Input Demand and Output Supply

Long Run Input Demand §

def. Ordinary Input Demand. Ordinary Demand for Inputs (=labor $l$, capital $k$)

• To derive: Profit Function
• Properties:
  1. HD0 in $(w,r,p)$
  2. Decreasing in own-price ←regardless of anything! (Unlike Utility Maximization)

def. Conditional Input Demand. Demand of input (=labor $l$, capital $k$) in order to produce a certain level out output $\bar{x}$

• HOWTO get: Cost Minimization
• Properties
  1. HD0 in input prices $w,r$
  2. Decreasing in own-price

Short Run Input Demand §

Short run (own-price) labor demand: $l_{\bar{k}}(w,p|\bar{k})$

• $\bar{k}$ is a parameter. Set it as the long-run input demand $k(w,r,p)$.
• In the short run, a change in $w$ or $p$ will only operate within $l_{\bar{k}}$ with no change in $\bar{k}$ possible.
• In the long run, we simply calculate the long-run input demand $l(w,r,p)$.
• Relationship between long-run input demand, visually:

Long Run Output Supply §

def. Ordinary Supply. Relates the prices of inputs and output with the quantity of output produced

$$x:(w,r,p)\mapsto x$$

Properties

1. HD0 in prices $w,r,p$
2. increasing in output price $x$
3. decreasing in input price $w,r,$

(HowTo) Derive Supply Function §

Method 1:

1. Get Input Demand and Output Supply from Profit Maximization
2. Substitute into the Production Function $x=f(l(w,r,p),k(w,r,p))$
3. Simplify to get $x(w,r,p)$. Method 2:
4. Get cost function from Cost Minimization

Short Run Output Supply §

Short run (own-price) output supply: $x_{\bar{k}}(w,p|\bar{k})$

• $\bar{k}$ is a parameter. Set it to the long-run input demand $k(w,r,p)$
• Use the Production Function $x=f(l,k)$ but with $\bar{k}$ fixed.
• In the short run, a change in $w$ or $p$ will only operate within $x_{\bar{k}}$ with no change in $\bar{k}$ possible.
• In the long run, we simply calculate the long-run output demand $l(w,r,p)$.
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