See also Cost Minimization

Short Run (One-input)

\text{max} ~ \pi=px-wl-r \bar{k} ~ \text{such that} ~ x=f(l,\bar{k}) $$ ...where $\bar{k}$ is fixed at the long run optimal point. Maximization of profit (reaching maximum iso-profit line) against the [[Production Function]] - Uses [[Constrained Optimization]] - Optimal is where the production function is tangent to the isoprofit line: $-\frac{w}{r}=TRS$ - reminder: [[Technical Rate of Substitution|TRS]] is the tangency of the isoprofit line. - the slope of the isocost line is $-\frac{w}{r}$ - Result is the [[Input Demand]]s and Output [[Supply Function]] - As in [[Utility Maximization]], you can substitute these results into profit formula $\pi-px-wl$ to get the [[Profit Function]]. Alternative Characterization

max_{x,l}~p\cdot f(l,\bar{k})-wl-r\bar{k}

- Where $c(w,x)$ is the reformulation of the production function into a function of input $l$, multiplied by $w$ (= $\text{input quantity}\times \text{input price}$ given a certain level of output $x$) - Solvable simply by finding where tangent is zero. - You will get short run [[Input Demand]]s ## Long Run (Multiple Input)

max_{x,l,k}~ \pi=px-wl-rk ~ \text{such that} ~ x=f(l,k)

- First Order Condition (FOC) is equivalent to $pMP_{L}=w,pMP_{k}=r$ - i.e. produce when the additional revenue is equal to price of inputs - i.e. $MR_{L}=MC_{L},MR_{k}=MC_{k}$ - Gets you [[Input Demand]] functions - ! Beware returns to scale: If Increasing Returns/Constant Returns to scale, then only one of the inputs are used. (See below for more details) ## Profit Maximization Problem There are two main ways for profit maximization: - One-step: - $\text{max}\ \pi=px-wl-rk \ \text{ s.t. } x=f(l,k)$ to get profit-maximization condition $p\cdot \text{MP}_l=w, \ \ p\cdot MP=r$ - This gets you the input demand functions $l(p,w,r), k(p,w,r)$ and output supply $x(p,w,r)$. - Two-step: 1. $\text{min} \ c=wl+rk \ \text{ s.t. } \ x=f(l,k)$ to get cost-minimization condition $-\frac{w}{r}=\frac{\text{MP}_l}{\text{MP}_k}(=\text{TRS})$ This gets you conditional input demand functions $l(x,w,r), k(x,w,r)$ 1. Solve $p=\text{MC}^{\text{LR}}$ by using above conditional input demand functions (recall $\text{MC}=\frac{\delta C}{\delta x}$) This gets you output supply $x(p,w,r)$ ### Special Case: Labor and Capital Are Perfect Compliments


- When $0<\alpha<1$ it has decreasing returns to scale - → simply solve two [[Unconstrained Maximization]] problems: - $max_{l}\pi =pl^a-wl-rk$, - $max_{k}\pi =pk^a-wl-rk$, - $l^\alpha = k^\alpha$ <- **dont forget!** ### Special Case: Labor and Capital Are Perfect Substitutes $$f(l,k)=(l^\alpha+k^\alpha)^\beta
  • when : decreasing returns to scale
  • when : Isoquant is bowed in the wrong direction
    • → Use only one of the inputs (corner solution)
  • when but : Isoquant may or may not be bowed in the wrong direction.
    • → May or may not use only one input (corner solution)

Example plot to demonstrate this

 Plot3D[(l^alpha + k^alpha)^beta, {l, 0, 5}, {k, 0, 5}, 
  AxesLabel -> {"l", "k", "x"}, PlotRange -> All], {alpha, 0.1, 
  5}, {beta, 0.1, 5}]