Consider a representative firm that aims at maximizing its profit by choosing the real wage (𝑤), capital input (

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Consider a representative firm that aims at maximizing its
profit by choosing the real wage
(𝑤), capital input (𝐾) and labour input (𝐿) optimally. The
profit function of the firm is
given by:
𝜋 = 𝐹(𝐾,𝑒(𝑤)𝐿) − 𝑟𝐾 − 𝑤𝐿
The effort function of the representative firm takes the
following form:
𝑒(𝑤) = (𝑤
̂
−𝑤
𝑎
𝑤
𝑎
)𝜃
, 0 < 𝜃 < 1
where 𝑤
̂ and 𝑤𝑎 denote the representative firm’s optimal real wage and
employees’
alternative wage, respectively.
(a)
Derive the first-order condition for profit maximization problem
with respect to
𝑤
̂, 𝐾 and 𝐿.
(b)
Solve the model for the Solow condition. Explain the Solow
condition and discuss
how the optimal wage depends on the model parameters.