14.02 Principles of Macroeconomics

Fall 2000

Prof. Ricardo Caballero

Problem Set 2

posted: 9/20

due: 9/27

Solutions:

1a This is true: An expansionary open market operation means buying bonds for money which was not in circulation previously. Therefore it increases the money supply, so in order to make people want to hold the increased amount of money, interest rates must go down. Alternatively, with less bonds in the market, bond prices rise. However, the interest rate on a $100 bond is i = (100-p_B)/p_B, so an increase in p_B means a decrease in the interest rate.

1b False: they might or might not decide to consume or save more; but they will definitely shift their portfolio allocation from cash to bonds. So money demand will go down.

1c: Even in a closed economy, investment is equal to the sum of private and public saving. Since public saving is rarely zero, the statement is false. In an open economy, investment can be also financed by foreigners (net exports).

1d: True. With credit cards, one does not need to keep cash at hand for daily transactions. You can keep your wealth in bonds, and then convert it to cash only once a month, when you have to pay your credit card bill. Of course, you will still keep some cash with you, but less.

2a If c₁ goes down, consumers are spending relatively less from each additional dollar. Therefore, they are saving relatively more of that extra dollar.

The equilibrium level of output is

Y*=1/(1- c₁)*( c₀+I+G- c₁ *T)= 1/(1- c₁)*(c₀+I)+G (using that T=G).

So if c₁ decreases, 1- c₁ increases, 1/(1- c₁) decreases, and all the rest stays the same -- therefore output goes down.

Note that the T=G assumption was necessary for this clear conclusion.

2b You want to save more, which is assumed to be a virtue for the economy (i.e., that is expected to increase output) -- and instead, you may end up with lower production. Moreover, the amount of saving will stay the same (see part c): you save a larger fraction (1-c₁ is larger) of a smaller pie (Y-T is smaller).

This is, however, only a short-run result: in the long-run, a higher saving rate will increase investment, will promote accumulation of capital, the economy will have more factories, machines etc, and that will serve as the basis for long-run growth.

2c S=Y-T-C=I+G-T; so saving stays the same. With G=T, S=I also holds. Investment, which is a constant, equals the sum of private and public saving, so if public saving does not change, neither does private saving.

3a Here consumption depends not only on current income, but on past income as well. One rationale might be that it takes some time to change your consumption behavior after a change in your income. Or an increase in your income might be transitory, so you do not immediately start spending much of it – next year, when you already have two years with increased income, you indeed spend a lot more.

3b With G going up, there is a stimulus to demand, hence output and also income rises. Of that increase, ½*c₁ goes into extra consumption. This gives a multiplication, but not as big as with c₁. Next year, however, the term for past income will rise, which shifts demand further up. So this consumption function implies a gradual response to fiscal expansion, the adjustment takes some time. (In terms of our diagrams, it slows down the vertical arrows.)

4a With higher interest rates, it is more costly to hold cash, since you are giving up more interest income. So you will keep smaller cash balances on average, and correspondingly, larger bond balances.

With higher nominal income, you are likely to do more transactions. This makes it necessary to have more cash available, therefore your cash holdings go up.

4b In the standard i-M graph, we have a downward-sloping money demand line and a vertical money supply line. Their intersection is the equilibrium interest rate, given by i^* = PY/M^S.

4c A contractionary open market operation pours bonds to the market and removes money in exchange. So M^S decreases, which implies an increase in the interest rate. In the graph, money supply shifts inwards, so the intersection point moves upward along the money demand line.

One can give two verbal arguments: One starts with lower money stock available. At current interest rates, people are not willing to reduce their cash balances, so the interest rate must rise enough to make people ready to deplete their cash holdings. From a bond market perspective, the argument is similar to question 1a: more bonds, therefore lower bond prices, which is the same as higher interest rates.