Sources: Wikipedia (CC BY-SA 4.0) · Hicks (1937), IS-LM model · Taylor (1993), Taylor rule
Governments spend and tax (fiscal). Central banks set interest rates and control money supply (monetary). The IS-LM model shows how these two levers interact to determine output and interest rates simultaneously.
The IS-LM model
The IS curve (Investment-Saving) shows combinations of interest rate r and output Y where the goods market clears. Higher r means less investment, so lower Y. The LM curve (Liquidity-Money) shows where money demand equals money supply. Higher Y means more money demand, so higher r. The intersection is general equilibrium.
Scheme
; IS-LM equilibrium; IS: Y = multiplier * (autonomous spending - b*r); LM: r = (1/h) * (k*Y - M/P); Parameters
(define C0 100) ; autonomous consumption
(define b 500) ; investment sensitivity to r
(define G 200) ; government spending
(define T 150) ; taxes
(define c 0.8) ; marginal propensity to consume
(define k 0.5) ; income sensitivity of money demand
(define h 1000) ; interest sensitivity of money demand
(define M/P 500) ; real money supply; IS curve: Y = (1/(1-c)) * (C0 - c*T + G - b*r); Multiplier
(define mult (/ 1 (- 1 c)))
(display "Multiplier: ") (display mult) (newline)
; Solve IS-LM simultaneously:; Y = mult * (C0 - c*T + G - b*r); r = (k*Y - M/P) / h; Substituting LM into IS:; Y = mult * (C0 - c*T + G - b*(k*Y - M/P)/h); Y * (1 + mult*b*k/h) = mult * (C0 - c*T + G + b*M/P/h)
(define A (+ C0 (- (* c T)) G))
(define denom (+ 1 (/ (* mult b k) h)))
(define Y-star (/ (+ (* mult A) (/ (* mult b M/P) h)) denom))
(define r-star (/ (- (* k Y-star) M/P) h))
(display "Equilibrium Y* = ") (display (round Y-star)) (newline)
(display "Equilibrium r* = ") (display (/ (round (* r-star 1000)) 1000))
If the government spends an extra dollar, GDP rises by more than a dollar. The multiplier = 1/(1-c), where c is the marginal propensity to consume. With c = 0.8, the multiplier is 5: each dollar spent becomes income, 80 cents of which is re-spent, and so on. In IS-LM, the multiplier is smaller because higher Y raises r, which crowds out private investment.
Scheme
; Spending multiplier: trace the rounds; MPC = 0.8, initial spending = $100
(define mpc 0.8)
(define (multiplier-rounds G rounds)
(define (loop n spending total)
(if (> n rounds)
total
(begin
(display "Round ") (display n)
(display ": $") (display (round spending)) (newline)
(loop (+ n 1) (* spending mpc) (+ total spending)))))
(loop 1 G 0))
(display "--- Multiplier rounds ---") (newline)
(define total (multiplier-rounds 1008))
(display "Total after 8 rounds: $") (display (round total)) (newline)
(display "Theoretical total (infinite): $")
(display (/ 100 (- 1 mpc)))
Central bank tools and the Taylor rule
Central banks control the money supply through open market operations, reserve requirements, and the policy rate. The Taylor rule gives a formula: r = r* + 0.5(inflation - target) + 0.5(output gap). It says: raise rates when inflation is above target or output is above potential.
Scheme
; Taylor Rule:; r = r* + 0.5*(pi - pi*) + 0.5*(y - y*); r* = neutral real rate, pi = inflation,; pi* = target, y-y* = output gap (% of potential)
(define (taylor-rate r-neutral inflation target output-gap)
(+ r-neutral
(* 0.5 (- inflation target))
(* 0.5 output-gap)))
; Scenario 1: inflation at target, no output gap
(display "Baseline: ")
(display (taylor-rate 0.020.020.020.0)) (newline)
; Scenario 2: inflation 5%, target 2%, output gap +1%
(display "High inflation: ")
(display (taylor-rate 0.020.050.020.01)) (newline)
; Scenario 3: recession, inflation 1%, output gap -3%
(display "Recession: ")
(display (taylor-rate 0.020.010.02-0.03))
IS-LM model