Isoquants, Isocosts and Cost Minimization Overheads

We define the production function as 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

y = f (x1, x2, x3, . . . xn ) y Holding other inputs fixed, the production function looks like this y = f (x1, x2, x3, . . . xn ) 350 300 Output -y y 250 200 150 100 50 2 4 6 8 10 12 Input -x

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 60 50 A MPP Output -y 40 30 20 10 -10 1 2 3 4 5 6 7 8 9 10 11 -20 -30 -40 Input -x

choose the least cost x’s The Cost Minimization Problem Pick y; observe w1, w2, etc; choose the least cost x’s

Isoquants The word “iso”means same while “quant” stand for quantity 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 x2 = 4, x1 = 2.636 Isoquant y = 2000 X X 12 10 8 2 4 6 8 10 12 14 X 2

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 -- 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 14 12 Isoquant y = 10000 x1 10 Isoquant y = 2000 8 6 4 2 2 4 6 8 10 12 14 x 2

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 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)  x1 2 4 6 8 10 12 14 x 14 Isoquant y = 10000 x1 10 Isoquant y = 2000 8  6  4 2  2 4 6 8 10 12 14 x 2

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 Workers Tractor-Wagons x1 x2 10 1 5.48 2 3.667 3 2.636 4 1.9786 5 Change x2 from 1 to 2

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

Change x2 from 5 to 6 More example calculations y = 10,000 x1 x2 12.469 4 9.725 5 8.063 6 6.883 7 5.990 8 Change x2 from 5 to 6

A declining marginal rate of substitution The marginal rate of substitution becomes larger in absolute value as we have more of an input. 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. 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

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

 The Marginal Rate of Substitution (MRTS) x1 2 4 6 8 10 12 14 x 14 12 Isoquant y = 2000 8  6 4 2  2 4 6 8 10 12 14 x 2

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, . . . 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. Quantities of inputs - x1, x2, x3, . . . Prices of inputs - w1, w2, w3, . . .

Graphical representation Cost = 120 w1 = 6 w2 = 20 22 x 20 1 18 16 14 12 10 8 6 4 2 1 2 3 4 5 6 7 x 2

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 Isoquant y = 10000 24 X Isocost 120 1 20 Isocost 180 16 12 8 4 3 4 5 6 7 8 9 10 11 12 13 X 2 At intersection there are opportunities for trade

Add C = 160 Isoquant y = 10000 X Isocost 120 Isocost 180 Isocost 160 X 24 Isoquant y = 10000 X 20 Isocost 120 Isocost 180 1 16 Isocost 160 12 8 4 4 5 6 7 8 9 10 11 12 13 X 2

Add C = 154.6 Isoquant y = 10000 X Isocost 120 Isocost 180 Isocost 160 24 Isoquant y = 10000 X 20 Isocost 120 1 Isocost 180 16 Isocost 160 Isocost 154.6 12 8 4 4 5 6 7 8 9 10 11 12 13 X 2

In review Isoquant y = 10000 X Isocost 120 Isocost 180 Isocost 160 24 Isoquant y = 10000 X 20 Isocost 120 1 Isocost 180 16 Isocost 160 Isocost 154.6 12 8 4 4 5 6 7 8 9 10 11 12 13 X 2

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 Slope of the isocost line = Slope of the isoquant Substituting we obtain The price ratio equals the ratio of marginal products

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

Example Table w1 = 6, w2 = 20 To get an x1, I can give up 3.33 x2 in terms of cost 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

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

w1 = 6 w2 = 20 C = 180 24 X 20 1 Isocost 180 16 12 10 8 6.66 4 3 4 5 6 7 8 9 10 11 12 13 X 2

The isoquant tells us the rate at which the firm can trade one input for the other and remain at the same production level 24 Isoquant y = 10000 x 20 1 16 12 8 4 4 5 6 7 8 9 10 11 12 13 x2

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

The isoquant tells us the rate at which the firm can trade one input for the other and remain at the same production level 24 Isoquant y = 10000 x 20 1 Isocost 180 16 Isocost 160 12 8 4 4 5 6 7 8 9 10 11 12 13 x2

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

When the slope of the isoquant is steeper than the isocost line, the firm will move down the line 24 Isoquant y = 10000 x 20 1 16 12 8 4 4 5 6 7 8 9 10 11 12 13 x2

The End