Please solve and include all steps.

Consider a two country (Home and Foreign) and two good (manufacturing and agriculture) economic framework. Both countries are endowed with fixed amount of labor, capital where L=1,K=4,L=4,K=1. Home and Foreign share the same production technology in the manufacturing and agriculture sectors: ym=(0.25Lm)0.25(0.75Km)0.75,ya=(0.75La)0.75(0.25Ka)0.25,ym=(0.25Lm)0.25(0.75Km)0.75,ya=(0.75La)0.75(0.25Ka)0.25. This implies the following marginal product: MPLm=(31LmKm)0.75,MPKm=(3KmLm)0.25,MPLa=(3LaKa)0.25,MPKa=(31KaLa)0.75. Workers and capital owners in both Home and Foreign earn factor return w/w,rK/rK, and spend the return on consuming the manufacturing and agricultural goods. The preference is given by U=cm0.5ca0.5,U=cm0.5ca0.5. In what follows, the subscript ' a ' refers to agriculture, and the superscript 'A' denotes autarky equilibrium. 1. International trade equilibrium without specialization: we are going to focus on the equilibrium where both countries are producing both goods under trade. a. Given the factor and world good prices w,w,rK,rK,pm, set up the manufacturing firm's profit maximization problem in both Home and Foreign. To derive the optimal factor demand in both and Foreign, what are the optimality conditions you plan to use? What is the economic meaning of those conditions? b. If you indeed solve the profit maximization problem in Home and Foreign under trade, you will get the following optimal factor demand ratios and world good prices in the manufacturing and agriculture sectors KmLm=31wrK,KmLm=31wrK,pm=w0.25rK0.75=w0.25rK0.75,KaLa=3wrK,KaLa=3wrK,pa=w0.75rK0.25=w0.75rK0.25. Show that in the international trade equilibrium without specialization, factor prices are equalized, w=w and rK=rK. c. If you indeed solve the utility maximization problem in Home and Foreign under trade for both workers and capital owners, you will get the following optimal aggregate consumption demand ("aggregate" means we are summing the demand from workers and capital owners), and notice we've applied the factor price equalization w=w and rK=rK, CmCa=0.5pmwL+rKK,Cm=0.5pmwL+rKK=0.5pmwL+rKK,=0.5pawL+rKK,Ca=0.5pawL+rKK=0.5pawL+rKK. Show that to satisfy the total consumption demand of manufacturing and agriculture goods in Home and Foreign, total labor demand in Home and Foreign are Ldemand+Ldemand=0.5wwL+rKK+wL+rKK. d. Impose the condition such that total demand of labor should equate total endowment of labor in both countries to show that relative factor returns wrK and wrK satisfy: wrK=wrK=K+KL+L. Now plug in the values of L,L,K,K, and compute wrK and wrK. e. Now if besides international trade, we also allow international flows of labor, e.g. more foreign labor enter the Home labor market, explain in words how would that change the relative factor returns in Home and Foreign and why? f. Compute the relative world good price papm. g. If your calculation is correct, you shall observe that relative world good price lies between the relative autarky prices in Home and Foreign paApmA