Find The Equilibrium Prices And Quantities For The Three Commodities:$[ \begin{aligned} Q_{d1} &= 110 - 4P_1 + 3P_2 - 4P_3 \ Q_{d2} &= 46 + 2P_1 - 4P_2 + 4P_3 \ Q_{d3} &= 20 - P_1 + 4P_2 - 2P_3 \ Q_{s1} &= 2P_1 - 20 \ Q_{s2} &= -14 + 2P_2

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Introduction

In economics, the concept of equilibrium is crucial in understanding the behavior of markets. It refers to a state where the supply and demand for a particular commodity are equal, resulting in a stable market price. However, in a multi-commodity market, the situation becomes more complex. The prices and quantities of one commodity can affect the demand and supply of other commodities. In this article, we will explore how to find the equilibrium prices and quantities for three commodities using the given demand and supply functions.

Demand and Supply Functions

The demand and supply functions for the three commodities are given as:

Qd1=110βˆ’4P1+3P2βˆ’4P3Qd2=46+2P1βˆ’4P2+4P3Qd3=20βˆ’P1+4P2βˆ’2P3Qs1=2P1βˆ’20Qs2=βˆ’14+2P2\begin{aligned} Q_{d1} &= 110 - 4P_1 + 3P_2 - 4P_3 \\ Q_{d2} &= 46 + 2P_1 - 4P_2 + 4P_3 \\ Q_{d3} &= 20 - P_1 + 4P_2 - 2P_3 \\ Q_{s1} &= 2P_1 - 20 \\ Q_{s2} &= -14 + 2P_2 \end{aligned}

where Qd1Q_{d1}, Qd2Q_{d2}, and Qd3Q_{d3} are the demand quantities for commodities 1, 2, and 3, respectively, and Qs1Q_{s1} and Qs2Q_{s2} are the supply quantities for commodities 1 and 2, respectively.

Equilibrium Conditions

To find the equilibrium prices and quantities, we need to set the demand and supply functions equal to each other for each commodity. This will give us a system of equations that we can solve to find the equilibrium prices and quantities.

For commodity 1, we have:

Qd1=Qs1Q_{d1} = Q_{s1}

Substituting the demand and supply functions, we get:

110βˆ’4P1+3P2βˆ’4P3=2P1βˆ’20110 - 4P_1 + 3P_2 - 4P_3 = 2P_1 - 20

Simplifying the equation, we get:

130+6P1βˆ’3P2+4P3=0130 + 6P_1 - 3P_2 + 4P_3 = 0

For commodity 2, we have:

Qd2=Qs2Q_{d2} = Q_{s2}

Substituting the demand and supply functions, we get:

46+2P1βˆ’4P2+4P3=βˆ’14+2P246 + 2P_1 - 4P_2 + 4P_3 = -14 + 2P_2

Simplifying the equation, we get:

60+2P1βˆ’6P2+4P3=060 + 2P_1 - 6P_2 + 4P_3 = 0

For commodity 3, we have:

Qd3=Qs1+Qs2Q_{d3} = Q_{s1} + Q_{s2}

Substituting the demand and supply functions, we get:

20βˆ’P1+4P2βˆ’2P3=2P1βˆ’20+βˆ’14+2P220 - P_1 + 4P_2 - 2P_3 = 2P_1 - 20 + -14 + 2P_2

Simplifying the equation, we get:

54βˆ’3P1+6P2βˆ’2P3=054 - 3P_1 + 6P_2 - 2P_3 = 0

Solving the System of Equations

We now have a system of three equations with three unknowns (P_1, P_2, and P_3). We can solve this system using various methods, such as substitution or elimination.

Using the substitution method, we can solve the first equation for P_1:

P1=130+3P2βˆ’4P36P_1 = \frac{130 + 3P_2 - 4P_3}{6}

Substituting this expression for P_1 into the second equation, we get:

60+2(130+3P2βˆ’4P36)βˆ’6P2+4P3=060 + 2\left(\frac{130 + 3P_2 - 4P_3}{6}\right) - 6P_2 + 4P_3 = 0

Simplifying the equation, we get:

260+3P2βˆ’4P3βˆ’12P2+24P3=0260 + 3P_2 - 4P_3 - 12P_2 + 24P_3 = 0

Combine like terms:

βˆ’9P2+20P3=βˆ’260-9P_2 + 20P_3 = -260

Now, substitute the expression for P_1 into the third equation:

54βˆ’3(130+3P2βˆ’4P36)+6P2βˆ’2P3=054 - 3\left(\frac{130 + 3P_2 - 4P_3}{6}\right) + 6P_2 - 2P_3 = 0

Simplifying the equation, we get:

324+3P2βˆ’4P3βˆ’18P2+12P3=0324 + 3P_2 - 4P_3 - 18P_2 + 12P_3 = 0

Combine like terms:

βˆ’15P2+8P3=βˆ’324-15P_2 + 8P_3 = -324

Now we have two equations with two unknowns (P_2 and P_3):

βˆ’9P2+20P3=βˆ’260-9P_2 + 20P_3 = -260

βˆ’15P2+8P3=βˆ’324-15P_2 + 8P_3 = -324

We can solve this system using the elimination method. Multiply the first equation by 5 and the second equation by 3:

βˆ’45P2+100P3=βˆ’1300-45P_2 + 100P_3 = -1300

βˆ’45P2+24P3=βˆ’972-45P_2 + 24P_3 = -972

Subtract the second equation from the first equation:

76P3=βˆ’32876P_3 = -328

Divide by 76:

P3=βˆ’32876P_3 = -\frac{328}{76}

P3=βˆ’4119β‰ˆβˆ’2.16P_3 = -\frac{41}{19} \approx -2.16

Now that we have found P_3, we can substitute this value back into one of the original equations to find P_2. Using the second equation, we get:

βˆ’15P2+8(βˆ’4119)=βˆ’324-15P_2 + 8\left(-\frac{41}{19}\right) = -324

Simplifying the equation, we get:

βˆ’15P2βˆ’32819=βˆ’324-15P_2 - \frac{328}{19} = -324

Multiply both sides by 19:

βˆ’285P2βˆ’328=βˆ’6164-285P_2 - 328 = -6164

Add 328 to both sides:

βˆ’285P2=βˆ’5836-285P_2 = -5836

Divide by -285:

P2=5836285P_2 = \frac{5836}{285}

P2=11672570β‰ˆ20.5P_2 = \frac{11672}{570} \approx 20.5

Now that we have found P_2 and P_3, we can substitute these values back into one of the original equations to find P_1. Using the first equation, we get:

110βˆ’4P1+3(20.5)βˆ’4(βˆ’2.16)=2P1βˆ’20110 - 4P_1 + 3\left(20.5\right) - 4\left(-2.16\right) = 2P_1 - 20

Simplifying the equation, we get:

110βˆ’4P1+61.5+8.64=2P1βˆ’20110 - 4P_1 + 61.5 + 8.64 = 2P_1 - 20

Combine like terms:

180βˆ’4P1+70.14=2P1βˆ’20180 - 4P_1 + 70.14 = 2P_1 - 20

Combine like terms:

βˆ’2P1=βˆ’149.86-2P_1 = -149.86

Divide by -2:

P1=149.862P_1 = \frac{149.86}{2}

P1=74.93P_1 = 74.93

Equilibrium Quantities

Now that we have found the equilibrium prices, we can substitute these values back into the demand functions to find the equilibrium quantities.

For commodity 1, we have:

Qd1=110βˆ’4P1+3P2βˆ’4P3Q_{d1} = 110 - 4P_1 + 3P_2 - 4P_3

Substituting the equilibrium prices, we get:

Qd1=110βˆ’4(74.93)+3(20.5)βˆ’4(βˆ’2.16)Q_{d1} = 110 - 4\left(74.93\right) + 3\left(20.5\right) - 4\left(-2.16\right)

Simplifying the equation, we get:

Qd1=110βˆ’299.72+61.5+8.64Q_{d1} = 110 - 299.72 + 61.5 + 8.64

Combine like terms:

Qd1=βˆ’119.58Q_{d1} = -119.58

However, since the quantity cannot be negative, we can conclude that the equilibrium quantity for commodity 1 is 0.

For commodity 2, we have:

Qd2=46+2P1βˆ’4P2+4P3Q_{d2} = 46 + 2P_1 - 4P_2 + 4P_3

Substituting the equilibrium prices, we get:

Qd2=46+2(74.93)βˆ’4(20.5)+4(βˆ’2.16)Q_{d2} = 46 + 2\left(74.93\right) - 4\left(20.5\right) + 4\left(-2.16\right)

Simplifying the equation, we get:

Qd2=46+149.86βˆ’82+βˆ’8.64Q_{d2} = 46 + 149.86 - 82 + -8.64

Combine like terms:

Qd2=104.22Q_{d2} = 104.22

For commodity 3, we have:

Qd3=20βˆ’P1+4P2βˆ’2P3Q_{d3} = 20 - P_1 + 4P_2 - 2P_3

Substituting the equilibrium prices, we get:

Q<br/>βˆ—βˆ—FrequentlyAskedQuestions(FAQs)βˆ—βˆ—=====================================βˆ—βˆ—Q:Whatistheconceptofequilibriumineconomics?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Ineconomics,equilibriumreferstoastatewherethesupplyanddemandforaparticularcommodityareequal,resultinginastablemarketprice.βˆ—βˆ—Q:Howdoyoufindtheequilibriumpricesandquantitiesformultiplecommodities?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Tofindtheequilibriumpricesandquantitiesformultiplecommodities,youneedtosetthedemandandsupplyfunctionsequaltoeachotherforeachcommodity.Thiswillgiveyouasystemofequationsthatyoucansolvetofindtheequilibriumpricesandquantities.βˆ—βˆ—Q:Whatarethedemandandsupplyfunctionsforthethreecommodities?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Thedemandandsupplyfunctionsforthethreecommoditiesaregivenas:Q<br/> **Frequently Asked Questions (FAQs)** =====================================

Q: What is the concept of equilibrium in economics?

A: In economics, equilibrium refers to a state where the supply and demand for a particular commodity are equal, resulting in a stable market price.

Q: How do you find the equilibrium prices and quantities for multiple commodities?

A: To find the equilibrium prices and quantities for multiple commodities, you need to set the demand and supply functions equal to each other for each commodity. This will give you a system of equations that you can solve to find the equilibrium prices and quantities.

Q: What are the demand and supply functions for the three commodities?

A: The demand and supply functions for the three commodities are given as:

\begin{aligned} Q_{d1} &= 110 - 4P_1 + 3P_2 - 4P_3 \ Q_{d2} &= 46 + 2P_1 - 4P_2 + 4P_3 \ Q_{d3} &= 20 - P_1 + 4P_2 - 2P_3 \ Q_{s1} &= 2P_1 - 20 \ Q_{s2} &= -14 + 2P_2 \end{aligned}

βˆ—βˆ—Q:Howdoyousolvethesystemofequationstofindtheequilibriumpricesandquantities?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Tosolvethesystemofequations,youcanusevariousmethods,suchassubstitutionorelimination.Inthisarticle,weusedthesubstitutionmethodtosolvethesystemofequations.βˆ—βˆ—Q:Whataretheequilibriumpricesandquantitiesforthethreecommodities?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Theequilibriumpricesandquantitiesforthethreecommoditiesare: **Q: How do you solve the system of equations to find the equilibrium prices and quantities?** -----------------------------------------------------------------------------------

A: To solve the system of equations, you can use various methods, such as substitution or elimination. In this article, we used the substitution method to solve the system of equations.

Q: What are the equilibrium prices and quantities for the three commodities?

A: The equilibrium prices and quantities for the three commodities are:

\begin{aligned} P_1 &= 74.93 \ P_2 &= 20.5 \ P_3 &= -2.16 \ Q_{d1} &= 0 \ Q_{d2} &= 104.22 \ Q_{d3} &= 20 - 74.93 + 4\left(20.5\right) - 2\left(-2.16\right) \end{aligned}

βˆ—βˆ—Q:Whatisthesignificanceoffindingtheequilibriumpricesandquantities?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Findingtheequilibriumpricesandquantitiesissignificantbecauseithelpstounderstandthebehaviorofmarketsandtheimpactofchangesindemandandsupplyonmarketpricesandquantities.βˆ—βˆ—Q:Howcantheequilibriumpricesandquantitiesbeusedinrealβˆ’worldapplications?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Theequilibriumpricesandquantitiescanbeusedinrealβˆ’worldapplications,suchas:βˆ—Predictingmarkettrendsandpricesβˆ—Makinginformedinvestmentdecisionsβˆ—Developingmarketingstrategiesβˆ—Understandingtheimpactofgovernmentpoliciesonmarketsβˆ—βˆ—Q:Whataresomecommonchallengesinfindingtheequilibriumpricesandquantities?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Somecommonchallengesinfindingtheequilibriumpricesandquantitiesinclude:βˆ—Complexityofthedemandandsupplyfunctionsβˆ—Limiteddataandinformationβˆ—Uncertaintyandriskβˆ—Interdependenceofmarketsandcommoditiesβˆ—βˆ—Q:Howcantheequilibriumpricesandquantitiesbeupdatedandrevised?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Theequilibriumpricesandquantitiescanbeupdatedandrevisedby:βˆ—Collectingnewdataandinformationβˆ—Analyzingchangesindemandandsupplyβˆ—Adjustingthedemandandsupplyfunctionsβˆ—Reβˆ’solvingthesystemofequationsβˆ—βˆ—Q:Whataresomefuturedirectionsforresearchonequilibriumpricesandquantities?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Somefuturedirectionsforresearchonequilibriumpricesandquantitiesinclude:βˆ—Developingmorecomplexandrealisticdemandandsupplyfunctionsβˆ—Incorporatinguncertaintyandriskintotheanalysisβˆ—Examiningtheimpactofgovernmentpoliciesandregulationsonmarketsβˆ—Developingnewmethodsandtoolsforsolvingthesystemofequations. **Q: What is the significance of finding the equilibrium prices and quantities?** --------------------------------------------------------------------------------

A: Finding the equilibrium prices and quantities is significant because it helps to understand the behavior of markets and the impact of changes in demand and supply on market prices and quantities.

Q: How can the equilibrium prices and quantities be used in real-world applications?

A: The equilibrium prices and quantities can be used in real-world applications, such as:

  • Predicting market trends and prices
  • Making informed investment decisions
  • Developing marketing strategies
  • Understanding the impact of government policies on markets

Q: What are some common challenges in finding the equilibrium prices and quantities?

A: Some common challenges in finding the equilibrium prices and quantities include:

  • Complexity of the demand and supply functions
  • Limited data and information
  • Uncertainty and risk
  • Interdependence of markets and commodities

Q: How can the equilibrium prices and quantities be updated and revised?

A: The equilibrium prices and quantities can be updated and revised by:

  • Collecting new data and information
  • Analyzing changes in demand and supply
  • Adjusting the demand and supply functions
  • Re-solving the system of equations

Q: What are some future directions for research on equilibrium prices and quantities?

A: Some future directions for research on equilibrium prices and quantities include:

  • Developing more complex and realistic demand and supply functions
  • Incorporating uncertainty and risk into the analysis
  • Examining the impact of government policies and regulations on markets
  • Developing new methods and tools for solving the system of equations.