24 Decision tree: which technique?
When you read an exam question, identify the type before you start computing. The right approach is fast; the wrong one wastes minutes.
24.1 Step 1: What is the question asking?
| if the question asks for… | use this approach |
|---|---|
| CPI in some year or inflation between two years | Laspeyres formula. Cost of base-year basket at year-t prices ÷ cost at base-year prices × 100. |
| Unemployment rate | u = U / (E + U). Note: discouraged workers are NOT in U. |
| Equilibrium Y^* in a closed economy | Y^* = (\overline{C_0} + \overline{I}) / (1-c). |
| Equilibrium Y^* in an open economy with government | Sum-of-multipliers form: Y^* = \frac{1}{1-c}(\overline{C_0} + \overline{I} + \overline{G} + \overline{X} - \overline{M}) - \frac{c}{1-c}\overline{T}. |
| The change in Y^* from a change in autonomous variable | Multiplier \times change. \Delta Y^* = K_X \cdot \Delta\overline{X}. |
| What change closes a recessionary/inflationary gap | \Delta\overline{X} = (Y^{FE} - Y^*) / K_X. |
| The MPC, given Y^* and autonomous values | Solve 1 - c = (\overline{C_0} + \overline{I}) / Y^* for c. |
| \Delta M^S from an OMO purchase | \Delta M^S = K_S \cdot \Delta\text{Reserves} where K_S = 1/rrr. |
| Required reserves vs. excess reserves | RR = rrr \cdot DD_p. ER = TR - RR. Bank can lend up to ER. |
| Bond present value | PV = FV/(1+r)^T. |
| Implied interest rate from a bond price | r = (FV/PV)^{1/T} - 1. |
| The PPP exchange rate | E_{\$/\pounds} = P_{US}/P_{UK}. |
| Direction of currency move from relative inflation | High inflation \Rightarrow depreciate. PPP. |
| Direction of currency move from relative interest rates | High interest rate \Rightarrow appreciate. Capital flows. (Opposite of inflation.) |
| Effect of currency depreciation on Y | \uparrow EX, \downarrow IM, \uparrow (X-M), \uparrow Y, \uparrow P. |
| Effect of monetary policy under fixed vs. floating ER | Floating: both r channel and FX channel. Fixed: only r channel; FX channel dead. |
24.2 Step 2: Mind the signs
These are the four most-failed questions on macro exams:
- Tax multiplier is negative. K_T = -c/(1-c). Cutting T raises Y^*.
- Bond price and r are inverse. Always.
- Inflation and interest rate effects on FX go OPPOSITE directions. High inflation depreciates; high interest rate appreciates.
- Currency appreciation lowers exports (home goods more expensive abroad).
24.3 Step 3: Beware composition
If the question mentions multiple changes (\overline{G} rising AND \overline{T} rising), do them separately and sum:
\Delta Y^* = K_G \Delta\overline{G} + K_T \Delta\overline{T} + \ldots
The balanced-budget multiplier K_{BB} = K_G + K_T = 1 is just this for the special case where \Delta\overline{G} = \Delta\overline{T}.
24.4 Step 4: Sanity check
After you compute, ask:
- Is the sign right? (Did Y^* go up when I expanded fiscal policy?)
- Is the magnitude in a reasonable range? (Multiplier at 0.75 MPC should give K \approx 4, not 40.)
- Did I plug the numbers in the right place? (T goes inside the bracket multiplied by -c, not -1.)
If anything fails, redo. Most computational errors are sign errors or fraction-flip errors. The interactive tools on this site can verify numerical answers in 30 seconds — use them.