Production possibility frontier and functions

See the attached file for the problem.

This question eventually allows us to "discover" some interesting facts about PPF's and trade theory. A closed economy produces X and Y with labor and capital. They currently have 20 units of labor and 10 units of capital.

(a) Suppose production of X and Y are governed by the following production functions: (see attached)
What condition must hold for input efficiency? Use this equation and the ones governing total labor and capital use to derive an equation for the input contract curve- thus find (a), derive an expression for the input contract curve here —again, find K_x as a function of L_x. Draw an input Edgeworth box and label the contract curve. Put labor on the horizontal of your box and put the origin for X production (Ox) in the southwest corner of your box. Now derive an expression for the PPF. Comment on its shape.

(b) Now suppose, instead, that the production of X and Y are governed by the following, different production functions:(see attached file)
What condition must hold for input efficiency here? Using similar techniques as in (a), derive an expression for the input contract curve here — again, find K_x as a function of L_x. This one is nastier. Draw another input Edgeworth box and for values of Lx of 4 and of 10, plot the input efficient points. Sketch the input contract curve and discuss why it lies where it does (you may prove this by comparing the values of K_x for the two contract curves you have found in this problem). Given this input contract curve's position, which good's production is relatively more capital intensive? labor intensive? Explain with reference to the capital/labor ratios in the two sectors.

(c) Now, using the Edgeworth box you drew in (b), plot the following points on the corresponding PPF [thus find X and Y corresponding to]:
i) L_x = 0 (Point A in the box and on the PPF diagram)
ii) L_x = 4 (Point B in the box and on the PPF diagram)
iii) L_x =10 (Point C in the box and on the PPF diagram)
iv) L_x = 20 (Point D in the box and on the PPF diagram)
Round of the numbers for X and Y to the nearest tenth. Confirm that the PPF is concave by finding its between A and B, between B and C, and between C and D. Were the production functions CRS (constant returns to scale) in (b) ? What, then, explains the concavity of the PPF you fmd here?