Learning by Doing

Higher the knowledge $\implies$ Higher capital stock
This is because the act of creating and using new capital goods provides firms with valuable insights and experience. In this model, knowledge isn't something that's simply discovered; it's a byproduct of economic activity.
The production function depending on the technology accumulation will have the form $$ Y_{t} = K^\alpha (A_{t}L_{t})^{1-\alpha} $$ where,
$Y_{t}$ output
$K$ capital
$A_{t}$ stock of knowledge
$L_{t}$ Labor force

The simplest case of LBD is found in those situations where learning occurs as a consequence of production of new capital, i.e., increase in the stock of knowledge $= f(\text{increase in stock of capital})$.¹

Though capital is exogenous, which is determined by savings and investments in the economy, the stock of knowledge being a direct function of capital, becomes endogenous²

\[ \underbrace{A_{t}}_{\text{Stock of knowledge}} = \underbrace{BK^\beta}_{\text{Stock of capital}} \]

\[ Y_{t} = K^\alpha(B\cdot K^\beta L_{t})^{1-\alpha} \]

where, - $A_{t}:$ stock of knowledge - $B$ and $\beta$ are coefficients that explain the direct functional relationship between $A_{t}$ and $K_{t}$

Full economy, $Y_{t} = \underbrace{K^\alpha}_{\text{Direct Contribution}} B^{1-\alpha} \underbrace{K^{\beta(1-\alpha)}}_{\text{Indirect impact}} L_{t}^{(1-\alpha)}$ has,

Direct contribution: infrastructure, machinery
Indirect impact: Capital accumulation indirectly boosts productivity by generating new knowledge.

This is the additional benefit of capital, through its role in generating knowledge.

Firm-Level vs. Economy-Wide Effects¶

While the model holds true for the entire economy, it also applies at the individual firm level. For a single firm, output ($Y_{t}$) depends on its own inputs ($K_{i}$ and $L_{i}$) but also on the overall stock of knowledge in the economy, which is a result of the collective capital stock ($K$).

\[ \boxed{ Y_{t} = A(K) F(\underbrace{K_{i}, L_{i}}_{\text{Inputs}}) } \]

It captures how an individual firm output depends not just on its own inputs ($K_{i},L_{i}$) but also on the economy wide capital stock , $A(K)$ which enhances productivity through LBD, and thus has a spillover effect.

OR

This highlights a crucial distinction: a firm's productivity is not solely determined by its own actions. Instead, it benefits from the broader technological progress driven by the collective capital accumulation of all firms in the economy. This is what's known as the spillover effect, where the knowledge created by one firm's investment becomes a public good that benefits all other firms. This creates a positive externality that fuels sustained economic growth

In exogenous, only capital stock is exogenous because it is created by savings. ↩
i.e. it can be determined within the system, rather that being an external unexplained factor. ↩