Motivating Idea

• We call a contractor to come perform a renovation in our home:
  • They provide a bid on the job 
  • We give them the job or not
• Does the bid given depend on how nice our house is?! 

Motivating Idea

Well, it depends...

• If we called multiple contractors would typically model this as an auction
  • Bids do not depend on the homeowners reservation
• If there's any chance we only called one contractor then bids will depend on homewner's reservation value

Model

\(\Longrightarrow\)

Each bidder:

• Draws cost \(c_i\in[\underline{c},\overline{c}]\)
• Chooses bid \(\beta(c)\)
• Contract awarded to lowest bidder
  • so long as bid below reservation \(\omega>\overline{c}\)

Notation

• \(p\)  - Probability only one bidder
• \(\omega\) - Reservation price for auction
• \(c\) -  Agent's cost to provide the service
• \(\beta(c)\) - The equilibrium bid function
• \(\left[\underline{c},\overline{c}\right]\) - The support for bids

Model

Symmetric theoretical solution :

• Assume bid function \(\beta(c)\) is strictly increasing
• Bidders expected profit is given by:\[\pi(b|c)=\Pr\left\{b\text { lowest bid}\left|\beta\right.\right\}\cdot (b-c)\]

Model

First-order condition leads to a differential equation:

\[\frac{\beta(c)-c}{\overline{c}-\underline{c}}=\beta^\prime (c)\frac{\Pr\left\{\text{Lowest }c\right\} }{ \Pr\left\{\text{Not alone} \right\} }\]

• To solve this we need a boundary condition on the highest cost type:
  • With \(p=0\) this is \(\beta(\overline{c})=\overline{c}\)
  • With \(p>0\) this is \(\beta(\overline{c})=\omega\)

Stylized Data

• Look at all permits in Allegheny Count
• Focus on electrical panel installations
  • Common residential upgrade
  • Minimize unobserved heterogeneity
  • Lists size of the panel in Amperes
• Have 2,196 permit applications

Stylized Data

• Regress permit project value \(\log(\$V)\) on

Stylized Data

Design Sketch

Costs drawn from \(U[\$500,\$1500]\) with \(2\times 2\) design over:

1. Reservation of
   • Low: \(\omega=\$2,500\)
   • High: \(\omega=\$3,500\)
2. Uncertainty over #bidders
   • Scenario 1: \( \tfrac{3}{4}\cdot 2\text{ bidders} \oplus \tfrac{1}{4}\cdot 1\text{ bidder}\)
   • Scenario 2: \( \tfrac{3}{4}\cdot 2\text{ bidders} \oplus \tfrac{1}{4}\cdot 3\text{ bidders}\)

Design Sketch

Design Sketch

Design Sketch

Theory Hypotheses

1. No response to reservation if \(\Pr(N=1)=0\)
2. Response to reservation if \(\Pr(N=1)>0\)
   1. Reservation pricing only by the highest cost type
   2. Higher conditional markups for higher cost types

Behavioral Hypotheses

1. Response to reservation even when \(\Pr(N=1)=0\)
2. Pooling on reservation by many cost types