8. Consider the flexible accelerator model of economic theory. Nerlove postulates the following hypothesis, known as the partial adjustment hypothesis $$ I_{t}=\partial\left(K_{t}-K_{t-1}\right) $$ Interpret the foregoing relationship between actual investment, $$ I_{t} $$, the coefficient of adjustment, $$ \partial,\{0

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8. Consider the flexible accelerator model of economic theory. Nerlove postulates the following hypothesis, known as the partial adjustment hypothesis $$ I_{t}=\partial\left(K_{t}-K_{t-1}\right) $$ Interpret the foregoing relationship between actual investment, $$ I_{t} $$, the coefficient of adjustment, $$ \partial,\{0<\partial \leq 1\} $$ and the desired investment, $$ \left(K_{t}-K_{t-1}\right) $$, for that period

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The equation given in the problem is:

$$
I_{t} = \partial (K_{t} - K_{t-1})
$$

where:
- $$ I_{t} $$ is the actual investment at time $$ t $$,
- $$ K_{t} $$ is the capital stock at time $$ t $$,
- $$ K_{t-1} $$ is the capital stock at time $$ t-1 $$,
- $$ \partial $$ is the coefficient of adjustment, with the condition $$ 0 < \partial \leq 1 $$.

### Interpretation of the Relationship

1. **Desired Investment**: 
   - The term $$ (K_{t} - K_{t-1}) $$ represents the change in capital stock from the previous period to the current period. This change is often interpreted as the desired investment, which is the amount of investment that would be needed to achieve the desired level of capital stock.

2. **Actual Investment**:
   - The actual investment $$ I_{t} $$ is the amount of investment that actually occurs in the period $$ t $$.

3. **Coefficient of Adjustment**:
   - The coefficient $$ \partial $$ indicates the proportion of the desired investment that is realized as actual investment. 
   - If $$ \partial = 1 $$, it means that the actual investment fully adjusts to the desired investment, meaning that all of the desired change in capital stock is achieved.
   - If $$ \partial < 1 $$, it indicates that only a fraction of the desired investment is realized. This reflects the idea that adjustments to investment levels may take time due to various constraints (financial, operational, etc.).

### Summary of the Relationship

- The equation shows that actual investment is a function of the desired change in capital stock, scaled by the adjustment coefficient $$ \partial $$.
- The relationship captures the idea that firms do not adjust their investment levels instantaneously to changes in desired capital stock. Instead, they adjust gradually, which is a key aspect of the partial adjustment hypothesis.
- This model implies that if there is a significant gap between the desired and actual capital stock, the actual investment will increase in subsequent periods as firms attempt to close that gap, but the adjustment will be incomplete in any single period due to the value of $$ \partial $$ being less than 1.

In conclusion, the partial adjustment hypothesis provides a framework for understanding how firms adjust their investment behavior over time in response to changes in desired capital stock.
The equation $$ I_{t} = \partial (K_{t} - K_{t-1}) $$ shows that actual investment $$ I_{t} $$ in period $$ t $$ is a fraction $$ \partial $$ of the desired investment change $$ (K_{t} - K_{t-1}) $$. Here, $$ \partial $$ is between 0 and 1, meaning firms adjust their investment gradually over time rather than making immediate changes.

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