GM Rebates: The Intuition “What price, P*, should GM post when it gives rebates of $1000 to old buyers and $500 to new buyers of its trucks?” In addition, –Old buyers receive rebate coupons which they can either use or sell to new buyers for price Q (= $200). –In addition to the $200 they must pay to old buyers, new buyers must pay a brokerage fee of k (=$200) for each coupon they buy. Solver “Set P*=$20,480.”If we present this question to Solver, we are told, “Set P*=$20,480.” Lets make sense of this answer in five easy steps.

GM Rebates: The Intuition Step 1) Base Case: X = x = Q = k = 0 Optimal Price to each group = $20,000  Profit = $10bil

GM Rebates: The Intuition 2) Imagine both old and new buyers get $1,000 rebates and new buyers don’t have to pay anything for coupons. X = x = $1000 Q = k = 0 Solution to this is a no - brainer … GM does best when each group faces an effective price of $20,000. –If each group gets the same rebate and there are no costs to transfer coupons from old buyers to new buyers, GM need only raise its posted price by the amount of the rebate. Everyone then continues to pay an effective price of $20,000. Raise price $1,000.

GM Rebates: The Intuition Step 3) In fact, old buyers do get a $1,000 rebate but new buyers only get a $500 rebate. X = 1000, x = 500 Q = k = 0 –GM would like to keep the effective price to old- buyers at $20,000. It would like to raise the posted price to this group by $1,000. –GM would like to keep the effective price to new- buyers at $20,000. It would like to raise the posted price to this group by $500. Solution: Raise the posted price between $500 and $1,000, weighted by the size of each group. The optimum: ΔP = $650 (=.7x x1,000) Raise price $650!

GM Rebates: The Intuition Step 4) Next suppose there are no rebates but new buyers must pay old buyers $200 for a coupon while the brokerage fee is zero. Q = 200 X = x = k = 0 For each group, the effective price of a truck is now $200 above the posted price, P –This is like a $200 tax on truck buyers Solution (explained on next slide): Reduce posted price by half of the “tax” Reduce price $100!

GM Rebates: The Intuition Step 4 Explained: Q = 200 X = x = k = 0 For each group, the effective price of a truck is now $200 above posted price P –Each group’s demand curve shifts down by $200. –Instead of letting demand fall, imagine GM reimburses buyers for the “cost” of the coupon. If  MC = + 200,  MR = at new optimum. Since MR rises twice as fast as P (= average revenue) for straight line demand curve, the effective price to buyers would need to rise only $100 for MR to rise by $200, if indeed GM did bear the $200 cost. For effective price to rise only $100 when Q = $200 and buyers bear this cost, GM must lower its posted price by $100. GM meets its customers half-way on this “tax.” Reduce posted price $100!

GM Rebates: The Intuition Step 5) Now suppose there are still no rebates but new buyers must get a free coupon from old buyers and pay a $200 brokerage fee. k = 200 X = x = Q = 0 –GM would like to see the effective price to new-buyers rise by only half of this “tax” or $100. GM would like to lower the posted price to new buyers by $100. –GM would like to keep the effective price to old-buyers where it is, at $20,000. –Solution: Reduce the posted price between $100 and zero dollars, weighted by the sizes of the two groups. The optimum: ΔP = - $70 (=.7x(-100) +.3x0) Reduce posted price $70.

GM Rebates: The Intuition General Case: X = 1000, x = 500, Q = 200, k = 200 From what we’ve learned –Want to raise P by $650 because of the rebates The closer the rebates are to each other, the closer is this offset to the amount of the rebates –Want to lower P by $100 (half of the coupon’s market value, Q) because of this equal “tax” on both old- and new-buyers. –Want to lower P by an additional $70 because of the brokerage fee on new-buyers Ideally would lower P by $100 (half of k) to new buyers, except for effect on old buyers. Summarizing Optimal P = 20, – 100 – 70 = 20,480