Suppose that a risk neutral competitive firm has to make its output (y) decision before it observes the uncertain output price, p. Assume that p is a random variable whose mean is E(p) = 2 and whose variance is Var(p) = 4. Assume that p is always greater than 1 (this ensures that there is always a strictly positive solution for y, regardless of what p is). Let the cost function be given by C = y + y². How much is the firm willing to pay for information about p?
Suppose that a risk neutral competitive firm has to make its output (y) decision before it observes the uncertain output price, p. Assume that p is a random variable whose mean is E(p) = 2 and whose variance is Var(p) = 4. Assume that p is always greater than 1 (this ensures that there is always a strictly positive solution for y, regardless of what p is). Let the cost function be given by C = y + y². How much is the firm willing to pay for information about p?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Question:
Suppose that a risk neutral competitive firm has to make its output (y) decision
before it observes the uncertain output price, p. Assume that p is a random variable
whose mean is E(p) = 2 and whose variance is Var(p) = 4. Assume that p is always
greater than 1 (this ensures that there is always a strictly positive solution for y,
regardless of what p is). Let the cost function be given by C = y + y². How much
is the firm willing to pay for information about p?
Use below provided example case to solve above question
8.2 Example of Calculation of Value of Information
. Consider the choice of output. Let the cost function be C = C(y).
• Case 1: choose output before the price is known.
• Case 2: choose output after the price is known.
• Let C = y².
• Case 1:
• Case 2:
Note: SOC OK.
. Calculate profits:
• Thus, expected profits are:
Note: the second-order condition (SOC) is OK.
• Calculate expected profits:
• But, remember that:
EVPI = E{max(py – C(y)}
Y
Thus, using this, we get:
EVPI = E{max(py – y²)} – max E(py - y²)
Y
Y
max E(py - y²)
Y
max uy —
Y
πT (μ)
=
max py - y²
Y
π(p)
ܝ ܕ | ܠ
var (p)
E(p²)
=
=
μ
Y
р =
Y =
=
P/²/2
TIT
-max E(py - C(y))
Y
p²
=
=
FOC
2y
NIE
=
22
E[π(p)] = = E(p²)
FOC
2y
=
E(p²) - μ²
var(p) + µ²
1
1
E[ñ(p)] − ñ(µ) = ½{var(p) + µ²³] — — µ² = ²/var(p) > 0
• This is the amount of money the firm is willing to pay for information on p.
● It is willing to pay this, although it is risk-neutral.
• What about the case of a risk-averse firm?
(80)
(81)
(82)
(83)
(84)
(85)
(86)
(87)
(88)
(89)
(90)
(91)
(92)
(93)
(94)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09238b20-1e79-43dc-b0a6-3d40e032e374%2Fe120a225-daf3-4a24-a4dd-e1d92a66bea0%2Fbxs7ke_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question:
Suppose that a risk neutral competitive firm has to make its output (y) decision
before it observes the uncertain output price, p. Assume that p is a random variable
whose mean is E(p) = 2 and whose variance is Var(p) = 4. Assume that p is always
greater than 1 (this ensures that there is always a strictly positive solution for y,
regardless of what p is). Let the cost function be given by C = y + y². How much
is the firm willing to pay for information about p?
Use below provided example case to solve above question
8.2 Example of Calculation of Value of Information
. Consider the choice of output. Let the cost function be C = C(y).
• Case 1: choose output before the price is known.
• Case 2: choose output after the price is known.
• Let C = y².
• Case 1:
• Case 2:
Note: SOC OK.
. Calculate profits:
• Thus, expected profits are:
Note: the second-order condition (SOC) is OK.
• Calculate expected profits:
• But, remember that:
EVPI = E{max(py – C(y)}
Y
Thus, using this, we get:
EVPI = E{max(py – y²)} – max E(py - y²)
Y
Y
max E(py - y²)
Y
max uy —
Y
πT (μ)
=
max py - y²
Y
π(p)
ܝ ܕ | ܠ
var (p)
E(p²)
=
=
μ
Y
р =
Y =
=
P/²/2
TIT
-max E(py - C(y))
Y
p²
=
=
FOC
2y
NIE
=
22
E[π(p)] = = E(p²)
FOC
2y
=
E(p²) - μ²
var(p) + µ²
1
1
E[ñ(p)] − ñ(µ) = ½{var(p) + µ²³] — — µ² = ²/var(p) > 0
• This is the amount of money the firm is willing to pay for information on p.
● It is willing to pay this, although it is risk-neutral.
• What about the case of a risk-averse firm?
(80)
(81)
(82)
(83)
(84)
(85)
(86)
(87)
(88)
(89)
(90)
(91)
(92)
(93)
(94)
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