Consider a simple economy that produces two goods, beer (denoted by x) and quiche (denoted by y), using labor and capital (denoted by L and K, respectively) that are supplied by two types of households, those consisting of wimps (denoted by W) and those consisting of hunks (denoted by H). Each household of hunks supplies 100 units of labor and no units of capital. Each household of wimps supplies 10 units of capital and no units of labor. There are 100 households of each type. Both beer and quiche are produced with technologies exhibiting constant returns to scale. The market supply curves for beer and quiche are Px = w^1/6.5/6
Py = w^3/4.1/4
where w denotes the price of labor and r denotes the price of capital. The market demand curves for beer and quiche are given by
Px = 201w + 901H/X
Py = 801w + 101H/Y
where X and Y denote the aggregate quantities of beer and quiche demanded in this economy and IW and IH are the household incomes of wimps and hunks, respectively. Finally, the market demand curves for labor and capital are given by
L = X/6(r/w)^1/2 + 3Y/4(r/w)^1/2
K = 5X/6(w/r)^1/2 + Y/4(w/r)^1/2
There are four unknowns in our simple economy: the prices of beer and quiche, Px and Py, and the prices of labor and capital, w and r. Write the four equations that determine the equilibrium values of these unknowns.