題組內容

三、A firm produces a product to sell in a competitive market using capital and labor. Given the production function \(Y=f(K,L)=K^{\alpha}L^{\beta}\) where \(Y\) is output, \(K\) is capital input and \(L\) is labor input. Suppose that \(p, r\) and \(w\) are prices of output (\(Y\)), capital (\(K\)) and labor (\(L\)), respectively. Assume that \(p, r, w\) are exogenous and \(\alpha>0\) and \(\beta>0\) and \(\alpha+\beta<1\).

(b) What are the first order conditions for this maximization problem? (10 分)