David Romer's name has come up several times in recent discussions of the IS-LM and IS-MP models. This is how Romer's new edition of his graduate level macroeconomics book derives the IS-LM and IS-MP curves:
Assume that firms produce labor using labor as the only input, i.e. Y = F(L), F'>0, F''≤0, and that government, international trade, and capital are left out of the model for convenience (so that Y=C+I+G+NX becomes Y=C).
Also assume that "There is a fixed number of infinitely lived households that obtain utility from consumption and from holding real money balances, and disutility from working. For simplicity, we ignore population growth and normalize the number of households to 1. The representative household's objective function is":
There is diminishing marginal utility (or increasing marginal disutility), as usual. (Note that assuming money is in the utility function is a standard short-cut. See Walsh for a more extensive discussion of this.)
Next, let the utility functions for consumption and real money balances take their usual constant relative risk aversion forms:
There are two assets in the model, money and bonds. Money pays no interest, while bonds receive an interest rate of it. Wealth evolves according to:
where At is household wealth at the start of period t, WtLt is nominal income, PtCt is nominal consumption, and Mt is nominal money holdings. This equation says that wealth in period t+1 is equal to the amount of money held at the end of time t plus (1+it) times the bonds help from t to t+1 (the term in parentheses is bonds).
Households take the paths of P, W, and i as given, and they choose the paths of C and M to maximize the present discounted value of utility subject to the flow budget
constraint and a no-Ponzi-game condition (for simplicity, the choice of L is set aside for the moment). Finally, the path of M is chosen by the monetary authority (later, when the MP curve is derived, this assumption will be changed).
The optimization condition (Euler equation) for the intertemporal consumption tradeoff is:
We now, in essence, have the New Keynesian IS curve. To see this, take logs of both sides:
And using the fact that Y=C, approximating ln(1+r) as r (which holds fairly well when r is small), and dropping the constant for convenience gives:
This is the New Keynesian IS curve. It's just like the ordinary IS curve, except for the lnYt+1 term on the right-hand side (in models with stochastic shocks, this becomes EtlnYt+1, where EtlnYt+1 is the expected value of Yt+1 given the information available at time t -- often the information set contains only lagged values of variables in the model).
Thus, the big difference between the old IS and the microfounded New Keynesian IS curve is the EtlnYt+1 term on the right-hand side. (Thus, it's relatively easy to amend the traditional model of the IS curve to incorporate the expectation term.)
It can also be shown (e.g. through a variations argument) that the first order condition for money holding is:
This implies that:
Money demand is increasing in output and decreasing in the nominal
interest rate. If this is set equal to (exogenous) money supply, then we have an LM curve. And if we graph the LM curve along with the New Keynesian IS curve, it looks just like the traditional formulation of the model (with the main difference being the expectation of future output term discussed above).
The ideas captured by the new Keynesian IS curve are appealing and useful... The LM curve, in contrast, is quite problematic in practical applications. One difficulty is that the model becomes much more complicated once we relax Section 6.1's assumption that prices are permanently fixed... A second difficulty is that modern central banks do not focus on the money supply.
The first problem is that the LM curve shifts when P changes, so if there is inflation it will be in constant motion making it hard to use as an anlytical tool. That can be overcome, but the second objection is harder to dismiss. However, it is easy to address. Simply assume that the central bank follows a rule for the interest rate such as:
If the central bank adjusts M to ensure this holds, then the money supply is now essentially endogenous (and the interest rate is set externally through the rule). This is an upward sloping curve in r-lnY space, and it is called the MP curve (for monetary policy). It replaces the LM curve in the IS-LM diagram giving us the IS-MP model.
However, it would still be possible to do the analysis with the IS-LM diagram, just put a horizontal line at the fixed interest rate and find the money supply that makes this an equilibrium, but as noted above in the presence of inflation the LM curve shifts out continuously making the model hard to use. Thus, in the presence of inflation and an interst rate rule, the IS-MP formulation is much simpler to use. But for other questions, e.g. quantitative easing at the lower bound or pedagogically examining a money rule, the IS-LM model is often more intuitive.
But the main point is that if you start from (very simple) microfoundations, the resulting model looks a lot like the old IS-LM model. It still needs to be able to handle price-changes, so it's necessary to add a model of supply to the model of demand provided by the IS-MP or the IS-LM diagrams, and the expectation term on the right-hand side of the IS curve is an important difference from the older modeling scheme, but the two models have a lot in common.