Question

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)

Answer #1

Consider a firm whose production technology can be represented
by a production function of the form q = f(x1, x2) = x α 1 x 1−α 2
. Suppose that this firm is a price taker in both input markets,
with the price of input one being w1 per unit and the price of
input two being w2 per unit. 1. Does this production technology
display increasing returns to scale, constant returns to scale,
decreasing returns to scale, or variable...

A firm’s production function is given as y=(x1)^(1/2) *
(x2-1)^(1/2) where y≥0 for the output, x1≥0 for the input 1 and
x2≥0 for the input 2. The prices of input 1 and input 2 are given
as w1>0 and w2>0,
respectively. Answer the following questions.
Which returns to scale does the production function
exhibit?
Derive the long-run conditional input demand functions and the
long-run cost function.

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 ))

Consider the technology of production f(K,L) = 0.3log(x) +
0.3log(y)
a) Check whether the production function exhibits constant,
decreasing or increasing returns to scale. Explain
b) Find the conditional demand functions. Use (p1, w1, w2) to
denote the exogenous prices of output x1 and x2 respectively
c) Find the cost function and verify Shephard's lemma
d) Find the profit function

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...

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

Question 2 (Technology). A local factory making greeting cards
employs only workers and machines. Let x1 represent workers and x2
represent machines. The firm’s production function is: f(x_1,
x_2)=10*min(x_1, 1/2x_2)
a) Draw an isoquant representing a quantity of 100 units and an
isoquant representing a quantity of 200 units. Label two points on
each isoquant.
b) Suppose the firm wants to produce 300 cards. The price of an
hour of labor is $20 and the price of a machine hour...

A firm produces a single output using two inputs x1, x2. Let p,
w1, w2 be the prices. The production function f is C2 (twice
continuously differentiable). Atp=5,w1 =1,w2
=2,theoptimalinputsarex∗1 =2,x∗2 =2. Ifεx1p =0.2 (the elasticity of
x1 w.r.t. p), εx1w1 = −0.4 (the elasticity of x1 w.r.t. w1), and
εx2 w2 = −0.5 (the elasticity of x2 w.r.t. w2 ), then, can you
derive εx1 w2 , εx2 p and εx2w1? If so, please find them

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?

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