1^st Stage: firm's profit maximization problem:

1. Choose: < output >

2. In order to maximize: < profits >

• We'll cover this later...first we'll explore:

2^nd Stage: firm's cost minimization problem:

1. Choose: < inputs >

2. In order to minimize: < cost >

3. Subject to: < producing the optimal output >

• Minimizing costs \iff maximizing profits

\begin{align*} \color{green}{\text{Isoquant curve slope}} &= \color{red}{\text{Isocost line slope}} \\ \end{align*}

\begin{align*} \color{green}{\text{Isoquant curve slope}} &= \color{red}{\text{Isocost line slope}} \\ \color{green}{MRTS_{l,k}} &= \color{red}{\frac{w}{r}} \\ \color{green}{\frac{MP_l}{MP_k}} &= \color{red}{\frac{w}{r}} \\ \color{green}{0.5} &= \color{red}{0.5 } \\\end{align*}

• Marginal benefit = Marginal cost

  • Firm would exchange at same rate as market

• No other combination of (l,k) exists at current prices & output that could produce q^\star at lower cost!

• One reason Cobb-Douglas functions are great: easy to determine returns to scale:
  q=Ak^\alpha l^\beta

• \alpha + \beta = 1: constant returns to scale

• \alpha + \beta >1: increasing returns to scale

• \alpha + \beta <1: decreasing returns to scale

• Note this trick only works for Cobb-Douglas functions!

• A common case of Cobb-Douglas is often written as:
  q=Ak^\alpha l^{1-\alpha} (i.e., the exponents sum to 1, constant returns)

• \alpha is the output elasticity of capital

  • A 1% increase in k leads to an \alpha% increase in q

• 1-\alpha is the output elasticity of labor

  • A 1% increase in l leads to a (1-\alpha)% increase in q