The demand and supply functions for two interdependent commodities are given by QD1=100-2 P1

The demand and supply functions for two interdependent...

Key Concepts

Market Equilibrium

Market equilibrium refers to the state in a market where the quantity demanded equals the quantity supplied. In the context of multiple interrelated commodity markets, equilibrium is achieved when each product’s demand matches its supply concurrently, taking into account any cross-dependencies between the goods.

Demand Function

The demand function expresses the relationship between the price of a commodity and the quantity demanded by consumers, often accounting for factors like the prices of related goods. It is typically downward sloping, meaning that as the price increases, the quantity demanded decreases. In systems involving interdependent commodities, cross-price terms indicate how the price of one good can affect the demand for another.

Supply Function

The supply function represents the relationship between the price of a commodity and the quantity supplied by producers, generally forming an upward sloping curve. A higher price usually incentivizes producers to supply a greater quantity. For multiple commodities, each supply function can be analyzed separately but must be considered together with demand functions when determining equilibrium.

Simultaneous Equations

Simultaneous equations are used to solve for multiple unknown variables that are interrelated. In a multi-commodity market, the equations obtained by setting demand equal to supply for each commodity form a system of linear equations. Solving this system yields the equilibrium prices and quantities for both commodities.

Interdependencies in Markets

Interdependencies in markets occur when the demand or supply of one commodity is influenced by the price or quantity of another. This is represented in the functions by cross-price coefficients, and it requires simultaneous analysis of multiple markets because changes in one market can shift the equilibrium in another. Understanding these interdependencies is crucial for accurately determining market equilibrium in a system of interrelated goods.

Transcript

00:02 The demand and supply functions for three interdependent quantities or commodities are given.

00:08 Where q sub d is quantity demanded, q sub s is quantity supplied, and p is the price.

00:14 We're going to determine the equilibrium price and quantity for this three commodity model.

00:19 It's important to remember that in equilibrium, quantity demanded equals quantity supply.

00:23 So for this first one, actually for all three of them, we can put the quantity demanded equal to the quantity supplied.

00:30 So for this first one, we get 15 minus p sub 1 plus 2 p sub 2 plus p sub 3 equals negative 7 plus p sub 1.

00:42 For the second one, i get 9 plus p sub 1 minus p sub 2 minus p sub 3 equals negative 4 plus 4 p sub 2.

00:51 And for this final one, i get 8 plus 2 p sub 1 minus p sub 2 minus 4 p sub 3 equals negative.

01:01 At 5 plus 2 p sub 3.

01:07 On this first one, we're going to go ahead and solve for p sub 1.

01:10 I'm going to start by adding 7 to both sides, and i'm also going to add p sub 1 to both sides.

01:18 I get 22 plus 2 p sub 2 plus p sub 2.

01:26 I'm going to divide everything by 2, and we get 11 plus p.

01:31 Sub 2, plus 1 half p .7 3 equals 2, equals p sub 2.

01:37 I'm going to go ahead and substitute.

01:40 Sorry, this is p sub 1.

01:41 Substitute p sub 1 in over here.

01:45 I get 9 plus 11 plus p sub 2 plus 1 half p sub 3 minus p sub 3 equals negative 4 plus 4 p sub 2.

01:57 I don't actually need these parentheses since there's nothing in front of it to distribute.

02:02 So i'm going to drop them.

02:03 I'm going to start combining my like terms.

02:06 9 plus 11, we get 20 plus p sub 2 plus 1 half p sub 3 minus p sub 2 equals negative 4 plus 4 p sub 2.

02:18 P sub 2 minus piece of 2 on the left hand side.

02:21 They're going to cancel out.

02:22 This gives me 20 plus 1 half piece of 3 minus p sub 3.

02:28 I have one full piece of 3 here.

02:31 So i'm going to go ahead and change this to 2.

02:34 Over two piece of three, negative four plus four piece of two over here.

02:39 This way i have a common denominator so i can combine that.

02:43 20 minus one half piece of three equals negative four plus four piece of two.

02:50 I'm going to go ahead and add four to both sides.

02:53 I get 24 minus one half piece of three equals four piece of two.

03:02 We're going to go ahead and solve this for piece of two.

03:05 And i'm going to do that by multiplying both sides by one four.

03:07 I'm choosing to multiply by the fraction instead of dividing by four because i have a fraction here to deal with and it's going to make my life easier.

03:17 That's going to cancel out.

03:18 I'm going to be left with p sub two over here.

03:20 One fourth of 24 is six.

03:23 One fourth times negative one half gives me negative one eighth.

03:27 So i get six minus one eighth piece of three equals p sub two.

03:31 I'm going to go down to this bottom equation here and substitute those things in.

03:36 I get 8 plus 2 times 11 plus p sub 2 plus 1 half p sub 3 minus 6 minus 1 8th piece of 3 minus 4 p sub 3 equals negative 5 plus 2 p sub 3.

03:54 I'm going to go ahead and distribute that 2 through and that negative sign through.

03:59 I get 8 plus 22 plus 2 p sub 2 plus p sub 2 plus p sub 3 minus 6.

04:06 Plus one eighth piece of three minus four piece of three equals negative five plus two piece of three.

04:15 Eight plus 22 is 30 minus six.

04:20 So eight plus 22 minus six.

04:24 We get 24.

04:25 I've got two piece of two here.

04:28 I'm going to substitute in this equation here for piece of two, six minus one eighth piece of three.

04:34 And then i have p sub three minus one eighth p sub three minus four...

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