Description of the economy: Consider a 2x2 production economy with two firms, X and Y, and two factors of production, labor L and capital K. The total endowment of factors in the economy is 100 units of labor and 100 units of capital, so (L, K) = (100, 100). Firm X can produce output good X using labor and capital. The firm's technology is described by the following continuous and differentiable production function: fx(lx, kx) = alx + (1 - alkx, = where lx > 0 is the quantity of labor and kx > 0 is the quantity of capital employed by firm X, and a (0,1) is a production parameter. Firm Y can produce output good Y using labor and capital. The firm's technology is described by the following continuous and differentiable production function: 1-B fy(ly, ky) = 1kj-B, = wherely > 0 is the quantity of labor and ky > 0 is the quantity of capital employed by firm Y, and BE (0,1) is a production parameter. This is a small open economy, so output prices for goods X and Y, (Px, py) >> 0, are given. For all of the following problems, the price of the output good Y is 1 (i.e., Py = 1). = = a (1) Suppose a = B. (i) Describe the set of production efficient factor allocations and illustrate them in a Bowley box di- agram. (ii) Describe the set of production efficient output allocation and illustrate the production possibility frontier in a diagram. (iii) Find the competitive equilibrium wage rate w, rental rate r, and allocations of factors and output goods for each firm, for any possible price px > 0 of the output good X. Description of the economy: Consider a 2x2 production economy with two firms, X and Y, and two factors of production, labor L and capital K. The total endowment of factors in the economy is 100 units of labor and 100 units of capital, so (L, K) = (100, 100). Firm X can produce output good X using labor and capital. The firm's technology is described by the following continuous and differentiable production function: fx(lx, kx) = alx + (1 - alkx, = where lx > 0 is the quantity of labor and kx > 0 is the quantity of capital employed by firm X, and a (0,1) is a production parameter. Firm Y can produce output good Y using labor and capital. The firm's technology is described by the following continuous and differentiable production function: 1-B fy(ly, ky) = 1kj-B, = wherely > 0 is the quantity of labor and ky > 0 is the quantity of capital employed by firm Y, and BE (0,1) is a production parameter. This is a small open economy, so output prices for goods X and Y, (Px, py) >> 0, are given. For all of the following problems, the price of the output good Y is 1 (i.e., Py = 1). = = a (1) Suppose a = B. (i) Describe the set of production efficient factor allocations and illustrate them in a Bowley box di- agram. (ii) Describe the set of production efficient output allocation and illustrate the production possibility frontier in a diagram. (iii) Find the competitive equilibrium wage rate w, rental rate r, and allocations of factors and output goods for each firm, for any possible price px > 0 of the output good X
