Question

Assume one input, one output production function which we know very little about. But over a period of three years we see that (1) with p = 1,w = 1 the firm choose x = 2,y = 5 (2) with p = 3,w = 0.5, the firm will choose x = 4 and y = 6 and (3) withp=4,w=2,thefirmwillchoosex=3,y=11. Graph the isoprofit lines associated with each pair of prices and quantity choices. Show on your graph what we have learned about the firm’s technology. Note we do not know the functional form of the production function.

Answer #1

Question 1: A firm produces one good with a technology given by
the production function y = f (x) = x1/3. The factor price w and
the price p for the good are fixed.
a) Explore whether the production function exhibits increasing
returns to scale.
b) Determine the cost function
c) Determine the demand function for the input factor.
d) How much will the firm produce?

production function Consider a firm that produces a single
output good Y with two input goods: labor (L) and capital (K). The
firm has a technology described by the production function f : R 2
+ → R+ defined by f(l, k) = √ l + √ k, where l is the quantity of
labor and k is the quantity of capital. (a) In an appropriate
diagram, illustrate the map of isoquants for the firm’s production
function. (b) Does the...

Consider a firm that produces a single output with a single
input, labor, using production function F(L)=100L−4(L^2), for
L∈[0,12.5]. The input price is W=2.
1. Determine the firm’s cost function C(Q), that is, the lowest
cost of producing Q units of output. State the associated labor
choice for a given required output, L(Q). Consider only the range Q
∈ [0, 12.5] .
2. What is the firm’s marginal cost curve, MC(Q)?

2. Suppose a firm is producing 200 widgets. The firm’s
production function is Cobb-
Douglas with decreasing returns to scale. (This means we have
normal, convex
isoquants). The firm uses K’ units of capital and L’ units of
labor. Initially, the input prices
are w’ and r’. However, an exogenous shock in the labor market
causes an increase in
the wage rate, resulting in an increase in input prices from w’
to w’’ where w’<w’’. Using
a graph (of isoquant...

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...

Problem 3. Consider the Leontiev (perfect complements)
production function f(x, y) = M in x 9.6 , y 5.2 .
(A) How many units of good y would be a perfect complement for 1
unit of good x? What is the equation of the firm’s kink line?
(B) Assume the firm has a production quota of q = 400 units.
Graph the firm’s level-400 isoquant. What are the coordinates of
the kink?
(C) Suppose the input prices are (px, py)...

The production function is given by y=L1/2, where y
is the output, and L is the amount of labor input. Assume the wage
rate is w so that the cost of using L unit of labor input is wL.
Let p denote the unit price of the output. Note that w and p are
exogenously given.
(1) Find the function for the profit (in terms of p, w and
L).
(2) Find the optimal choice of labor input and the...

Consider the Leontiev (perfect complements) production function
f(x, y) = M in x 9.6 , y 5.2 .
(A) How many units of good y would be a perfect complement for 1
unit of good x? What is the equation of the firm’s kink line?
(B) Assume the firm has a production quota of q = 400 units.
Graph the firm’s level-400 isoquant. What are the coordinates of
the kink?
(C) Suppose the input prices are (px, py) = (16,...

Suppose a firm’s production function is given by Q = L 1/2 , K
1/2.
a) Suppose the firm has a fixed cost FC=6, the price
of labor is w = 64 and the price of capital is r = 4. Derive the
firm’s total cost function, TC(Q).
b) What is the firm’s marginal cost?
c) Graph the firm’s isoquant for Q = 20 units of
output. On the same graph, sketch the firm’s isocost line
associated with the total...

Suppose that we have perfectly competitive input markets (for
both capital and labour) and output markets. Firm Tomato Harvesting
produces canned tomatoes which it sells at $50 (it is a big can of
tomatoes!). Suppose that initally the firm’s production technology
is given by: f(k,l)= √l
A technological innovation has occured however! A new tomato
harvester has been invented by a professor at UC Davis. If the firm
employs the tomato harvester, the new production technology is
given by: f(k,l)...

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