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 - dip and q = -so + sip, respectively, where p is the market price of the product, qd 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 +$₁). Economists typically assume that markets are in equilibrium and justify this assumption with help of stability arguments. For example, consider the simple price adjustment equation dp dt λ(9a-9s), where >> 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(t) satisfies p(t) → p* as t → ∞. (a) Find the price adjustment path: Substitute the expressions for qd and q, into the price adjustment equation and show that the solution to the resulting differential equation is p(t) = [p(0) - p*]et + p", where c = −(d₁ + $₁). (b) Is the market equilibrium globally stable?
G Market Equilibrium: Stability and Time Paths Courtesy of James E. Foster, George Washington University A perfectly competitive market is made up of many buyers and sellers of an economic product, each of whom has no control over the market price. In this model, the overall quantity demanded by the buyers of the product is taken to be a function of the price of the product (among other things) called the demand function. Similarly, the overall quantity supplied by the sellers of the product is a function of the price of the product (among other things) called the supply function. A market is in equilibrium at a price where the quantity demanded is just equal to the quantity supplied. The linear model assumes that the demand and supply functions have the form qa = do - dip and q = -so + sip, respectively, where p is the market price of the product, qd is the associated quantity demanded, q, is the associated quantity supplied, and do, d₁, so, and s₁ are all positive constants. The functional forms ensure that the "laws" of downward sloping demand and upward sloping supply are being satisfied. It is easy to show that the equilibrium price is p" = (do + so)/(d₁ +$₁). Economists typically assume that markets are in equilibrium and justify this assumption with help of stability arguments. For example, consider the simple price adjustment equation dp dt λ(9a-9s), where >> 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(t) satisfies p(t) → p* as t → ∞. (a) Find the price adjustment path: Substitute the expressions for qd and q, into the price adjustment equation and show that the solution to the resulting differential equation is p(t) = [p(0) - p*]et + p", where c = −(d₁ + $₁). (b) Is the market equilibrium globally stable?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 6EQ: Redo Exercise 5, assuming that the house blend contains 300 grams of Colombian beans, 50 grams of...
Related questions
Question
100%
Fundamentals of Differential Equation
Please specify the steps one by one
Market Equilibrium: Stability and Time Path
![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%2F8e9eac79-72b5-41d8-98d1-090a72b6b025%2Fn6ctndl_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?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning