2,
(z4 − x4)/y2. If the amount of each input is multiplied by 3, its output will be multiplied by
a.
0.30.
b.
3.
c.
9.
d.
27.
e.
The answer depends on the original choice of x, y, and z.
24. A firm uses 3 factors to produce its output. Its production function is f(x, y, z) = minx3/y, y2,
(z4 − x4)/y2. If the amount of each input is multiplied by 2, its output will be multiplied by
a.
2.
b.
4.
c.
8.
d.
0.40.
e.
The answer depends on the original choice of x, y, and z.
25. A firm has a production function f(x, y) = 1.40(x0.60 + y0.60)2 whenever x 0 and y 0. When the
amounts of both inputs are positive, this firm has
a.
increasing returns to scale.
b.
decreasing returns to scale.
c.
constant returns to scale.
d.
increasing returns to scale if x + y 1 and decreasing returns to scale otherwise.
e.
increasing returns to scale if output is less than 1 and decreasing returns to scale if output
is greater than 1.
26. A firm has a production function f(x, y) = 1.80(x0.80 + y0.80)2 whenever x 0 and y 0. When the
amounts of both inputs are positive, this firm has
a.
decreasing returns to scale.
b.
constant returns to scale.
c.
increasing returns to scale if x + y 1 and decreasing returns to scale otherwise.
d.
increasing returns to scale.
e.
increasing returns to scale if output is less than 1 and decreasing returns to scale if output
is greater than 1.
27. A firm has a production function f(x, y) = 1.80(x0.80 + y0.80)3 whenever x 0 and y 0. When the
amounts of both inputs are positive, this firm has
a.
increasing returns to scale if x + y 1 and decreasing returns to scale otherwise.
b.
decreasing returns to scale.
c.
constant returns to scale.
d.
increasing returns to scale.
e.
increasing returns to scale if output is less than 1 and decreasing returns to scale if output
is greater than 1.
28. The production function Q = 50K0.25L0.25 exhibits
a.
increasing returns to scale.
b.
constant returns to scale.
c.
decreasing returns to scale.
d.
increasing, then diminishing returns to scale.
e.
negative returns to scale.
29. The production function Q = 50K0.25L0.75 exhibits
a.
increasing, then diminishing returns to scale.
b.
increasing returns to scale.
c.
decreasing returns to scale.
d.
constant returns to scale.
e.
negative returns to scale.
30. The production function Q = 50K0.25L0.75 exhibits
a.
increasing returns to scale.
b.
decreasing returns to scale.
c.
constant returns to scale.
d.
increasing, then diminishing returns to scale.
e.
negative returns to scale.
31. The UJava espresso stand needs two inputs, labor and coffee beans, to produce its only output,
espresso. Producing an espresso always requires the same amount of coffee beans and the same
amount of time. Which of the following production functions would appropriately describe the
production process at UJava, where B represents ounces of coffee beans, and L represents hours of
labor?
a.
Q = B0.60L0.40.
b.
Q = B/2 + L/2.
c.
Q = min(2B, 60L).
d.
Q = 0.5B + 0.5L0.5.
e.
None of the above.
32. The UJava espresso stand needs two inputs, labor and coffee beans, to produce its only output,
espresso. Producing an espresso always requires the same amount of coffee beans and the same
amount of time. Which of the following production functions would appropriately describe the
production process at UJava, where B represents ounces of coffee beans, and L represents hours of
labor?
a.
Q = min(2B, 60L).
b.
Q = B0.40L0.60.
c.
Q = B/2 + L/30.
d.
Q = 0.5B + 0.5L0.5.
e.
None of the above.
33. The UJava espresso stand needs two inputs, labor and coffee beans, to produce its only output,
espresso. Producing an espresso always requires the same amount of coffee beans and the same
amount of time. Which of the following production functions would appropriately describe the
production process at UJava, where B represents ounces of coffee beans, and L represents hours of
labor?
a.
Q = 0.5B + 0.5L0.5.
b.
Q = B0.80L0.20.
c.
Q = min(2B, 60L).
d.
Q = B/2 + L/30.
e.
None of the above.
PROBLEM
1. On separate axes, draw typical production isoquants for each of the following production functions.
a. f(x, y) = min2x, x + y.
b. f(x, y) = xy.
c. f(x, y) = x + minx, y.
d. (x, y) = x + y1/2.
2. For each of the following production functions, comment on the ability to substitute capital for labor.
a. Q = K + L.
b. Q = K0.5L0.5.
c. Q = min(K, L).
3. For each of the following production functions, draw a diagram showing the general shape of its
corresponding isoquant. Comment on the ease at which labor and capital can be substituted for one
another relative to the other two production functions.
a. Q = K + L.
b. Q = K0.5L0.5.
c. Q = min(K, L).