Paradox of savings in an open economy

Hi, I modified investment but otherwise took the open economy equation from Tutorial 5 to get: Y = (c0 + c1Y) + (b0 + b1Y - b2i) + G + (m*Y - E*mY).

Solving for i:
i = [c0 + c1Y + (b0 + b1Y) + G + (m*Y - E*mY) - Y]/b2

In open economy Y - C - G = S = I + NX, so ∂S/∂c0 = ∂(NX+I)/∂c0 = ∂(I+NX)/∂i * ∂i/c0 by the chain rule.

∂(I+NX)/∂i * ∂i/c0 = (-b2)*(1/b2) = -1 < 0.

So the paradox of savings does not hold in the open economy and a one unit fall in c0 leads to a one unit rise in savings (i.e. investment + net exports).

Is this correct?

Hi,

This is not quite right but a good effort. A few comments:

1. As I understand what you've done, this is an open-economy IS model (or goods market model). As in, there is no LM curve and no UIP condition; both i and E* are exogenous. i.e. this is not an open-economy IS-LM model. Which is fine, as long as you're clear on what model you've set up.

2. Since i is exogenous, we need to solve for Y (not i), since Y is endogenous. Here is how I'd go about it:
- Original equation (Note: Exports are m*Y*)
Y = (c0 + c1Y) + (b0 + b1Y - b2i) + G + (m*Y* - E*mY)
- Get all the Y's on the LHS (everything on the RHS is exogenous):
Y = (1/(1-c1-b1+E*m)) x (c0 + b0 - b2i + G + m*Y*)
- calculate dY/dc0:
dY/dc0 = (1/(1-c1-b1+E*m))
- Definition of private savings:
S = Y - C = I + G + NX
= (b0 + b1Y - b2i) + G + (m*Y* - E*mY)
(Note: We now have savings as a function of Y, our only endogenous variable, and a whole bunch of exogenous stuff)
- Calculate dS/dc0, using the chain rule:
dS/dc0 = b1dY/dc0 - E*mdY/dc0
= (b1 - E*m) x dY/dc0
- Now we can substitute in our earlier result for dY/dc0 to get:
dS/dc0 = (b1 - E*m) / (1-c1-b1+E*m))

3. How could we have checked to make sure that your answer made sense? First we could have "turned off" all the open economy stuff by setting m*=m=0. Then we should get back to the closed economy result. In your case, that is not what we get - so we should already be worried that something has gone wrong. In my case, I do get the closed economy result:
dS/dc0 = b1 / (1-c1-b1)
which is at least comforting (although of course not a guarantee that I've done my algebra right!)

4. What have we learnt?
In the closed economy world, dS/dc0>0 ---- a fall in c0 leads to a fall in S --- we get the savings paradox.
In the open economy world, as long as b1>E*m, we also get the savings paradox. But, suppose the marginal propensity to import is really large such that b1<E*m ("the investment multiplier is weaker than the leakage from the circular flow from imports). Then we get the opposite result.

Keep practicing - because this is how we learn - learning by doing!

Best
Oliver

Hi,

This type of question would not be an unreasonable question. You have solved several savings paradox problems in tutorials. The steps in the derivation here are exactly the same - the only difference that the model is a little more complicated.

Best
Oliver