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Given:

A ten year comparison between the United States and the Soviet Union in terms of crop yields per acre revealed that when only planted acreage is compared, Soviet yields were equal to 68 percent of United States yields. When total agricultural acreage (planted acreage plus fallow acreage) is compared, however, Soviet yield was 114 percent of US yield. From the information above, show that it cannot be proven that The United States had more fallow acreage than planted acreage.

Reference:Prove that it cannot be proven that "The United States had more fallow acreage than planted acreage"

Proof:

Let $p_u,f_u$ be the planted/fallow acreage in the U.S. and $p_s,f_s$ be the planted/fallow acreage in the Soviet Union. Also, let $y_u$ be the yield per planted acre in the U.S. and $y_s$ be the yield per planted acre in the Soviet Union. The information given says that

$$\frac{y_s}{y_u} = 0.68 \; \iff \; y_s = 0.68y_u \tag{1}\label{eq1}$$

The total yield in the U.S. is $y_u p_u$, so the yield per all of the acres would be

$$y_{au} = \frac{y_u p_u}{p_u + f_u} \tag{2}\label{eq2}$$

Similarly, for the Soviet Union, it's yield per all of the acres would be

$$y_{as} = \frac{y_s p_s}{p_s + f_s} \tag{3}\label{eq3}$$

By cross-multiplying and combining the terms for $f_s$ and $p_s$, you get

\begin{align} y_{as}(p_s + f_s) & = y_s p_s \\ y_{as}p_s + y_{as}f_s & > = y_s p_s \\ y_{as}f_s & = y_s p_s - y_{as}p_s \\ f_s & = \frac{p_s(y_s - y_{as})}{y_{as}} \tag{4}\label{eq4} \end{align}

It's also given that

$$\frac{y_{as}}{y_{au}} = 1.14 \; \iff \; y_{as} = 1.14y_{au} > \tag{5}\label{eq5}$$

This is the only information provided, so if $2$ sets of values are found which are consistent with the above equations but with one showing that $f_u \gt p_u$ and the other showing that $f_u \lt p_u$, then this would answer what's requested.

Let's set $y_u = 100$. Then from \eqref{eq1}, you get $y_s = 68$. Next, let $p_u = 10,000,000$ and $f_u = 11,000,000$. Substituting these into \eqref{eq2} gives $y_{au} = 47.619\ldots$. From \eqref{eq5}, this gives $y_{as} = 54.285\ldots$. From \eqref{eq4}, you get

$$f_s = \frac{p_s(68 - 54.285\ldots)}{54.285\ldots} > \tag{6}\label{eq6}$$

Note you can plug any value of $p_s$ you want to get a specific value of $f_s$, e.g., if $p_s = 10,000,000$, then $f_s = > 2,526,315.789\ldots$.

Next, consider $f_u = 9,000,000$. Then \eqref{eq2} gives $y_{au} = > 52.631\ldots$. From \eqref{eq5}, this gives $y_{as} = 60$. From \eqref{eq4}, you get

$$f_s = \frac{p_s(68 - 60)}{60} \tag{7}\label{eq7}$$

If you use $p_s = 10,000,000$ again, then $f_s = 1,333,333.333\ldots$.

All of these values are consistent with the equations relating the only information which was provided, but with one set showing more fallow acreage than planted acreage in the U.S. (i.e., $f_u = > 11,000,000 \gt p_u = 10,000,000$) and the other one showing the opposite (i.e., $f_u = 9,000,000 \lt p_u = 10,000,000$).

A main reason why you can't prove which of the fallow and planted acreage in the U.S. is greater is because there are $6$ unknowns of $p_u,f_u,p_s,f_s,y_u$ and $y_s$, but only $4$ equations relating them. As these equations are consistent with each other, it's an under-determined system of equations, with $2$ degrees of freedom in this case (in general, you would have more than $2$ if any of the equations are linearly dependent). Also, note the question numeric value restrictions are for comparing values between the U.S. and the Soviet Union, meaning there are fewer constraints among values within the U.S. (and the Soviet Union as well).

The proof above is too complicated to understand, or maybe it's more important to see a big picture than rather know how to prove this. Is there a more concise, shorter way, simpler way to prove? I don't know, with pictures, WLOG?

  • What is it that you don't understand? – saulspatz Jul 18 '19 at 19:52
  • "so if 2 sets of values are found which are consistent with the above equations but with one showing that $f_u>p_u$ and the other showing that $f_u<p_u$, then this would answer what's requested." -- Actually, no, in order to show that $f_u>p_u$ cannot be concluded, it suffices to exhibit one valid set of values, namely one with $f_u\le p_u$. – Hagen von Eitzen Jul 18 '19 at 19:55
  • @saulspatz The notation of $y_{as}$ or $y_{au}$ is what I don't understand. What is "as" or "au"? –  Jul 18 '19 at 20:08
  • @HagenvonEitzen I misread the question to think it was asking for being able to prove either the fallow acreage was either more or less than the planted acreage. As such, you're correct in that I only needed to use one set of values which exhibit $f_u \le p_u$. – John Omielan Jul 18 '19 at 20:10
  • They mean the the for all acres, for the Soviet Union and the United States, respectively. – saulspatz Jul 18 '19 at 20:11
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    @Germania The "$as$" and "$au$" are just subscripts I used to denote values for "all Soviet" and "all US" in terms of yields among all the acres. Noted that I tried to give a more detailed explanation of why you can't conclude certain details but, as Hagen von Eitzen's answer below shows, I could have made it considerably simpler for just what you were specifically asking. – John Omielan Jul 18 '19 at 20:11
  • @John Omielan Is "all Soviet" or "all US" which is $y_{as}$ or $y_{au}$ the proportion of yield/total land area? –  Jul 18 '19 at 20:23
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    @Germania Yes, that is correct. The "a" stands for "all" land, although perhaps using "t" for "total" land would have been more appropriate, and perhaps a bit easier to follow. – John Omielan Jul 18 '19 at 20:26
  • @JohnOmielan You listed the "6 unknowns". What are the 4 equations we have? What are the "4 knowns"? You listed up to (6) but we clearly don't have 6 equations. –  Jul 19 '19 at 01:45
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    @Germania Note the "$6$ unknowns are the input variables. In hindsight, perhaps that would've been a better phrase to use than unknowns. In general, you can define everything else in terms of $6$ values, as I showed in my answer. As you can see, Hagen von Eitzen's answer also specifies $6$ input values, although it uses the total crop harvests instead of the yield per planted acre. The $4$ equations are (1), (2), (3) and (5). I'm not sure what you mean by "4 knowns". The point is there aren't 6 independent equations. – John Omielan Jul 19 '19 at 01:54
  • @JohnOmielan Why aren't $y_{as}$ or $y_{au}$ two of the unknown variables? How aren't there 8 unknowns as input variables? –  Jul 19 '19 at 01:59
  • @Germania The other equations come from these $4$. Note (4) is a reformulation of (3), while (6) & (7) come from specific values in the earlier equations. In summary, there're $4$ original, independent equations, with $6$ input values, so there're $2$ degrees of freedom, resulting in certain things not being specified, such as the relation of the # of fallow acres to the # of planted acres in the U.S. $y_{as}$ & $y_{au}$ are not unknown variables as they're defined in terms of the $6$ input variables, as you can see in (2) & (3), so once you have those $6$ inputs, $y_{as}$ & $y_{au}$ are set. – John Omielan Jul 19 '19 at 02:02
  • @JohnOmielan How do you know $y_s$ and $y_u$ are both unknown variables or "input variables"? If and only if I know $y_s$ then I know $y_u$ so I think either $y_s$ or $y_u$ is unknown but not both. –  Jul 19 '19 at 02:04
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    @Germania As you say, $y_s$ and $y_u$ are directly related to each other through (1). If you wish, you could just use $y_u$ as an input value and replace all uses of $y_s$ by $0.68y_u$. In this case, you then have $5$ input variables, but since you no longer need to use (1), you can remove it, giving you only $3$ other equations relating those variables. There are still $2$ degrees of freedom. It all depends on what you want to consider as input initially & what is determined by the equations, keeping in mind depending on how you define things, not all input values may be independent. – John Omielan Jul 19 '19 at 02:09
  • @Germania I used both $y_s$ and $y_u$ as initial input values, even though they're not independent of each other, to allow the equations to be basically agnostic to the U.S. or the Soviet Union, so the variables for the U.S. would have a $1-1$ correspondence with variables for the Soviet Union. If you do as suggested as a possibility above, then there would be no $y_s$ corresponding to $y_u$. – John Omielan Jul 19 '19 at 02:12

1 Answers1

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The information given is consistent with the following values:

  • US planted acres $=34$
  • US fallow acres $=23$
  • US total crop harvest $=100$
  • SU planted acres $=34$
  • SU fallow acres $=0$
  • SU total crop harvest $=68$

To check: $$\frac{\text{SU crop yield per planted acre}}{\text{US crop yield per planted acre}} =\frac{\frac{68}{34}}{\frac{100}{34}}=\frac{68}{100}=68\,\%,$$ $$\frac{\text{SU crop yield per total acre}}{\text{US crop yield per total acre}} =\frac{\frac{68}{34}}{\frac{100}{57}}=\frac{57}{50}=114\,\%.$$

As this consistent set of values has less fallow than planted US acres, no proof of the contrary is possible.

Hagen von Eitzen
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