In: Economics

There are two kinds of factors of production, labor L and
capital K, which are only available in non-negative quantities.
There are two ﬁrms that make phones, Apple and Banana. To make qA
phones, Apple’s input requirement of (L,K) is given by production
function f(L,K) = L0.6K0.2. To make qB phones, Banana’s input
requirement of (L,K) is given by production function g(L,K) =
L0.75K0.25.

(a) (Time: 3 minutes) How many phones can Apple make with factor
bundle (L1,K1) = (1,1)? And how many phones can Banana make with
this factor bundle? How many phones can Apple make with factor
bundle (L2,K2) = (2,2)? And how many phones can Banana make with
this factor bundle?

(b) (Time: 2 minutes) More generally, for any factor bundle (L,K) where L > 1 and L > 1, would you agree that Banana can make more phones than Apple? Explain your answer.

In the case of Apple and Banana, suppose that they both face the
same input prices for labor and capital, w and r, respectively, and
that the price at which they can sell their output is the same as
well, equal to p. Also, assume they are price takers in factor
markets as well as the output market. Suppose w = 3 and r =
1.

(c) (Time: 6 minutes) From your answer above, it should be clear
that Apple could produce 1 unit of output by using factor bundle
(L1,K1). Show why this input bundle is actually cost-minimizing,
i.e. the cheapest possible input bundle that ensures an output of
1.

(d) (Time: 2 minutes) Explain why, at output level of 1, Apple’s marginal cost is 5.

Whether or not an output level of 1 maximizes proﬁt will depend
upon the price of output. Continue to assume w = 3 and r = 1, and
now also suppose p = 5.

(e) (Time: 2 minutes) Explain why output level qA = 1 is
proﬁt-maximizing for Apple.

Part (a)

Part (b)

if inputs are doubled

Doubling of inputs leads to doubling of output for banana phones. but less than proportionate increase in output is observed for apple phones,

production function for apple phones shows decreasing returns to scale whereas production function for banana phones shows constant returns to scale

Part (c) cost is minimized at point where above condition is fulfilled

we have to find cost minimizing input bundle for an output of 1 unit of apple phones

Part (d)

we know from the cost minimizing condition

Part (e)

given price of an apple phone is 5

maximizing profit by equating first derivative with respect to output to 0

Consider an economy that uses two factors of production, capital
(K) and labor (L), to produce two goods, good X and good Y. In the
good X sector, the production function is X = 4KX0.5 + 6LX0.5, so
that in this sector the marginal productivity of capital is MPKX =
2KX-0.5 and the marginal productivity of labor is MPLX = 3LX-0.5.
In the good Y sector, the production function is Y = 2KY0.5 +
4LY0.5, so that in this sector...

The production function has two input, labor (L) and capital
(K). The price for L and K are respectively W and V.
q = L + K a linear production function
q = min{aK, bL} which is a Leontief production function
1.Calculate the marginal rate of substitution.
2.Calculate the elasticity of the marginal rate of
substitution.
3.Drive the long run cost function that is a function of input
prices and quantity produced.

A firm produces output y using two factors of production
(inputs), labour L and capital K. The firm’s production function is
?(?,?)=√?+√?=?12+?12. The wage rate w = 6 and the rental price of
capital r = 2 are taken as parameters (fixed) by the firm. a. Show
whether this firm’s technology exhibits decreasing, constant, or
increasing returns to scale. b. Solve the firm’s long run cost
minimization problem (minimize long run costs subject to the output
constraint) to derive this...

Imagine a firm that only uses capital (K) and labor (L). Use an
isocost / isoquant diagram to illustrate the firm’s equilibrium
input mix for given prices of capital and labor and a given rate of
output. Now illustrate what happens if the price of labor falls,
and the firm wants to produce the same rate of output. What happens
to the cost of production? Compare the relative marginal products
of labor and capital (the MRTS) at the two equilibria.

Suppose a country’s production function has three factors of
production: K—capital, L—the number of workers, and H—human
capital. The production function is Y = (K^0.3)(L^0.4)(H^0.3). (a)
What is the marginal product of labor? How does human capital
affect the MPL?
(b) What is the marginal product of human capital? How does
human capital affect the MPH?
c) What share of total income is paid to labor? What share of
total income is paid to human capital?
(d) What share of...

Capital and labor are the only two inputs for the following
production process. Capital is fixed at 4 units, which costs 50
dollars each unit per day. Workers can be hired for 100 each per
day. Complete the following table and plot the marginal cost (MC),
average total cost (ATC), average variable cost (AVC), average
fixed cost (AFC) on the same graph.
The quantity of labor input
Total output per day
AFC
AVC
ATC
MC
0
0
1
100...

1. A country’s production function depends on labor
(L), physical capital (K), human capital (H), and natural resources
(N). When L = 200, K = 10, H = 30, and N = 4, output is
80. What would output be if L = 700, K = 35, H = 105,
and N = 14?
A. 30 B. 60 C. 240 D. 280 E. 320
F. More than one of A-E is
possible G. None
of A-G is
possible ______
2. A nation’s real GDP is 2,000,000 and its GDP
deflator is 125. What is its nominal GDP?
A. 1,600,000 B. 1,999,875 ...

1. A country’s production function depends on labor (L) and
capital (K) and shows constant returns to scale. When L = 140 and K
= 150, output is 330. Based on this information, what is a possible
value for output when L = 70 and K = 50? Briefly explain your
reasoning.
2. In 1960, CPI in the United States was 29.6. In
2017, CPI in the United States was 245.1. If a pair of
pants cost $35 in 2017, how much...

A firm produces output using capital (K) and labor (L). Capital
and labor are perfect complements and 1 unit of capital is used
with 2 units of labor to produce 1 unit of output. Draw an example
of an isoquant. If wages and rent are $2 and $3, respectively, what
is the Average Total Cost?
A firm has a production function given by Q=4KL where K, L and Q
denote capital, labor, and output, respectively. The firm wants to
produce...

Consider a production function of two inputs, labor and capital,
given by Q = (√L + √K)2. Let w = 2 and r = 1. The marginal products
associated with this production function are as follows:MPL=(√L +
√K)L-1/2MPK=(√L + √K)K-1/2
a) Suppose the firm is required to produce Q units of output.
Show how the cost-minimizing quantity of labor depends on the
quantity Q. Show how the cost-minimizing quantity of capital
depends on the quantity Q.
b) Find the equation...

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