# Savings "identically" equal to investment?

So there's a multiple choice question that says, savings varies positively with the level of income and savings is IDENTICALLY equal to investment. Find the slope of the IS curve.

Options are :

Positive sloping

Negative sloping

Doesn't exist

Vertical

^this is the only information given.

According to my knowledge, savings is always identically to investment from the macroeconomic equations Y = C + S and Y = C + I. Hence it should be downward sloping. However according to my teacher, IS curve should not exist because they have specified identically.I was under the impression this always happens. Can someone please fill in where my gap in understanding is? What am I missing? Is savings not always identically equal to investment? Or is there something in the question itself that I am missing ?

This is indeed a strange explanation. I can only speculate, but my guess is that your teacher reasons as follows:

(i) The IS curve, as the name says, is the set of points in $$(Y,r)$$-space where $$I=S$$. (ii) But $$I\equiv S$$ by definition, so this holds for all points. (iii) Therefore there is no such thing as an IS curve.

If this is really his reasoning, then he mixes up demand for investment resp. savings with actual investment resp. savings.

• Yup, this is exactly her reasoning. So I assume this is wrong with respect to the original question? What should the actual answer be, downward sloping? I’m having a bit of trouble understanding :( Commented May 1 at 15:02
• @viktornikiforov, yes, downward sloping, as usual. Commented May 2 at 23:08

Yes total saving is by definition always equal to investment. This, is definitional because we define savings in macroeconomics as $$S=Y-C$$. If you work with $$Y=C+I$$ then its easy to see that $$Y-C=S=I$$. You can prove similar result with by adding government or opening the economy, just by adding public savings ($$T-G$$).

It is unclear what your teacher is talking about you should confront your teacher and ask for clarification but IS exists even when $$S=I$$. To derive IS curve assuming no government spending you can start with:

$$Y = C(Y)+I(Y,r) \tag{1}$$

To derive IS curve you simply have to solve the (1) for $$Y$$. Lets assume linear consumption and investment function such that $$C= c_0 +c_1Y$$ and $$I=i_1Y -i_2r$$. Substitute, expressions for $$C$$ and $$I$$ into (1) and solve for $$Y$$ which gives you:

$$Y = \frac{c_0}{1 -c_1 - i_1} -\frac{i_2}{1 -c_1 - i_1}r$$

Which gives you classic downward sloping IS curve.

The identity still holds since again by definition $$S=Y-C(Y)$$, substitute it to (1) and you will find that $S=I(Y,r). However, you should still confront your teacher because maybe he is using the word "identically" not as meaning $$S=I$$ but as something else, also maybe there are some other hidden background assumptions mentioned in class that you omit. • Thanks so much for your reply. So we have these lectures uploaded on the portal and I went back to check it and according to our teacher , because savings is identically equal to investment, they are the same curve in the S - Y space. Usually they are not the same curve and thus from the points they intersect we can derive the IS curve. But because they are the same curve, the goods market is always in equilibrium. However I’m still having some trouble understanding this explanation cause why doesn’t that mean that the IS curve is every point in the plane? Commented Apr 21 at 5:21 • @viktornikiforov frankly I do not understand what your teacher is talking about. 1. Usually IS curve is derived from so called Keynesian cross (what I did above). 2. As shown above the IS curve can be derived while having$S=I(Y,r)$. 3. You can also find IS curve as an intersection of demand for savings and demand for investment but note$S\$ is not demand for savings but the total amount of savings/quantity of savings. Hence S =I without saving demand completely overlapping investment demand. 4. If saving and investment demand are both vertical and overlap then IS
– 1muflon1
Commented Apr 21 at 11:39
• curve would also be just vertical line. In Y,r space a vertical line is not a function (maybe that is what your teacher means), but you can always reorient the graph to work in r,Y space, putting Y on x axis is just convention, and in r,Y space they are both function
– 1muflon1
Commented Apr 21 at 11:41
• okay yes I somewhat understand what youre saying. In that case it will be either vertical or downward sloping? For example, if both S and I are linear positive functions of x and they overlap, in that case will it still be vertical or downward sloping in the r-Y plane? I will ask my teacher once Commented Apr 21 at 12:34
• @viktornikiforov it will have 0 slope and be horizontal in r-Y plane, it will be vertical in Y,r plane but it won't be a function. The slope approaches negative infinity as function becomes more vertical but I am not sure it is appropriate to call it downward sloping you would have to consult someone who specializes in mathematics
– 1muflon1
Commented Apr 21 at 12:59