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- Isoquants, Isocosts and Cost Minimization

Jack M.

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Isoquants, Isocosts and Cost Minimization Overheads We define the production function as 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 y represents output y = f (x1, x2, x3, . . . xn ) 0 50 100 150 200 250 300 350 0 2 4 6 8 10 12 Input -x Output -y y Holding other inputs fixed, the production function looks like this 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 Graphically marginal product looks like this -40 -30 -20 -10 0 10 20 30 40 50 60 1 2 3 4 5 6 7 8 9 10 11 Input -x Output -y A MPP The Cost Minimization Problem Pick y; observe w1, w2, etc; choose the least cost x?s 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 Another view Yet another view (low x?s) With a horizontal plane at y = 250,000 With a horizontal plane at 100,000 Contour plot Another contour plot There are many ways to produce 2,000 bales of hay per hour Workers Tractor-Wagons Total Cost AC 10 1 80 0.04 6.45 1.66 71.94 .03597 5.48 2 72.8658 0.0364 3.667 3 82.0015 0.041 2.636 4 95.8167 0.0479 1.9786 5 111.872 .0559 Plotting these points in x1 - x2 space we obtain Or Isoquant y = 2,000 0 2 4 6 8 10 12 0 2 4 6 8 10 12 14 X 2 X 1 Isoquant y = 2000 x2 = 4, x1 = 2.636 Cutting Plane for y = 10,000 Isoquant for y = 10,000 Only the negatively sloped portions of the isoquant are efficient Isoquant for y = 10,000 x1 x2 output y -- 1 10,000 -- 2 10,000 -- 3 10,000 12.469 4 10,000 9.725 5 10,000 8.063 6 10,000 6.883 7 10,000 5.990 8 10,000 5.290 9 10,000 y = 10,000 y = 2, 000 y = 10,000 Graphical representation Isoquants y = 2,000, y = 10,000 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 x 2 Isoquant y = 2000 Isoquant y = 10000 x1 y = 2,000, y = 5,000, y = 10,000 More levels And even more Comparison to full map 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 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 Higher isoquants represent greater levels of production 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 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 x 2 Isoquant y = 2000 Isoquant y = 10000 x1 The Marginal Rate of Substitution (MRTS) ? ? ? Algebraic formula for the MRS The marginal rate of (technical) substitution of input 1 for input 2 is We use the symbol - | y = constant - to remind us that the measurement is along a constant production isoquant Example calculations y = 2,000 Change x2 from 1 to 2 Workers Tractor-Wagons x1 x2 10 1 5.48 2 3.667 3 2.636 4 1.9786 5 Example calculations y = 2,000 Change x2 from 2 to 3 Workers Tractor-Wagons x1 x2 10 1 5.48 2 3.667 3 2.636 4 1.9786 5 More example calculations y = 10,000 Change x2 from 5 to 6 x1 x2 12.469 4 9.725 5 8.063 6 6.883 7 5.990 8 A declining marginal rate of substitution The marginal rate of substitution becomes larger in absolute value as we have more of an input. When the firm is using 10 units of x1, it can give up 4.52 units with an increase of only 1 unit of input 2, and keep production the same But when the firm is using only 5.48 units of x1, it can only give up 1.813 units with a one unit increase in input 2 and keep production the same The amount of an input we can to give up and keep production the same is greater, when we already have a lot of it. Slope of isoquants and 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 Marginal physical product and isoquants All points on an isoquant are associated with the same amount of production Hence the loss in production associated with ?x1 must equal the gain in production from ?x2 , as we increase the level of x2 and decrease the level of x1 Rearrange this expression by subtracting MPPx2 ? x2 from both sides, Then divide both sides by MPPx1 Then divide both sides by ? x2 The left hand side of this expression is the marginal rate of substitution of x1 for x2, so we can write So the slope of an isoquant is equal to the negative of the ratio of the marginal physical products of the two inputs at a given point The isoquant becomes flatter as we move to the right, as we use more x2 (and its MPP declines) and we use less x1 ( and its MPP increases) So not only is the slope negative, but the isoquant is convex to the origin 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 x 2 Isoquant y = 2000 x1 The Marginal Rate of Substitution (MRTS) ? ? Approx x1 x2 MRS MPP1 MPP2 12.4687 4.0000 --- 664.6851 2585.7400 11.8528 4.1713 -3.5946 739.5588 2465.2134 9.7255 5.0000 -2.5672 1010.5290 2050.0940 9.3428 5.1972 -1.9411 1063.1321 1975.4051 8.0629 6.0000 -1.5941 1254.9695 1724.5840 6.9792 6.9063 -1.1959 1447.3307 1508.9951 6.8827 7.0000 -1.0291 1466.3867 1489.5380 Approx x1 x2 MRS MPP1 MPP2 12.4687 4.0000 --- 664.6851 2585.7400 11.8528 4.1713 -3.5946 739.5588 2465.2134 9.7255 5.0000 -2.5672 1010.5290 2050.0940 9.3428 5.1972 -1.9411 1063.1321 1975.4051 8.0629 6.0000 -1.5941 1254.9695 1724.5840 6.9792 6.9063 -1.1959 1447.3307 1508.9951 6.8827 7.0000 -1.0291 1466.3867 1489.5380 x2 rises and MRS falls Isocost lines Quantities of inputs - x1, x2, x3, . . . Prices of inputs - w1, w2, w3, . . . An isocost line identifies which combinations of inputs the firm can afford to buy with a given expenditure or cost (C), at given input prices. Graphical representation Cost = 120 w1 = 6 w2 = 20 0 2 4 6 8 10 12 14 16 18 20 22 0 1 2 3 4 5 6 7 x 2 x 1 Slope of the isocost line So the slope is -w2 / w1 Example C = $120, w1 = 6.00, w2 = 20.00 Intercept of the isocost line So the intercept is C / w1 With higher cost, the isocost line moves out Isoquants and isocost lines We can combine isoquants and isocost lines to help us determine the least cost input combination The idea is to be on the lowest isocost line that allows production on a given isoquant Combine an isoquant with several isocost lines Isocost lines for $20, $60, $120, $180, $240, $360 Consider C = 120 and C = 180 At intersection there are opportunities for trade 0 4 8 12 16 20 24 3 4 5 6 7 8 9 10 11 12 13 X 2 X 1 Isoquant y = 10000 Isocost 120 Isocost 180 Isocost 160 Isoquant y = 10000 0 4 8 12 16 20 24 4 5 6 7 8 9 10 11 12 13 X 2 X 1 Isocost 120 Isocost 180 Add C = 160 Isocost 160 Isocost 154.6 Isoquant y = 10000 0 4 8 12 16 20 24 4 5 6 7 8 9 10 11 12 13 X 2 X 1 Isocost 120 Isocost 180 Add C = 154.6 Isocost 160 Isocost 154.6 Isoquant y = 10000 0 4 8 12 16 20 24 4 5 6 7 8 9 10 11 12 13 X 2 X 1 Isocost 120 Isocost 180 In review The least cost combination of inputs The optimal input combination occurs where the isoquant and the isocost line are tangent Tangency implies that the slopes are equal Slope of the isocost line -w2 / w1 Slope of the isoquant Optimality conditions Substituting we obtain The price ratio equals the ratio of marginal products Slope of the isocost line = Slope of the isoquant We can write this in a more interesting form Multiply both sides by MPPx1 and then divide by w2 Graphical representation Statement of optimality conditions a. The optimum point is on the isocost line b. The optimum point is on the isoquant c. The isoquant and the isocost line are tangent at the optimum combination of x1 and x2 d. The slope of the isocost line and the slope of the isoquant are equal at the optimum e. The ratio of prices is equal to the ratio of marginal products f. The marginal product of each input divided by its price is equal to the marginal product of every other input divided by its price Approx x1 x2 MRS MPP1 MPP2 MPP1/w1 MPP2/w2 MRS -w2 / w1 1.0000 -3.3333 2.0000 -3.3333 3.0000 -3.3333 12.4687 4.0000 664.6851 2585.7400 110.7809 129.2870 -3.8902 -3.3333 11.8528 4.1713 -3.5946 739.5588 2465.2134 123.2598 123.2607 -3.3334 -3.3333 9.7255 5.0000 -2.5672 1010.5290 2050.0940 168.4215 102.5047 -2.0287 -3.3333 9.3428 5.1972 -1.9411 1063.1321 1975.4051 177.1887 98.7703 -1.8581 -3.3333 8.0629 6.0000 -1.5941 1254.9695 1724.5840 209.1616 86.2292 -1.3742 -3.3333 6.9792 6.9063 -1.1959 1447.3307 1508.9951 241.2218 75.4498 -1.0426 -3.3333 6.8827 7.0000 -1.0291 1466.3867 1489.5380 244.3978 74.4769 -1.0158 -3.3333 5.9898 8.0000 -0.8929 1663.9176 1305.9560 277.3196 65.2978 -0.7849 -3.3333 5.2904 9.0000 -0.6994 1855.3017 1155.0760 309.2169 57.7538 -0.6226 -3.3333 4.7309 10.0000 -0.5595 2044.1823 1026.1700 340.6971 51.3085 -0.5020 -3.3333 4.2773 11.0000 -0.4535 2232.4134 912.4680 372.0689 45.6234 -0.4087 -3.3333 3.9071 12.0000 -0.3702 2420.9761 809.4240 403.4960 40.4712 -0.3343 -3.3333 3.6042 13.0000 -0.3029 2610.3937 713.8360 435.0656 35.6918 -0.2735 -3.3333 Example Table w1 = 6, w2 = 20 To get an x1, I can give up 3.33 x2 in terms of cost Intuition for the conditions The isocost line tells us the rate at which the firm is able to trade one input for the other, given their relative prices and total expenditure For example in this case the firm must give up 3 1/3 units of input 1 in order to buy a unit of input 2 0 4 8 12 16 20 24 4 5 6 7 8 9 10 11 12 13 X 2 X 1 Isocost 180 w1 = 6 w2 = 20 C = 180 3 6.66 10 Isoquant y = 10000 0 4 8 12 16 20 24 4 5 6 7 8 9 10 11 12 13 x2 x 1 The isoquant tells us the rate at which the firm can trade one input for the other and remain at the same production level If there is any difference between the rate at which the firm can trade one input for another with no change in production and the rate at which it is able to trade given relative prices, the firm can always make itself better off by moving up or down the isocost line Isocost 160 Isoquant y = 10000 0 4 8 12 16 20 24 4 5 6 7 8 9 10 11 12 13 x2 x 1 Isocost 180 The isoquant tells us the rate at which the firm can trade one input for the other and remain at the same production level When the slope of the isoquant is steeper than the isocost line, the firm will move down the line When the slope of the isoquant is less steep than the isocost line, the firm will move up the line 0 4 8 12 16 20 24 4 5 6 7 8 9 10 11 12 13 x2 x 1 Isoquant y = 10000 When the slope of the isoquant is steeper than the isocost line, the firm will move down the line The End

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