This is a long difficult economics question about transportation costs. This first part will go over the math for
calculating the equilibrium, and future posts will discuss the implications of
the other parts.

Assume that, when sellers have no transaction cost, buyers will pay the
market price plus $18 in travel cost. If transaction cost were zero, demand
would be D1 which intersect the SS curve at $19 and 31 units. With transaction
cost of $18 per unit, the net demand facing sellers is D2, which is $18 below
D1. The market clears at $19 units and sellers receive $13 for each unit. Now
also assume that, an outsider tells buyers that for $8 per unit she will go to
the market for them and eliminate their former $18 transaction costs. Sellers
now perceive demand curve D3, $8 below D1 and $10 above D2. Sellers get $16.33
for each of the 25.67 units transacted. Both sides of the market benefit from
lower transaction cost.

Now the problem:

Demand with zero transaction costs is Q1d = 50 - P and Supply is Qs = -7 + 2P.

Demand with zero transaction costs is Q1d = 50 - P and Supply is Qs = -7 + 2P.

a) Verify all of the prices and quantities calculated in the above
assumption/discussion.

b) Now assume that intermediaries come from a competitive market with an
equilibrium price of $8 per unit for their services, that is any buyer or
seller who wants an intermediary's services must pay $8 for them What is the
maximum per unit that sellers are willing to pay intermediaries if hiring them
saves buyers $8 in transaction costs?

c) Does your answer to Q16a change if buyers pay $8 per unit to the
intermediary but sellers offer to rebate part of that expense to buyers?

First equilibrium occurs at p*=$19, and q*=31 units. Second equilibrium occurs at

p*=$13+$18=$31,
and q*=19 units. Third equilibrium
occurs at P*=$16.22+$8=$24.22, and q*=22.67.

These prices and quantities make sense, because:

Qd = Qs = 50 – P = -7 + 2P

Or

50 – P = -7 + 2P, add P and 7 to both sides to get:

57 = 3P +> P = 19, so this is right, we can confirm this
by plugging in 19 into both equations to double check the equilibrium quantity
and we find that it holds for a Q of 31.

Now we need to “shift” the demand curve by the amount of the
transaction cost (or travel cost) in order to change the demand curve as
perceived by the suppliers. We can do
this easily by changing the intercept value of the demand function, which
essentially “shifts” the demand curve.

If we add $18 to the transaction cost, this would shift the
demand curve down, or subtract this $18 from the intercept of $50 (because
consumer would be willing to pay $18 less for the good in order to account for
this new cost).

So the new equation for the demand curve is:

Qd = 50 – 18 – P = 32 – P

When we set this new quantity demanded function equal to the
quantity supplied function we get:

32 – P = - 7 + 2p, and if we add P and 7 to both sides we
get:

3P = 39 => P = 13

After plugging this into our quantity demanded and supplied
functions we do confirm that equilibrium quantity is 19.

Finally, we can shift our original demand curve by 8, to
account for this “new” transaction cost, but the procedures are identical to
the one above.