Question

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 returns to scale? Justify your answer. 2 2. What is the firm’s long-run cost minimisation problem? 3. Derive the firm’s output-conditional input demand functions for each of the two inputs. 4. Verify that each of the output conditional input demand functions are homogeneous of degree zero in input prices. 5. Derive the firm’s (minimum total) cost function. 6. Verify Shephard’s lemma for each of the inputs. 7. Verify that the cross-price effects in the output-conditional input demand functions are symmetric.

Answer #1

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

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)

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 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 ??(∙).

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

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:

A firm has the following production function: ?(?, ?) = ?
3/4?1/4 .
A) What is the firm’s Technical Rate of Substitution?
B) What is the optimality condition that determines the firm’s
optimal level of inputs?
C) Suppose the firm wants to produce exactly ? units and that
input ? costs $?? per unit and input ? costs $?? per unit. What are
the firm’s conditional input demand functions? D) Using the
information from part C, write down the firm’s...

ABC Corp wants to produce whatever output it wishes to sell at
minimum cost. Its production function is y = z1/2z1/2, where y is
output and z ≥ 0 and z ≥ 0 are inputs. It 1212 is a price taker in
input markets so its total cost, C, equals w1z1 + w2z2, where w1
> 0 and w2 > 0 are input prices.
(i) Fix output at y0 > 0 and use the production
function to write z2 as...

2. ABC Corp wants to produce whatever output it wishes to sell
at minimum cost. Its production function is y = z 1/3 1 z 2/3 2 ,
where y is output and z1 ≥ 0 and z2 ≥ 0 are inputs. It is a price
taker in input markets so its total cost, C, equals w1z1 + w2z2,
where w1 > 0 and w2 > 0 are input prices. Fix output at
y0.
(i) Use Lagrange’s Method to find...

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 ))

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