Learning by Doing

  • Higher the knowledge \(\implies\) Higher capital stock
  • 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

As capital is exogenous , which grows through savings, since the knowledge is a function of capital, knowledge also becomes endogenous.

\[ \underbrace{A_{t}}_{\text{Stock of knowledge}} = \underbrace{BK^\beta}_{\text{Stock of capital}} \]
\[ Y_{t} = K^\alpha(B\cdot K^\beta L_{t})^{1-\alpha} \]

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: helps in generating stock of knowledge

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

For a single firm,

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

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.

  1. In exogenous, only capital stock is exogenous because it is created by savings. ↩