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, ( $$ K_{t}-K_{t-1} $$ ), for that period (4 marks)

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1. Define the terms:
   - $$ I_t $$ is the actual investment made during period $$ t $$.
   - $$ K_t - K_{t-1} $$ represents the desired change in the capital stock (i.e., the target increase in capital) during period $$ t $$.
   - $$ \partial $$ is the coefficient of adjustment, where $$ 0 < \partial \leq 1 $$, indicating the fraction of the gap between actual and desired capital formation that is closed in each period.

2. Interpret the equation:
   - The equation
     $$
     I_t = \partial \left( K_t - K_{t-1} \right)
     $$
     indicates that the actual investment $$ I_t $$ is only a fraction $$ \partial $$ of the desired investment. In other words, firms do not fully adjust their capital stock to the new desired level immediately, but instead gradually move toward it.
   
3. Explain the role of $$ \partial $$:
   - If $$ \partial = 1 $$, the entire desired adjustment is implemented during the period, meaning the firm fully adjusts its capital stock.
   - If $$ 0 < \partial < 1 $$, only a portion of the desired increase is realized in the current period. The smaller the value of $$ \partial $$, the slower the adjustment process.
   - This gradual adjustment may be due to various factors such as adjustment costs, information delays, or other market frictions.

4. Summarize the economic implications:
   - The model captures the idea that firms face constraints or costs that prevent instant adjustment to their ideal capital stock.
   - The partial adjustment hypothesis explains why investment processes in the economy are gradual rather than instantaneous, which has important implications for the dynamics of capital accumulation and economic fluctuations.

Thus, the relationship $$ I_t = \partial \left( K_t - K_{t-1} \right) $$ reflects a scenario where actual investment is only a partial step toward the optimal or desired level of capital accumulation, with $$ \partial $$ quantifying the speed of adjustment.
In the flexible accelerator model, actual investment $$ I_t $$ is a fraction $$ \partial $$ of the desired change in capital $$ K_t - K_{t-1} $$. This means firms adjust their investment gradually, not immediately reaching the desired level. The coefficient $$ \partial $$ (where $$ 0 < \partial \leq 1 $$) represents how much of the desired change is implemented each period. If $$ \partial $$ is closer to 1, adjustments are faster; if closer to 0, adjustments are slower. This gradual adjustment reflects real-world constraints and explains why investment changes over time are not instantaneous.

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