PRIN MICROECONOMICS ECON 101
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Ms. Ari Lesch
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This 2 page Class Notes was uploaded by Ms. Ari Lesch on Saturday September 26, 2015. The Class Notes belongs to ECON 101 at Iowa State University taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/214443/econ-101-iowa-state-university in Economcs at Iowa State University.
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Date Created: 09/26/15
Cost Minimization and Isoquants Handout The production function is a function that gives the maximum output attainable from a given combination of inputs The production function is defined as fx max y x y is an element of the production set y max 1 y 8 PM where y represents output x is the quantity used of the jth input x1 x2 x3 x is the input bundle n is the number of inputs used by the firm and f represents the functional relationship between y and x1 x2 x3 x Marginal physical product is defined as the increment in production that occurs when an additional unit of one 1 7 0 particular input is employed It is defined as M I g yl yo where y1 and x1 are the level of output and 1 x1 7x1 input after the change in the input level and y0 and x0 are the levels before the change in input use An isoquant curve in 2 dimensions represents all combinations of two inputs that produce the same quantity of output The word iso means the same and quant stands for quantity of output lsoquants are contour lines of the production function lsoquants 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 Properties of isoquants lsoquants slope down have a negative slope Higher further from the origin isoquants represent greater levels of output than lower isoquants lsoquants are convex to the origin The slope of an isoquant is called the marginal rate of technical substitution between input 1 and input 2 and 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 two x2 in order to keep production the same We denote this rate of substitution as MRSXIY or MRTSXIYX2 Algebraically we define this as MRS 1 x2 xm y constant sz Slope of isoquants MRS and marginal physical products All points on an isoquant are associated with the same amount of production Hence the loss in production associated with Ax1 must equal the gain in production from sz as we increase the level of X2 and decrease the level of XI Using algebra we can express this as MPP1C Ax1 MPPC sz 0 We can rearrange this expression by l 2 subtracting MPP1C2 sz from both sides and then dividing both sides first by MPP1C1 and then by x2 This will give WP Axl WP Ax 0 X gt9 2 2 WP Axl illIPP Ax X gt9 2 7MP sz Axl 2 MPPX Ax rMPP X1 MRS sz MPP W The cost minimization problem For each possible level of output the firm will choose the one of several combinations of inputs that has the lowest cost We can write this problem as n Cy w1w2r min 2 wlxl such Ihaly fx1x2wxn I 1 161162 36 For each y the firm will choose different levels of each of the inputs Isocost lines An isocost line identifies the combinations of inputs the firm can afford to buy with a given expenditure or cost C at given input prices With 11 inputs the isocost line is given by wlx1 wzx2 w3x3 wnxn C With two wz W1 inputs the isocost line is given by wlx1 wzxZ C The slope of the isocost line is given by We can obtain this by solving the isocost equation for x1 Statement of optimality conditions The optimum point is on the isocost line The optimum point is on isoquant The isoquant and the isocost line are tangent at the optimum combination of x1 and x The slope of the isocost line and the slope of the isoquant are equal at the optimum which implies that wz 7MPPX2 MRS w 1192 Mpp 1 61 w MPPX The ratio ofmarginal products is equal to the ratio of prices ie 72 2 w1 MPPXl The marginal product of each input divided by its price is equal to the marginal product of every other input MP MP divided by its price ie 2 1 2 WI Example table 0 10000 xl xZ Approx MRS MP1l MPPZ MPPlWl MPPZWz MRS wZW1 7 10000 7 00000 170000 00000 08500 7 33333 7 20000 7 2000000 480000 333333 24000 02400 33333 7 30000 7 4000000 730000 666667 36500 01825 33333 124687 40000 7 6646851 25857400 1107809 1292870 38902 33333 118528 41713 35946 7395588 24652134 1232598 1232607 33334 33333 97255 50000 25672 10105290 20500940 1684215 1025047 20287 33333 93428 51972 19411 10631321 19754051 1771887 987703 18581 33333 80629 60000 15941 12549695 17245840 2091616 862292 13742 33333 69792 69063 11959 14473307 15089951 2412218 754498 10426 33333 68827 70000 10291 14663867 14895380 2443978 744769 10158 33333 59898 80000 08929 16639176 13059560 2773196 652978 07849 33333 52904 90000 06994 18553017 11550760 3092169 577538 06226 33333 47309 100000 05595 20441823 10261700 3406971 513085 05020 33333 42773 110000 04535 22324134 9124680 3720689 456234 04087 33333 39071 120000 03702 24209761 8094240 4034960 404712 03343 33333 36042 130000 03029 26103937 7138360 4350656 356918 02735 33333