Commodity bundles as vectors

By definition,vectors are quantities having magnitude as well as direction ,especially as determining the position of one point in space relative to another

We study consumption bundles by taking them as vectors.My question is why can't we think of them as scalar quantities to study them.Why do we think of consumption bundles as locations in commodity space.What is its use?

In real world, whenever we think of commodities , only their quantities come to our mind.And when we compare two commodity bundles, we only compare their quantities.Then,why do we have to bring in the concept of vectors here...are there any cases in the real world where we are concerned about the displacement of the commodity bundles.I am unable to understand this.

Thanks.

• For one, scalar is of only one dimension. So when you want to study a consumer's preference between a bundle $A$ consisting of 5 apples and 3 oranges and a bundle $B$ consisting of 2 apples and 8 oranges, it's more natural to compare the vectors $(5,3)$ and $(2,8)$ than one scalar measure of each bundle – Herr K. May 12 '16 at 2:51
• @ Herr K..yes scalers have only magnitude but no direction.So we use vectors and also vectors are two dimensional. – Glitteringstar May 12 '16 at 3:26
• Vectors don't have to be just two-dimensional; they can be of any dimension. For example, you can use a 3-vector to represent a bundle of three goods (apple, orange, banana). Or, more generally, an $n$-vector $(q_1,q_2,\dots,q_n)$ may be used to represent a bundle of $n$ goods. – Herr K. May 12 '16 at 4:20
• That is fine, I agree that vectors can be n dimensional.But I am asking what role displacement has to play as far as consumption bundles are concerned ?Suppose we are talking about only two dimensional consumption bundles as given above..and we have drawn them as I have shown above.The displacement arrows shown in red,where can we apply this concept in real world...can you cite me an example?.. – Glitteringstar May 12 '16 at 4:31
• I guess the notion of displacement doesn't have much economic meaning. That's probably why I had to look up this term to know what it means (I don't have a background in physics/engineering). As I mentioned earlier, economists use vectors to represent consumption bundles partly due to its convenience and well-understood mathematical properties. But not every mathematical (or physical) property of vector has to have an economic meaning. – Herr K. May 12 '16 at 4:46

First of all, you need to expand the definition of a vector. It is not only quantities having maginitude and direction. While I agree that is the representation in $\mathbb R^2$, it is not always the case. Think of a vector as a representation of any ordered group of numbers $(x_1,x_2,...,x_n)$ and forget completely that you ever thought these implied magnitud and direction.

We think of commodity bundles as vectors and not as quantities by themselves because we want to be able to explain consumption choices as bundles. Let me give you an example with a bundle of 2 goods, $(x_1, x_2)$. Think that $x_1$ is the amount of coffees that you can buy whereas $x_2$ is the amount of packets of sugar that you can buy. We want to be able to compare, say, $(2,1)$ with $(1,2)$. I agree, we could analyze both the choice of both goods separately, but ultimately we want to know what is the optimal consumption bundle and not the optimal amount of sugar and coffee separated. Now, did I say something that entails magnitude? Or direction? No.

Bottomline, yes, we are thinking about quantities of goods, but when we say vectors, it means bundles of quantities of goods. That is ultimately what happens in the real world.

• I am sorry, didn't get your point.You are trying to say that by taking commodity bundles as vectors, we mean their quantities and nothing else?so is the case in the real world? – Glitteringstar May 16 '16 at 8:43
• My first question was : Why commodity vectors or for that matter, optimal choice of the consumer i.e the optimal bundle is taken as a vector quantity and not scalar,I was told since consumption bundles can be taken in 2 or more dimensional space ( suppose it is two dimensional )and not in one dimensional space we take them as vector quantity and not scalar. – Glitteringstar May 16 '16 at 8:59
• By taking commodity bundles as vectors we mean the quantities of each of these elements of the vector. I don't get your second sentence. – MathUser May 16 '16 at 18:54

In physics it is useful to think of a vector as being defined by its magnitude and direction, but I think this is not the most useful intuition in Economics, rather the better intuition is as an ordered group of numbers as MathUser said. We can think of a vector's magnitude as its norm and think of its direction as where it points to in $n-space$. The main distinction of a scalar, $\alpha$, and a vector, $\mathbf{x}$ is what space they are in.

$$\alpha \in \mathbb{R}^1; \; \; \mathbf{x} \in \mathbb{R}^n$$

An equivalent way to represent a vector / consumption bundle is as a column matrix

$$\mathbf{x} = (x_1, x_2, \dots, x_n) = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$$

Where each row indicates a different commodity. Using the two vectors you drew in your question as examples:

$$\mathbf{v_1} = \begin{bmatrix} 5 \\ 3 \\ \end{bmatrix}; \; \; \mathbf{v_2} = \begin{bmatrix} 2 \\ 4 \\ \end{bmatrix}$$

We could compare their consumption bundles by saying Person 1 consumed 5 units of coffee and 3 units of sugar, and Person 2 consumed 2 units of coffee and 4 units of sugar. But what if we want to look at the utility gained of consuming both coffee and sugar? We need a function that takes both of these as inputs. A simple utility function could multiply the two.

$$u(x_1, x_2) = x_1 x_2$$

So Person 1 has utility

$$u_1 (5,3) = 5*3 = 15$$

and Person 2 has utility

$$u_2 (2,4) = 2*4 = 8$$

Using vectors has allowed us to calculate utility by take into account all of the information we are given, rather than looking at things one at a time.