Equilibrium and the optimum are found at 278 units of food and 253 units of clothing, with a distribution of the factors of production of av/ak = 299/101 and kv/kk = 210/390.
The allocation can be shown using two figures. Figure 36 confronts the social welfare function with the Production Possibility Curve (PPC).
Figure 36: Social Welfare and the Production Possibility Curve
The PPC gives those combinations of food and clothing that can be produced with the scarce resources. The choice of the highest possible value of the SWF generates a tangent of a contour of the SWF with the PPC. The tangent gives the optimal price ratio (thus trading ratio) of food and clothing.
Figure 37 confronts the production functions of the separate industries in an Edgeworth-Bowley diagram. The food industry has its origin in the lower left-hand corner, and the clothing industry has its origin in the top right-hand corner. The amounts of capital and labour that are not allocated to the food industry are allocated to the clothing industry. The drawn contour for the food industry gives those combinations of capital and labour that produce the same amount of food. That contour is touched in a tangent by a contour of the clothing industry. The collection of all tangency points is called the contract curve. The tangent drawn here passes through the optimum selected by the SWF. This tangent thus also determines the price ratio of wages and capital rent.