undamentals of
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Fundamentals of
Market Equilibrium Problem.
![G
Market Equilibrium: Stability
and Time Paths
Courtesy of James E. Foster, George Washington University
A perfectly competitive market is made up of many buyers and sellers of an economic product,
each of whom has no control over the market price. In this model, the overall quantity demanded
by the buyers of the product is taken to be a function of the price of the product (among other
things) called the demand function. Similarly, the overall quantity supplied by the sellers of the
product is a function of the price of the product (among other things) called the supply function.
A market is in equilibrium at a price where the quantity demanded is just equal to the quantity
supplied.
The linear model assumes that the demand and supply functions have the form qa = do – dịp
and q, = - so + sıP, respectively, where p is the market price of the product, qa is the associated
quantity demanded, q, is the associated quantity supplied, and do, d1, so, and s1 are all positive
constants. The functional forms ensure that the "laws" of downward sloping demand and upward
sloping supply are being satisfied. It is easy to show that the equilibrium price is p* =
(do + so)/(d1 + s1).
Economists typically assume that markets are in equilibrium and justify this assumption with
the help of stability arguments. For example, consider the simple price adjustment equation
dp
1(qd – 9,) ,
dt
where A > 0 is a constant indicating the speed of adjustments. This follows the intuitive require-
ment that price rises when demand exceeds supply and falls when supply exceeds demand. The
market equilibrium is said to be globally stable if, for every initial price level p(0), the price
adjustment path p(1) satisfies p(t) → p* as t → o.
(a) Find the price adjustment path: Substitute the expressions for qa and q, into the price
adjustment equation and show that the solution to the resulting differential equation is
p(t) = [p(0) – p°]e“ + p*, where c = –a(dı + s1).
(b) Is the market equilibrium globally stable?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec2a006c-864a-4a83-bba5-aff4ff880f07%2Ffd6ae5ff-2d38-470f-bcea-2ee5837eaf8b%2Foobdf4o_processed.png&w=3840&q=75)
Transcribed Image Text:G
Market Equilibrium: Stability
and Time Paths
Courtesy of James E. Foster, George Washington University
A perfectly competitive market is made up of many buyers and sellers of an economic product,
each of whom has no control over the market price. In this model, the overall quantity demanded
by the buyers of the product is taken to be a function of the price of the product (among other
things) called the demand function. Similarly, the overall quantity supplied by the sellers of the
product is a function of the price of the product (among other things) called the supply function.
A market is in equilibrium at a price where the quantity demanded is just equal to the quantity
supplied.
The linear model assumes that the demand and supply functions have the form qa = do – dịp
and q, = - so + sıP, respectively, where p is the market price of the product, qa is the associated
quantity demanded, q, is the associated quantity supplied, and do, d1, so, and s1 are all positive
constants. The functional forms ensure that the "laws" of downward sloping demand and upward
sloping supply are being satisfied. It is easy to show that the equilibrium price is p* =
(do + so)/(d1 + s1).
Economists typically assume that markets are in equilibrium and justify this assumption with
the help of stability arguments. For example, consider the simple price adjustment equation
dp
1(qd – 9,) ,
dt
where A > 0 is a constant indicating the speed of adjustments. This follows the intuitive require-
ment that price rises when demand exceeds supply and falls when supply exceeds demand. The
market equilibrium is said to be globally stable if, for every initial price level p(0), the price
adjustment path p(1) satisfies p(t) → p* as t → o.
(a) Find the price adjustment path: Substitute the expressions for qa and q, into the price
adjustment equation and show that the solution to the resulting differential equation is
p(t) = [p(0) – p°]e“ + p*, where c = –a(dı + s1).
(b) Is the market equilibrium globally stable?
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