Isoquants Isocosts and Cost Minimization Overheads We define the production function as f x max y x y is an element of the production set y f x1 x2 x3 max y y P x y represents output f represents the relationship between y and x xj is the quantity used of the jth input x1 x2 x3 xn is the input bundle n is the number of inputs used by the firm Holding other inputs fixed the production function looks like this Output y y f x1 x2 x3 xn 350 300 250 y 200 150 100 50 0 0 2 4 6 8 10 Input x 12 Marginal physical product Marginal physical product is defined as the increment in production that occurs when an additional unit of a particular input is employed Mathematically we define MPP as f x1 x2 x3 xn MPPxj MPPj xj Output y Graphically marginal product looks like this 60 50 A MPP 40 30 20 10 0 10 1 2 3 4 5 6 7 8 9 10 20 30 40 Input x 11 The Cost Minimization Problem C y w1 w2 min x1 x2 xn n wi xi such that y i 1 Pick y observe w1 w2 etc choose the least cost x s f x1 x2 xn Isoquants An isoquant curve in two dimensions represents all combinations of two inputs that produce the same quantity of output The word iso means same while quant stand for quantity Isoquants are contour lines of the production function If we plot in x1 x2 space all combinations of x1 and x2 that lead to the same level height for the production function we get contour lines similar to those you see on a contour map Isoquants are analogous to indifference curves Indifference curves represent combinations of goods that yield the same utility Isoquants represent combinations of inputs that yield the same level of production Production function for the hay example 75 0 10 20 40 30 x1 60 50 45 x2 30 60 15 0 70 300000 200000 100000 0 y Another view 75 60 x2 45 30 15 200000 0 0 y 0 10 20 30 x1 40 50 60 70 Yet another view low x s 25 20 15 x2 10 5 0 75000 y 30000 15000 0 0 5 15 10 x1 20 25 With a horizontal plane at y 250 000 75 60 x2 45 30 15 0 300000 200000 100000 0 y 0 10 20 40 30 x1 50 60 70 With a horizontal plane at 100 000 75 60 x2 45 30 15 0 300000 200000 100000 0 y 0 10 20 40 30 x1 50 60 70 Contour plot 80 x2 60 40 20 0 0 10 20 30 40 x1 50 60 70 Another contour plot 80 x2 60 40 20 0 0 10 20 30 40 x1 50 60 70 There are many ways to produce 2 000 bales of hay per hour Workers Tractor Wagons 10 1 6 45 1 66 5 48 2 3 667 3 2 636 4 1 9786 5 Total Cost 80 71 94 72 8658 82 0015 95 8167 111 872 AC 0 04 03597 0 0364 0 041 0 0479 0559 Plotting these points in x1 x2 space we obtain 10 8 x2 6 4 2 0 0 2 4 6 x1 8 10 Or Isoquant y 2 000 12 X1 10 8 Isoquant y 2000 6 4 x2 4 x1 2 636 2 0 0 2 4 6 8 10 12 X2 14 Cutting Plane for y 10 000 75 60 x2 45 30 15 0 300000 200000 100000 0 y 0 10 20 40 30 x1 50 60 70 Isoquant for y 10 000 40 x2 30 20 10 0 0 10 20 x1 30 40 Only the negatively sloped portions 40 of the isoquant are efficient x2 30 20 10 0 0 10 20 x1 30 40 Isoquant for y 10 000 x1 12 469 9 725 8 063 6 883 5 990 5 290 x2 1 2 3 4 5 6 7 8 9 output y 10 000 10 000 10 000 10 000 10 000 10 000 10 000 10 000 10 000 y 10 000 10 8 x2 6 4 2 0 0 2 4 6 x1 8 10 y 2 000 y 10 000 10 8 x2 6 4 2 0 0 2 4 6 x1 8 10 Graphical representation Isoquants y 2 000 y 10 000 14 x1 12 10 Isoquant y 2000 8 Isoquant y 10000 6 4 2 0 0 2 4 6 8 10 12 14 x2 y 2 000 y 5 000 y 10 000 10 8 x2 6 4 2 0 0 2 4 6 x1 8 10 More levels 30 25 x2 20 15 10 5 0 0 5 10 15 x1 20 25 30 And even more 60 50 x2 40 30 20 10 0 0 10 20 30 x1 40 50 60 Comparison to full map 80 x2 60 40 20 0 0 10 20 30 40 x1 50 60 70 Slope of isoquants An increase in one input factor requires a decrease in the other input to keep total production unchanged Therefore isoquants slope down have a negative slope Properties of Isoquants Higher isoquants represent greater levels of production Isoquants are convex to the origin This means that as we use more and more of an input its marginal value in terms of the substituting for the other input becomes less and less Slope of isoquants The slope of an isoquant is called the marginal rate of technical substitution MR T S between input 1 and input 2 The MRS tells us the decrease in the quantity of input 1 x1 that is needed to accompany a one unit increase in the quantity of input 2 x2 in order to keep the production the same The Marginal Rate of Substitution MRTS 14 x1 12 10 Isoquant y 2000 8 6 4 2 Isoquant y 10000 0 0 2 4 6 8 10 12 14 x2 Algebraic formula for the MRS The marginal rate of technical substitution of input 1 for input 2 is x1 MRSx1 x2 y constant x2 We use the symbol y constant to remind us that the measurement is along a constant production isoquant Example calculations y 2 000 Workers Tractor Wagons Change x2 from 1 to 2 x1 x2 10 1 x1 5 48 2 MRSx1 x2 y constant 3 667 3 x2 2 636 4 10 5 48 1 9786 5 1 4 52 1 2 4 52 Example calculations y 2 000 Change x2 from 2 to 3 Workers Tractor Wagons x1 x2 x1 10 1 MRSx1 x2 y constant 5 48 2 x2 3 667 3 5 48 3 667 2 636 4 1 9786 5 2 3 1 813 1 1 813 More example calculations y 10 000 x1 12 469 9 725 8 063 6 883 5 990 x2 4 5 6 7 8 Change x2 from 5 to 6 MRSx1 x2 x1 y constant x2 9 725 8 063 5 …
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