Final answer:
A production possibility frontier represents the maximum possible combinations of goods and services that can be produced given the available resources and technology. The PPF expression in this specific-factors model can be derived by equating the total available resources to the outputs of each sector. The concavity of the PPF can be determined by analyzing the derivatives of the production function.
Step-by-step explanation:
A production possibility frontier (PPF) represents the maximum possible combinations of goods and services that can be produced given the available resources and technology. In this specific-factors model, there are two sectors - one sector collects wild berries and the other sector produces a manufactured good. The production functions for each sector are given by yᵦ = Lᵦ and yₘ = K¹/²L¹/²ₘ respectively. The expression of the PPF can be derived by equating the total available resources to the outputs of each sector, resulting in the expression yᵦ + yₘ = Lᵦ + K¹/²L¹/²ₘ.
The concavity of the PPF can be determined by analyzing the first and second derivatives of the production function yᵦ with respect to yₘ. If the second derivative is negative, then the PPF is concave to the origin.
When trade is allowed and the price ratio pₘ/pᵦ increases relative to autarky, it means that the manufactured good becomes more expensive relative to berries. In this scenario, the variables Lᵦ , Lₘ , w/pᵦ , w/pₘ , r/pᵦ, and r/pₘ can be analyzed. The labor input in the wild berries sector (Lᵦ) and the manufactured goods sector (Lₘ) would depend on the changes in relative prices and the opportunity cost of labor. The variables w/pᵦ , w/pₘ , r/pᵦ, and r/pₘ would also be affected by the changes in relative prices, reflecting the changes in the real wage and rental rate of capital.