Thursday, May 14, 2009
THE LM CURVE
The LM curve, named because it shows positions at which the demand for money (L for liquidity preference) equals money supply (M), completes the model. In the quantity theory of money we have already met a model of the market for money balances. The quantity theory asserted that velocity was constant, or
(1) MV = Y
which can be rewritten as:
(2) M= (1/V)Y = kY.
Equation 2 says that the amount of money that people hold is a fraction of income. This equation is always true; k will take whatever value needed to make it true. In England in the early 19th century, this equation was altered and made into a demand-for-money equation. The average amount of money people want to hold depends on the amount of spending they expect to do. Thus people who expect to spend a great deal will, on the average, want to hold larger cash balances than those who expect to spend only a little. The "on the average" in the last sentence is important. If a person holds $1400 on Monday and nothing the rest of the week, he has an average weekly holding of $200.
Making these alterations in equation 2 gives:
(3) Money demand = k(expected income)
The k in this equation should not move much or else the equation does not tell us much about how people act. This equation is unlike equation 2 because it makes a statement about how people want to act, while equation 2 tells us how they do act.
The writings of John Maynard Keynes made economists reconsider the traditional demand-for-money function. Keynes argued that there were three reasons why people hold money. They hold cash for transactions purposes, which is what the quantity theory had always said. They also hold money for precautionary reasons, so that in an emergency they would have a ready source of funds. Finally, they hold money for speculative purposes. The speculative motive arose from the effects of interest rates on the price of bonds. When interest rates rise, the price of bonds falls. Thus when people think interest rates are unusually low, they would prefer to hold their assets in the form of money. If they invested in bonds and the interest rate rose, they would suffer a loss. Hence the amount of money people would want to hold should be inversely related to the rate of interest. People will want to hold more money (liquidity) when interest rates are low than when they are higher.
Keynes' introduction of the interest rate into the demand for money has survived, but not for the reasons he gave. Keynes was thinking in terms of a two-asset world: money, which earned no interest but which was liquid and had no danger of a capital loss, and bonds, which earned interest but which were not as liquid and which could yield a capital loss. If one thinks not in terms of a two-asset world, but in terms of the range of assets that actually exist in the world, there is no reason to hold cash balances for either precautionary or speculative purposes. There are assets that are both very liquid and that earn interest, such as savings accounts and Treasury bills, and these are a better form in which to hold assets for these purposes.
Though Keynes' explanation of why interest rates influence the demand for money is flawed, other explanations are sound. Money held for transactions purposes is much like inventory that businesses hold. Holding inventories either ties up funds on which a business could earn interest, or uses borrowed funds on which it must pay interest. Thus if a firm can sell $100,000 of its inventory, it has $100,000 in cash that it can either invest to earn interest or pay off debt on which it must pay interest. The cost of inventories increases as interest rates rise or as the size of inventories increases.
However, there are also costs to holding inventories that are too low. If inventories are too small, a business may run out of items and lose sales. Further, if inventories are held at low levels, the business will need to reorder often, and there are usually costs to reordering. Thus the business must balance these costs that rise as inventories increase with the other costs that fall as inventories increase. The problem can be solved elegantly using calculus, but you should be able to see intuitively that a rise in interest rates will decrease the optimal size of inventories, and a rise in the cost of reordering will increase the optimal size.
When people hold cash balances, they hold their assets in a form that earns either no interest (coin and currency and some deposits on which checks can be written) or less interest than is possible in accounts on which no checks can be written.1 If interest rates rise on non-money assets relative to money, the cost of holding money in terms of interest foregone rises, and one would expect people to try to economize on cash. A business, for example, could shift money from checking accounts into t-bills. It would be worthwhile to make more transactions into and out of interest-bearing assets to take advantage of the higher interest rates. When interest rates are very low, these transactions may not be worthwhile, and the business may be willing to let money lie idle for short periods in checking accounts.
In a nutshell, the argument boils down to the store-of-value function of money. Money becomes a less desirable way to hold wealth when interest rates on other assets rises, and as a result people will hold smaller cash balances. These considerations lead us to a revised demand for money function. Instead of equation 3, the demand for money should be:
(4) Md = kYe + wi.
The demand for money, or the average amount of money people want to hold, depends positively on expected transactions and negatively on the interest rate. The coefficient w should be a negative number because with higher interest rates people should want to hold smaller cash balances.
To complete this part of the model, we need a money-supply equation and an equilibrium condition. A simple money-supply equation is that money stock is determined outside the system by policy. The logical equilibrium condition is that the market for money balances is in equilibrium when money supply equals money demand.
To see how this part of the model functions, imagine that interest rates are very low. When interest rates are very low, people have no special reason to avoid holding idle cash, and will hold considerable amounts. If they hold lots of cash idle, the fixed amount of money cannot support very much spending. Lots of idle cash means that the representative dollar is not being spent very frequently.
On the other hand, if interest rates are very high, holding idle cash is costly, and people will try to keep their holdings low. This means that they will spend money rapidly, or that the velocity of money will be high. With higher interest rates the same fixed quantity of money will support more spending than it did when interest rates were low and people were holding idle cash balances.
The LM curve illustrated above shows the relationship discussed in the last two paragraphs. The curve tells how much spending some fixed amount of money will support. When interest rates are high, as at i*, money is spent rapidly and supports a lot of spending, y*. When interest rates are low, at i#, the money stock supports less spending or y#. Connecting these two points to represent what happens at other interest rates generates the LM curve.
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