International Trade: Two country trade problem A country (home) is pouplated with workers who produce either Food or Clothing. There are 200 workers producing food and 100 producing clothing. Each food worker produces 6 units of food and each clothing worker produces 3 units of clothing. Workers own the output they produce and trade with other workers. All workers share the same preferences over food and clothing represented by the utility function $U(D_c,D_f)=D_c(D_f)^2$.
  
  (a) What is the aggregate endowments of food and clothing in this economy?


Would the answer for this just be Ef= 200*6=1200 and Ec=100*3=300?


  (b) Now assume a foreign country with 600 food workers, 300 clothing workers with productivity levels 1 food worker produces 1 unit of food and 1 clothing worker produces 2 units of clothing. Describe the pattern of trade and verify that export supply matches import demand.


Firstly, I'm really confused over what the trade relative price equilibrium should be. I know that it will lie somewhere between these two country's autarky prices. Should I just assume that the trade relative price for the home country will be foreign country's autarky price?

My calculations are as follows:

1. Setting MRS = relative trade price (denoted by subscript w, I'm assuming it is the foreign country's autarky price)

$ \frac{D_f}{2D_c}=\frac{P_f}{P_c}^w = \frac{1}{2} \\ D_f=D_c $

2. Setting the budget constraint

$ \frac{P_c}{P_f}^w D_c + D_f = \frac{P_c}{P_f}^w E_c + E_f \\
\frac{1}{2}D_c + D_f = \frac{1}{2} \cdot 300 + 1200 = 1350 $

3. Plug (1) into (2)

$\frac{3}{2}D_f = 1350 \\  D_f = D_c = 900$

This somehow does not seem to make sense to me, and I feel that my answer is intuitively wrong. I would think home would export more of the good in which they have abundance (food) in order to import more clothing.
Part (a) is correct. Your error in (b) is to implicitly assume that relative prices are equal to $1/2$.

The best way to understand the problem is using the Edgeworth box (excuse the sloppy picture; will add a custom one when had the time):



The axes represent the two goods. Their lengths are 1800 (global endowment of food, x axis) and 900 (global endowment of clothing, y axis). Point 3 represents the endowment point, which is (1200,300) for Home and (600,600) for Foreign. As the figure shows, that point is not an optimal allocation. Such optimal allocation is where the MRS for the two countries is the same (point 1), MRS which is equal to the relative price of goods (same for both countries, i.e. they are international prices).

Therefore, this allocation has the following properties:

$$ \frac{H_f^2}{2H_c} = \frac{F_f^2}{2F_c} = \frac{P_c}{P_f}$$

(I replaced $D$ with home and foreign letters, for clarity).

Notice that the ratios of consumption are the same across countries. Yet, before trade, production is such that the ratio of food versus clothing is larger in home than in foreign. Therefore, home will export food and import clothing, and vice versa for foreign. Yet, since imports of one country are the exports of the other country, you get:

$$ \frac{1200 - A}{300 + B} = \frac{600 + A}{600 - B} $$

where $A$ are exports of food and $B$ exports of clothing.

Additionally, you have that the trade balance must be in equilibrium. Therefore:

$$ P_fA = P_cB $$

Here you have all the equations you need to find the solution in terms of final consumption levels and relative prices. Let me know if this is enough. I can guide you further if you need.