Accelerator

The accelerator principle links investment to changes in output: when output rises, firms want more productive capacity and invest more; when output growth slows or falls, investment can drop sharply.

A simple accelerator relationship

One reduced-form version is:

[ I_t = v,(Y_t - Y_{t-1}) = v,\Delta Y_t, ]

where:

(I_t) is (net) investment,
(Y_t) is output,
(v) is an accelerator coefficient (how much capital firms want per unit of output).

The key feature is that investment responds to (\Delta Y_t) (the change in output), not just the level of output.

A more structural view: desired capital and adjustment

A common way to motivate the accelerator is:

firms have a desired capital stock (K_t^) proportional to output, (K_t^ = vY_t),
investment is how the firm closes the gap between desired and actual capital:

[ I_t = \delta,(K_t^* - K_{t-1}). ]

This “flexible accelerator” highlights adjustment costs and gradual capital accumulation.

Why it can amplify the business cycle

If output growth slows even a little, (\Delta Y_t) can fall a lot relative to its previous value. Because the accelerator ties (I_t) to (\Delta Y_t), investment becomes highly procyclical and can magnify booms and recessions.

Practical example

If a firm plans capacity based on sales growth, then a slowdown from 4% growth to 1% growth can cause a large cut in planned expansion spending, even if sales are still rising.

Knowledge Check

### In the simple accelerator \(I_t = v\,\Delta Y_t\), investment responds to: - [x] Changes in output (growth), not just the level of output - [ ] The unemployment rate only - [ ] The inflation rate only - [ ] The trade balance only > **Explanation:** The accelerator ties investment to how fast output is changing; when growth slows, \(\Delta Y_t\) falls and investment can drop sharply. ### In the “desired capital” motivation \(K_t^* = vY_t\), what does \(K_t^*\) represent? - [x] The capital stock firms would like to hold given current output - [ ] The amount of money households want to save - [ ] The government’s budget deficit - [ ] The equilibrium interest rate > **Explanation:** The accelerator logic comes from firms targeting a roughly stable capital-output relationship. ### Why can the accelerator amplify business cycles? - [x] Small slowdowns in output growth can produce large declines in investment spending - [ ] Investment becomes constant by definition - [ ] Output stops responding to demand shocks - [ ] It eliminates adjustment costs > **Explanation:** Because investment is sensitive to \(\Delta Y_t\), fluctuations in growth translate into large swings in investment and aggregate demand.