Government spending on goods and services (7) EX = EXP + vq

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1 ulic Affairs 854 enzie D. Chinn Spring 2011 Social Sciences 7418 University of Wisconsin-adison The IS-L odel This set of notes expands the Keynesian model with effects to incorporate the monetary sector. This model is popularly known as the IS-L model, and allows for oth fiscal and monetary policy multipliers. 1 Derivation To allow for a role for money, let s first modify the model. On the real side of the economy, everything is the same, except for euation (5), for investment. (1) = AD Output euals aggregate demand, an euilirium condition (2) AD C + I + G + EX I Definition of aggregate demand (3) C = CO + c( T) Cons. function, c is the marginal propensity to consume (4) T = TA + t Tax function; TA is lump sum taxes, t is tax rate. (5 ) I = IN i Investment function (6) G = GO Government spending on goods and services (7) EX = EX + v Export spending (8) I = I + m n Import spending The only essential difference is that investment spending now depends on the interest rate. The coefficient is the interest sensitivity of investment. Since income now depends on interest rates, which is endogenous, then solving euations (1)-(8) yields an euation of a line. (25) = α[ A + EX I + ( n+ v) i] <IS curve> A + EX I 1 c( 1 t) + m (25 ) i = <IS curve> This expression means that for lower levels of interest rates, investment, a component of aggregate demand, is higher, and thus income is also higher. We introduce the monetary sector y setting out money supply and money demand. E.No. Euation Description (26) d s = Euilirium condition (27) s = oney supply (28) d = k hi oney demand

2 Sustitute (27) and (28) into (26), to otain: (29) = k hi -h (k) is the interest (income) sensitivity of real money demand Solving for the interest rate yields: (30) i = 1 k + <L curve> There are two unknowns, and two euations. To figure out euilirium income and euilirium interest rates, one would need to solve the system. This is shown in the appendix. For now, I ll merely show the answer, and relate it to the graphical depiction. (31) 0 = A + EX I + n+ v + $ 1 ( ) where $ α 1 c( 1 t) + m+ k Notice that euilirium income now depends on the level of autonomous spending, the real exchange rate, and the money stock (in real terms). The euilirium interest rate is a complicated function of autonomous spending, real exchange rate and the money stock. To otain this value, one would sustitute (25) into (30). This is done in the appendix. The euilirium income level and interest rate is depicted in the figure elow: ( A + EX I) L, Slope = k/h 1 c(1 t) + slope = m EX, I, 1 0 2

3 One can see that the IS curve has a slope that depends upon the parameters y solving (25) with i on the left hand side, as in euation (25 ). Note: The position of the IS curve depends upon A, I, EX, and. The position of the L curve depends upon the real money stock, (/). Only at the comination of and 0 is it true that oth the goods market and the money market are in euilirium (=AD and s = d respectively). 2 olicy in the IS-L odel Notice that if one increases A, EX, or, then the vertical intercept increases, which is the same as the IS curve shifting out rightward. If one increases when is constant, then / rises, and the L vertical intercept shifts down, which is the same as the L curve shifting out rightward. In the former case, output increases, and interest rates rise. In the latter, interest rates fall, and output rises. ( A + ΔA + EX I) α ΔA i 1 L, αˆ ΔA IS A', EX, I, EX, I, Notation: A'= A +ΔAand = αˆ ΔA 1 0 3

4 Notice that the increase in output $αδa is smaller than that which would have een implied y the simple Keynesian multiplier (αδa ). The reason the increase in output is less in this system is ecause of crowding out. Higher output leads to higher money demand which, given a constant money supply, results in a higher euilirating interest rate. The higher interest rate depresses investment, thus offsetting in part the increase in output. ( A + EX I) L, L ', i 2 EX, I, ' Note: '= +Δ, and = ˆ( α / n) Δ( / ) 2 0 Interest rates fall, resulting in a higher level of investment, thus a higher level of aggregate demand and hence output. How can one solve for the change in income resulting from changes in policy variales analytically? Take euation (31), and reak it up into the constituent changes (i.e., take a total differential): (31) 0 = $ A + EX I + n+ v + ( ) 4

5 (32) Δ = $ ΔA + ΔEX ΔI + n + v Δ + Δ ( ) For changes in government spending only: (33) Δ = Δ $α GO Δ ΔGO = $α For changes in money only (with prices constant): (34) d s = Δ Δ ( / ) = $ α Appendix Solving the IS-L System Algeraically To solve for euilirium income, sustitute the euation (30) <L> into (25) <IS>: (25) = α[ A + EX I + ( n+ v) i] <IS Curve> (30) i = 1 k + <L Curve> (A1) = A + EX I + k n + v 1 + ( ) Bring the multiplier and the term to the left hand side. (A2) c t m k = A + EX I + n+ v + ( ) ( ) Divide oth sides y the term in parentheses to otain euation (31): (31) 0 = A + EX I + n+ v + $ 1 ( ) where $ α 1 c( 1 t) + m+ k To otain the euilirium interest rate, sustitute (25) into (30). It will depend upon all the variales that output depended upon. a854_islmt_s