Question

1. A ﬁrm has two variable factors of production, and its production function is f(x1,x2) = x1/2 1 x1/4 2 . The price of the output is 6. Factor 1 receives the wage $2, and factor 2 receives the wage $2. a. How many units of each factor will the ﬁrm demand? b. How much output will it produce?

2. Beth produces software. Her production function is f(x1,x2) = 3x1 + 2x2, where x1 is the amount of unskilled labor and x2 is the amount of skilled labor she employs. (a) Draw the isoquants representing combinations of inputs needed to produce 30 units, and 60 units. (b) Does the production function exhibit increasing, decreasing or constant returns to scale? (c) If Beth faces factors prices (2, 1), what is the cheapest way for her to produce 30 units of output?

Answer #1

1.

using standard marshellian condition

mpl/mpk = pk/pl

1/2x^{-1/2} .x2^{1/4} / . 1/4x2^{-3/4} .
x1^{1/2} = 2/2

2x2/ x1 =1

2x2 = x1

f= 2x2^{1/2} * x2^{1/4} = 2x2^{3/4}

2.

**30 = 3x1+2x2**

x1 = 0 , x2 = 15

x2= 0 , x1 =10

**60 = 3x1+2x2**

x1= 0 , x2 = 30

x2= 0 , x1 = 20

b) Constant returns to scale

c) since the production function is subsitutes

x1 = 0 (cost of producting is higher than cost of producing x2)

x2 = output/price = 30/1 = 30

Consider production function f (x1, x2) =
x11/2x21/3. The price
of factor 1 is w1 = 12
and the price of factor 2 is w2 = 1.
With x̄2 = 8, find the short-run cost function c(y).
Find short-run AC(y), AVC(y), and MC(y) based on the answer to
a.
Write out the long-run cost minimization problem to find the
cheapest way to produce y units of output.
Write out the Lagrangian for the long-run cost minimization
problem.
Solve the long-run...

A competitive firm’s production function is f(x1,x2)=
24x1^1/2x2^1/2. The price of factor 1 is 1, the price of factor 2
is 2 and the price of output is 4. (a) Write down the cost function
in terms of both the inputs. (b) What is the long-run cost
minimization condition for this firm? (c) In what proportions
should x1 and x2 be used if the firm wants to minimize its
costs?

Suppose a firm has production function f(x1, x2) = x1 + x2. How
much output should the firm produce in the long run?

Suppose the production function of a firm is given by f (x1; x2)
= min{x1, x2}
(a) Calculate the conditional demand functions of the firm
assuming w1 = 2; w2 = 4, and y = 8
(b) Calculate the minimum cost of the firm to produce 8 units of
the good when w1 = 2 and w2 = 4:

Consider a firm with production function given by f(x1, x2) =
(x1)^1/4 (x2)^1/2 :
Assume the prices of inputs 1 and 2 are w1 and w2, respectively,
and the market price of the product is p.
(a) Find the levels of the inputs that maximize the profits of
the firm (X1, X2)
(b) Derive the supply function of the firm (i.e., y = f (x 1 ; x
2 ))

4. Al’s production function for deer is f(x1, x2) = (2x1 +
x2)1/2, where x1 is the amount of plastic and x2 is the amount of
wood used. If the cost of plastic is $4 per unit and the cost of
wood is $4 per unit, then what is the cost of producing 8 deer
?

2 .Suppose the production function of a firm is given by f (x1,
x2) = 2x1 + 4x2
(a) Calculate the conditional demand functions of the firm
assuming w1 = 2; w2 = 3, and y = 8
(b) Calculate the minimum cost of the firm to produce 8 units of
the good when w1 = 2 and w2 = 3

In Problem 12, Al’s production function for deer is
f(x1, x2) =
(2x1 + x2)1/2,
where x1 is the amount of plastic and
x2 is the amount of wood used. If the cost of
plastic is $8 per unit and the cost of wood is $1 per unit, then
the cost of producing 7 deer is
a.
$49.
b.
$28.
c.
$196.
d.
$7.
e.
$119.
step by step, please

An industry has 50 identical firms. These firms have the same
production function, which is Y=3x1+x2. Each
firm has $0 fixed cost. The cost of factor 1 is $4 and the cost of
factor $2 is $1. Both factors are variable. If the firm acts
optimally (aka cost minimized), The total production cost for the
industry if each firm produces 15 units is:
750
1000
3750
Not enough information to determine

1. Consider a firm with technology that can be represented by
the following production function: f(x1, x2) = min {x1, x2} +
x2
Input 1 costs w1 > 0 per unit and input 2 costs w2 > 0 per
unit.
(a) Draw the isoquant associated with an output of 4. Make sure
to label any intercepts and slopes.
(b) Find the firm’s long-run cost function, c(w1, w2, y)

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