NAME 245
(b) The marginal product of x1(increases, decreases, remains constant)
(c) The marginal product of factor 2 is 3/2x1/2
1x1/2
2, and it (in-
(d) An increase in the amount of x2(increases, leaves unchanged, de-
(f) Does this technology have diminishing technical rate of substitution?
(g) This technology demonstrates (increasing, constant, decreasing)
19.4 (0) The production function for fragles is f(K, L)=L/2+√K,
where Lis the amount of labor used and Kthe amount of capital used.
(b) In the short run, capital is fixed at 4 units. Labor is variable. On the
graph below, use blue ink to draw output as a function of labor input in
the short run. Use red ink to draw the marginal product of labor as a
246 TECHNOLOGY (Ch. 19)
04812
16
2
4
6
Labour
Fragles
8
Black line
Blue
line
Red line
19.5 (0) General Monsters Corporation has two plants for producing
juggernauts, one in Flint and one in Inkster. The Flint plant produces
according to fF(x1,x
2)=min{x1,2x2}and the Inkster plant produces
(a) On the graph below, use blue ink to draw the isoquant for 40 jugger-
NAME 247
0204060
80
20
40
60
X2
80
a
b
c
Blue isoquant
Red
isoquant
Black isoquant
X1
(b) Suppose that the firm wishes to produce 20 juggernauts at each plant.
How much of each input will the firm need to produce 20 juggernauts
(c) Label with a bon your graph the point that shows how much of each
of the two inputs is needed in toto if the firm is to produce 10 juggernauts
in the Flint plant and 30 juggernauts in the Inkster plant. Label with a
cthe point that shows how much of each of the two inputs that the firm
needs in toto if it is to produce 30 juggernauts in the Flint plant and
10 juggernauts in the Inkster plant. Use a black pen to draw the firm’s
isoquant for producing 40 units of output if it can split production in any
manner between the two plants. Is the technology available to this firm
19.6 (0) You manage a crew of 160 workers who could be assigned to
make either of two products. Product A requires 2 workers per unit of
output. Product B requires 4 workers per unit of output.
(a) Write an equation to express the combinations of products A and
248 TECHNOLOGY (Ch. 19)
the combinations of A and B that could be produced with 160 workers.
(Assume that it is also possible for some workers to do nothing at all.)
0204060
80
20
40
60
B
80
A
Red
shading
Blue
shading
Black
shading
a
(b) Suppose now that every unit of product A that is produced requires
the use of 4 shovels as well as 2 workers and that every unit of product B
(c) On the same diagram, use black ink to shade the area that repre-
(d) On your diagram locate the feasible combination of inputs that use up
(e) If you have 160 workers and 180 shovels, what is the largest amount of
19.7 (0) A firm has the production function f(x, y)=min{2x, x +y}.
On the graph below, use red ink to sketch a couple of production isoquants
for this firm. A second firm has the production function f(x, y)=x+
min{x, y}. Do either or both of these firms have constant returns to scale?
NAME 249
Both do. On the same graph, use black ink to draw a couple of
isoquants for the second firm.
0102030
40
10
20
30
40
y
x
Black
isoquants
Red
isoquants
19.8 (0) Suppose the production function has the form
f(x1,x
2,x
3)=Axa
1xb
2xc
3,
where a+b+c>1. Prove that there are increasing returns to scale.
19.9 (0) Suppose that the production function is f(x1,x
2)=Cxa
1xb
2,
where a,b,andCare positive constants.
(a) For what positive values of a,b,andCare there decreasing returns
(b) For what positive values of a,b,andCis there decreasing marginal
250 TECHNOLOGY (Ch. 19)
(c) For what positive values of a,b,andCis there diminishing technical
(xa
1+xa
2)b,whereaand bare positive constants.
(a) For what positive values of aand bare there decreasing returns to
19.11 (0) Suppose that a firm has the production function f(x1,x
2)=
√x1+x2
2.
(a) The marginal product of factor 1 (increases, decreases, stays constant)
(b) This production function does not satisfy the definition of increasing
returns to scale, constant returns to scale, or decreasing returns to scale.
amount of both inputs will more than double the amount of output.