Deck 17: Technology

Question 1

The production function f(x, y) = x + y has constant returns to scale.

Answer 1

True

Question 2

If there are constant returns to scale, then doubling the amount of any input will exactly double the amount of output.

Answer 2

False

Question 3

If the production function is f(x₁, x₂) = x₁x₂, then there are constant returns to scale.

Answer 3

False

Question 4

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.

Question 5

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.

Question 6

The production function f(x, y) = + has increasing returns to scale.

Question 7

If the marginal product of each factor decreases as the amount of that factor used increases, then there must be decreasing returns to scale.

Question 8

A firm's production function is f(x₁, x₂) = x₁ + 2x₂. This means that x₂ is twice as expensive as x₁.

Question 9

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 = . (2) y = 3 . (3) y = . (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

Question 10

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.

Question 11

The production set of a firm is the set of all products the firm can produce.

Question 12

If the production function is f(x, y) = x + min{x, y}, then there are constant returns to scale.

Question 13

If the production function is f(x, y) = min{ 12x, 3y}, then there is convexity in production.

Question 14

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

Question 15

If the production function is f(x, y) = min{ 2x + y, x + 2y}, then there are constant returns to scale.

Question 16

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.

Question 17

A firm has two variable factors and a production function f(x₁, x₂) = . The technical rate of substitution between x₁ and x₂ is constant.

Question 18

A production isoquant is a locus of combinations of inputs that are equally profitable.

Question 19

It is possible to have decreasing marginal products for all inputs, and yet have increasing returns to scale.

Question 20

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.

Question 21

A firm has the production function f(x₁, x₂) = x¹¹x^0.50₂. The isoquant on which output is has the equation

A) x₂ = 30x^-21.
B) x₂ = 30x²¹.
C) x₂ = 30x^-0.50₁.
D) = 2.
E) x₁ = 0.50x^-0.50₂.

Question 22

A firm has the production function f(X, Y) = , 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.

Question 23

A firm has the production function f(x, y) = x^1.40y¹. 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.

Question 24

A firm has the production function f(x, y) = x^1.40y^0.90. 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.

Question 25

A firm uses 3 factors to produce its output. Its production function is f(x, y, z) = min{ , 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.

Question 26

A firm has the production function f(x, y) = x + min{x, 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.

Question 27

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.

Question 28

A firm has the production function f(x, y) = . 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.

Question 29

A firm has the production function f(x₁, x₂) = (x^b₁ + x_b²)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.

Question 30

Suppose that the production function is f(x₁, x₂) = (x^a₁ + x^a₂)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.

Question 31

A firm has the production function f(X, Y) = , 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.

Question 32

A firm has the production function f(x₁, x₂) = x^0.60₁x^0.30₂. The isoquant on which output is has the equation

A) x₂ = 80x^-2₁.
B) x₂ = 80x^3.33₁.
C) = 2.
D) x₂ = 80x^-0.30₁.
E) x₁ = 0.30x^-0.70₂.

Question 33

A firm has the production function f(x, y) = . 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.

Question 34

A firm has the production function f(X, Y) = , 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.

Question 35

A firm has the production function f(x, y) = x^0.90y^0.80. 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.

Question 36

A firm has the production function f(x, y) = . 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.

Question 37

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.

Question 38

A firm uses 3 factors to produce its output. Its production function is f(x, y, z) = min{ , 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.

Question 39

A firm uses 3 factors to produce its output. Its production function is f(x, y, z) = min{ , y², }. If the amount of each input is multiplied by 2, its output will be multiplied by

A)27
B)4
C)8
D) 0.40.
E) The answer depends on the original choice of x, y, and z.

Question 40

A firm has the production function f(x₁, x₂) = x^0.80₁x^0.20₂. The isoquant on which output is has the equation

A) x₂ = 70x⁵₁.
B) = 4.
C) x₂ = 70x^-4₁.
D) x₂ = 70x^-0.20₁.
E) x₁ = 0.20x^-0.80₂.

Question 41

A firm has a production function f(x, y) = 1.80(x^0.80 + y^0.80)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.

Question 42

The production function Q = 50K^0.25L^0.75 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.

Question 43

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 = B^0.40L^0.60.
C) Q =+.
D) Q = 0.5B + 0.5L^0.5.
E) None of the above.

Question 44

The production function Q = 50K^0.25L^0.25 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.

Question 45

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 = K^0.5L^0.5.
c. Q = min(K, L).

Question 46

On separate axes, draw typical production isoquants for each of the following production functions.
a. f(x, y) = min{ 2x, x + y}.
b. f(x, y) = xy.
c. f(x, y) = x + min{x, y}.
"d. (x, y) = x + ."
a. f(x, y) = min{ 2x, x + y}.
b. f(x, y) = xy.
c. f(x, y) = x + min{x, y}.
d. (x, y) = x + .

Question 47

A firm has a production function f(x, y) = 1.80(x^0.80 + y^0.80)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.

Question 48

The production function Q = 50K^0.25L^0.75 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.

Question 49

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 = B^0.60L^0.40.
B) Q =+.
C) Q = min(2B, 60L).
D) Q = 0.5B + 0.5L^0.5.
E) None of the above.

Question 50

For each of the following production functions, comment on the ability to substitute capital for labor.
a. Q = K + L.
b. Q = K^0.5L^0.5.
c. Q = min(K, L).

Question 51

A firm has a production function f(x, y) = 1.40(x^0.60 + y^0.60)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.

Question 52

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 + 0.5L^0.5.
B) Q = B^0.80L^0.20.
C) Q = min(2B, 60L).
D) Q =+.
E) None of the above.