Supply and Demand Let the supply and demand functions for...

Key Concepts

Supply Curve

The supply curve depicts the relationship between the price of a good and the quantity that producers are willing to offer for sale. It is typically upward sloping, indicating that producers supply more at higher prices, and in this context, it is represented by a linear equation which simplifies the analysis of supply behavior.

Demand Curve

The demand curve illustrates the relationship between the price of a good and the quantity that consumers are willing to purchase, usually showing a downward trend as higher prices lead to lower demand. It is modeled by a linear equation, making it easier to understand how changes in price affect consumer demand.

Market Equilibrium

Market equilibrium is reached when the quantity supplied equals the quantity demanded, which occurs at the intersection of the supply and demand curves. At this point, both the equilibrium price and the equilibrium quantity are determined, providing a snapshot of market stability where there is no inherent tendency for change.

Transcript

00:01 So this problem gives us the supply and demand functions for butterscotch ice cream.

00:10 And they're given by these two formulas right here.

00:14 P represents the price and dollars and q is the number of 10 gallon types.

00:20 So the first thing that we're asked to do is to graph these two functions on the same axis.

00:28 So since both of these are linear functions, we need only find.

00:31 Two points on each of them and that's how we're going to strive.

00:38 So for this supply function, first thing we're going to note is we can rewrite the function and this is going to be or this is the slope intercept form which means that in this case b equals and since the x, i'm sorry, the y intercept is zero, then that means that this point is on this line.

01:22 Second, for the second point we are going to set q to a specific number.

01:32 We're going to pick 300 and then we're going to find out to the p.

01:42 Coordinate is or s of q is so when p is 300 p is s of q when s of q when this case is 300 and now we're going to use this formula that we're given and this is a constant 5 over 7 and q times q when this case is 300 and this gives us approximately 214.

02:40 .29.

02:43 So then this is the value of p when q is 300, which means that the point with coordinates 300 and 214.

03:02 Point 29 is on this graph also.

03:10 Here is our first point.

03:14 This is the second.

03:19 Now we're going to do the same thing for this other, for the demand function.

03:27 So here we already have it in slope intercept form.

03:33 So the b or y intercept.

03:41 Is 200.

03:45 So this means that the point 0 comma 200 is on this function, on the graph for this function.

04:00 So this is their first point.

04:02 And next we're going to do something similar.

04:07 Instead of setting a value to q, we are going to set a value for p.

04:18 So if we set a value for p, if we set p to be zero, this will be equivalent to finding the x or in this case of q intercept.

04:35 So let's do that.

04:38 So p is equal to d of q and d of q is equal to this formula.

04:45 So we're going to set all of this to zero.

04:56 And then we are going to solve for q.

05:01 And that's going to be the q intercept.

05:05 We can call it that.

05:08 So first thing, we're going to move this guy over here.

05:14 To do that, we're going to subtract 3 over 7 q on both sides.

05:24 Q.

05:25 Don't forget that.

05:29 Oh, i'm sorry.

05:30 This is negative, so we need to add this on both sides.

05:39 And then on this side, this cancels with this.

05:44 So we are left with 200 on one side.

05:50 And on the other side, we have 3 over 7 times q.

05:58 Now, last step to solve for q is we want to make this one.

06:06 So we're going to multiply this side by 7 over 3.

06:14 And if we do it on one side we have to do it on the other which means 200 times 7 over 3.

06:26 So as you wanted, these two cancel out and we are left with q equal to whatever this number is.

06:40 If you plug it into the calculator we get approximately 467.

06:51 So this is the value of q when p is zero, which means that the point with coordinates 467 and 0 is on this graph.

07:11 So these are our two points that we're going to use to plot the demand function.

07:22 So we have now two points for each with the supply and demand function.

07:26 We can go ahead and plot that in here.

07:29 So we're going to start with the supply function.

07:36 Supply s of q.

07:41 So the first point is origin.

07:54 That will be over here.

07:56 The second point is 300 and 214 .29...

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