Consider a financial market with two assets: peaches and lemons. The fraction of lemons in the economy is $=0\mathrm{.}4.$ Buyers value peaches at $v_p^b=$$$20$ and lemons at $v_l^b=$$$10.$ Sellers value peaches at $v_p^s=$$$16$ and lemons at $v_l^s=$$$8.$ Sellers have all the bargaining power when setting prices at which assets trade.

a$)$ Assume both buyer and sellers can perfectly observe the quality of assets in the market. At what price will lemons and peaches trade?

b$)$ Now assume that the quality of assets is unobservable, but that buyers and sellers have symmetric information. That is$,$ neither sellers or buyers can tell whether a particular asset is a lemon or a peach. What is the $($pooling$)$ price $P^{**}$ at which assets trade?

c$)$ Now assume that only sellers can observe the quality of assets, so that there is asymmetric information between buyers and sellers. Explain $($intuitively$,$ without equations$)$ why the quality of assets traded in equilibrium will now depend on the price.

d$)$ Calculate the share of lemons that trade in the market as a function of the price, and buyers' expected valuation for the asset as a function of the price. Show that the pooling price $P^{**}$ clears the market.

e$)$ Assume the fraction of lemons increases to $=0\mathrm{.}5.$ Show that the pooling price $P^{**}$ no longer clears the market. Calculate the new equilibrium price. Comment.