Question

Q1) Define and derive the IS-LM model. More specifically, how we obtain the IS-LM equilibrium? Please explain every step in detail. (40 pts.)

What happens if autonomous government spending increases? Use graphs to illustrate your points. (60 pts.)

Answer 1

The IS-LM model, which stands for "investment-savings" (IS) and "liquidity preference-money supply" (LM) is a Keynesian macroeconomic model that shows how the market for economic goods (IS) interacts with the loanable funds market (LM) or money market. It is represented as a graph in which the IS and LM curves intersect to show the short-run equilibrium between interest rates and output.

The LM curve can be derived from the Keynesian theory from its analysis of money market equilibrium. According to Keynes, demand for money to hold depends upon transactions motive and speculative motive. It is the money held for transactions motive which is a function of income.

Equilibrium

Combining the discussion of the LM and the IS curves will generate equilibrium levels of interest rates and output. Note that both relationships are combinations of interest rates and output. Solving these two equations jointly determines the equilibrium.

Algebraically, we have an equation for the LM curve:

r = (1/L2) [L0 + L1Y – M/P].

And we have an equation for the IS curve:

Y = mE0(r),

where we let m = (1/(1 – β)) denote the multiplier. If we assume that the dependence of spending in the interest rate is linear, so that E0(r) = e0 – e1r, then the equation for the IS curve is

Y = m (e0-e1r),

To solve the IS and LM curves simultaneously, we substitute Y from the IS curve into the LM curve to get

r = (1/L2) [L0 + L1 m(e0-e1r) – M/P].

Solving this for r we get

r = Ar – BrM/P.

where both Ar and Br are constants, with Ar = (L0 + L1me0)/(L1me1 + L2) and Br = 1/(L1me1 + L2). This equation gives us the equilibrium level of the real interest rate given the level of autonomous spending, summarized by e0, and the real stock of money, summarized by M/P.

To find the equilibrium level of output, we substitute this equation for r back into the equation for the IS curve. This gives us

Y = Ay + By(M/P),

where both Ay and By are constants, with Ay = m(e0 – e1Ar) and By = me1Br. This equation gives us the equilibrium level of output given the level of autonomous spending, summarized by e0, and the real stock of money, summarized by M/P.

The classical economic theory states that any rise in autonomous expenditures will create at least an equivalent rise in aggregate output, such as GDP, if not a greater increase.