Decimal ExponentsDate: Wed, 02 Nov 1994 14:36:16 -0800 From: Caryl Lynn Segal Subject: problem Hope you'll help a graduate student trying to understand an example in an economics textbook. This involves food production and the equation is: Q = 100(K[superscript.8]L[superscript.2]) K involves machines and L involves labor. The diagram shows Q(output) as 13,800 L as 760 K as 90 I have been unable to figure out how to make the equation reveal those numbers. Any help you can provide will be appreciated. P.S. I know this is set up for K-12 but am hoping you are not too busy to help out someone beyond this level Date: Wed, 2 Nov 1994 15:40:28 -0500 From: Melissa D. Binde Subject: Re: problem I had trouble too--is there any more information you can give us? What kind of economics (micro, macro, etc.) is this? I'm not an expert in economics, but perhaps some of the other "doctors" will be able to help you later this evening. >P.S. I know this is set up for K-12 but am hoping you are not too busy to >help out someone beyond this level We'll do our best! Melissa _____________ From: Dr. Ethan Subject: Re: problem Date: Wed, 2 Nov 1994 15:50:06 -0500 (EST) 760 to the .8 power times 90 to the .2 power times 100 equals 13789.8, which rounds to 13800 so I think we are just dealing with multiplication and exponents. Ethan Doctor-On Call Date: Wed, 02 Nov 1994 17:24:42 -0800 From: Caryl Lynn Segal Subject: Re: problem I guess what I'm trying to discover but didn't explain is how you get from K.8 power to learn it is 90. My problem is being unable to figure out how to work with .8 power or .2 power etc. Date: Wed, 2 Nov 1994 18:28:44 -0500 From: Gabe Farmboy Cavallari Subject: decimal exponents Here's a possible explanation. A decimal exponent could also be thought of as a fractional exponent, so .8 would equal 4/5. The fractional expression, at least for me, is easier to use, because raising a number to the 4/5's power is like raising that number to the 4th power and then taking the 5th root. For any number raised to the m/nth power, raise the number to the mth (from the numerator)power and take the nth (from the denominator) root. As a more familar example, consider the square root. The exponential equivalent would be to raise a number to the 1/2 power, or equivalently the .5 power. Hope this helps. Gabe Cavallari, math doc Date: Wed, 2 Nov 1994 18:34:21 -0500 From: Heather Stickney Subject: Re: problem From what you've told us, you can't do the problem. What I'm thinking is that this equation is perhaps a standard equation to use to determine output when you know K and L, or to determine L when you know K and how much output you want, or vice-versa. I think the diagram is just showing an example of how they used the equation. When K=90 and L=760, there is an output of 13,800. You are not expected to use the equation to determine the values of all the variables at the same time. If this doesn't make sense, or I've misunderstood what you are asking, write back.> Date: Wed, 2 Nov 1994 22:46:21 -0500 (EST) From: "Michael W. S. Morton" I think there may be another way of looking at this problem (that may be more in depth than you care to go, or even want to think about. :) Having taken a little econ, I know there are such graphs as isoquants, that is, graphs where the output of a firm stays constant as you move along the curve, varying K and L. These curves show the relationship between capital (K) and labor (L) at different points, holding the output (Q) constant. So, say..... Q = 13800 The function then looks like: .8 .2 13800 = 100 * K * L .8 .2 138 = K * L .8 .2 K = 138 / L This function can be graphed (most preferably by a computer). It represents the way in which capital and labor can be interchanged at a constant output of 13800 units. The slope of this line at any given point is equal to the Marginal Rate of Technical Substitution (MRTS) at that point. More curves can be put on this graph by letting Q = different values. This may be going further than you wanted to, or maybe it helps. If you have any questions, just lemme know! -MORTON, A math Doctor w/a little econ :) |
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