Question

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 *w*_{1}>0 and *w*_{2}>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.

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...

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 ))

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

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)

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

Let the input prices be w = (w1, w2) and output price be p.
Derive the cost function c (w; y) and the output supply function y
(w, p) for firms with the following production functions:
a] f (x1; x2) = sqrt(x1) + 2sqrt(x2)
b] f (x1; x2) = min [sqrt(x1), 2 sqrt(x2)]

Suppose that a firm’s production function is given by
Φ(?1,?2)=?1?2. The firm incurs per-unit input costs of ?1 and ?2
when employing inputs ?1 and ?2, respectively. Derive the firm’s
conditional input demand functions ?1?(∙) and ?2?(∙) and the firm’s
total cost function ??(∙).

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

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...

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