1.2 Refer to Figure 1.6 and the discussion of Pareto optimality for Bob above. When Bob is forced to consume (85/56) units of Y, how much X is he able to consume? What level of utility does he achieve? dUB = of . dx + dy of . dy (1.6) 1.3 Using Equation 1.28, calculate XMRTS at the cost minimization point shown in Figure 1.8 XMRTSK,L = (2 ( K. (1.28) ar = = . 10 . x i . yi = 5 . (2)= >0 forall(x, y) (1.8) of dy = 4 . 10. x1 .y * = 5(=) >0 for all(x, y) 1.4 Using Equation 1.29, calculate the slope of the cost function at the cost minimization point shown in Figure 1.8 =r . Kz + w . Lz 1 . 10 . x 1 . y = 5 . (2) >0 forall(x, y) = rent, The price of capital (1.29) (1.8) = wages, The price of labor " = 4 . 10 . x1 .y ! = 5(=) >0 for all(z,y) 1.5 Using Equation 1.14 and 1.15 calculate Bob's MRSX, Y if his MUy = 5, and his MUx remains 25/4 MU, = = 4 . 10 . x7 . y i MRSzy = -MU, (25 25 MUy (1.15) = 5. (1.14) = 5 0.8 1.26 1.6 Assume nothing else changes for Sue or Bob. Why is this not a Pareto optimum? 1.7 Now bring into play the concepts of overall efficiency, and assume PX = $100, PY = $50, Bob's MRSX, Y is -1.25, and Sue's MRSX, Y is -1.5625. Directionally, what adjustments will happen in this economy to move toward overall efficiency?A 14 Production of goods xand1.5 General equilibrium and overall efficiency The conditions for production efficiency are not very restrictive. We simply must be at a point of allocation of the factors of production K, and L between the production of X and Y so that the marginal rate of technical substitution is equalized. It does not say that the levels of X and Y produced must be equalized or balanced in any way. In an economy made up of two commodities: assault rifles and toys, production efficiency can be achieved with a lot of weapons and few toys or with a lot of toys and few weapons. As long as the producers of X and Y face the same competitively determined prices for factors of production K and L, we know production efficiency will automatically be reached. Similarly, exchange efficiency can be achieved between consumers Bob and Sue as long as their marginal rates of substitution of X for Y are equalized, it doesn't matter how much of X and Y Bob and Sue each have. Overall efficiency is a condition that ties production efficiency together with exchange efficiency by considering the technical ability of converting commodity X into commodity Y via the production functions, and how that jibes with consumer preferences (utility functions) that govern the exchange of those commodities. We started the exploration of production with a given output objective (isoquant) for X and similarly for Y had a starting notion of how much Y to produce. The exercise then was to determine how much K and L to employ in each production function with the producers acting independently and minimizing their respective costs of labor and capital. The resulting efficient equilibrium was presented in Figures 1 8 and 1.9, which is summarized as follows: . Ky = 100, Ky = 26.277, K = 126.277 . Ly = 5.604, Ly = 8.83, L = 14.434 . X= 44/35, Y = 170/56 There are a number of other isoquants for X and Y that satisfy the production efficiency condition: XMRTS, = YMRTS, . In fact, there are an infinite number of different X and Y equilibria production levels that meet this criteria. Overall efficiency determines not only how to make X and Y (which point on the isoquant), but also how much X and Y to make (which isoquant). if we consider the total amount of K and L available in an economy as finite numbers, then something, some underlying force, must be guiding how much K to devote to X and Y. If we take some K and L from X and use them to make more Y we are moving to a different point on the economy's production possibility frontier. Figure 1.10 shows the production possibility frontier for our economy with the current output of X and Y shown as one point on the frontier. As we move along the frontier toward the X-axis we are moving to other possible output combinations of X and Y by taking capital and labor from Y and applying it to X. Doing so yields a rate of return that declines as we become more and more biased toward X. The tradeoff along the frontier is not a straight line, but inherits the diminishing marginal product of labor or capital in the production of X. The rate at which the economy can trade off X for Y or the slope of the production possibilities frontier is known as the Marginal Rate of Transformation of X and Y. MRTy Production Possibility Frontier 1.26. 3.09 Figure 1.10: The production possibility frontier for our example with the equilibrium combination of X and Y shown on the frontier. Source: Author's Image The rate at which we can convert X into Y is the slope of this production possibility frontier. Ay MRTy= A (1.38) As we take a resource, such as labor, from production of X, it will reduce the output of X by AL . XMPL = AX (1.39) The added labor to production of Y will increase Y output by AL . YMPL = Ay (1.40) Combining these equations we have the following relationship between the marginal product of labor in the production of X or Y and the marginal rate of transformation of X and Y. MRTz,y = Ay YMPL Ar XMPL (1.41) A similar identity is easily derived for the other factor of production. MRTz,y = Ay YMPK Ar XMPK (1.42) Now, let's put a pin in that and consider the profit maximizing behavior of the producers. If all producers are price-takers and face the same market prices for each commodity as all other producers of the same product, and if each producer faces the same prices for factors of production regardless of commodity they produce. Profit maximizing behavior governs that producers will make product X and Y up to the point where marginal revenue equates marginal cost, or P, . XMP, = w Py . YMPL = w (1.43) The marginal revenue from the sale of X created by hiring one more unit of labor is the marginal product of labor in the production of X times the price of X, and that must equal the wage or marginal cost of labor. A similar profit maximizing rule applies to the production of Y and hiring of labor to that end. Wages (w) are the same for both producers of X and Y, allowing the following rearrangement of the identities as YMP L P2 . XMPL = w = Py . YMPL, XMPL Pz Py (1.44) A similar derivation could be done concerning the other factor of production, K. Finally, combining Equations 1.44, 1.41, and 1.12, we have the condition for overall efficiency MRTzy = YMPL Pz MU. XMPL - P, MU, = MRS,,y (1.45) This is a truly remarkable condition representing the coordination of an infinite number of consumption and production choices in such a way that Pareto optimality is achieved in consumption, optimal efficiency is achieved in the use of production resources, and overall efficiency is achieved in determining the mix of products to produce. All of these things happen automatically if the conditions of perfect competition underlie factors and commodity markets. The conditions of Equation 1.45 are what we would expect for overall efficiency when we think through the meaning of MRSx y and MRTy Suppose we are in a state of the economy where MRTy = 5 and MRSxy = 4. This means that along the production possibility frontier, we could gain five units of Y by giving up one unit of X via redeploying Ky, Ly toward Ky, Ly. The consumers are willing to give up one unit of X as long as they can get four units of Y as they move along their indifference curves and keep utility constant (MRS). Therefore, it is possible to increase consumer utility levels by changing production of X and Y since there will be an extra unit of Y available now, which can be distributed to the consumers and move them each to a higher indifference curve