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How do I graph the following conditions? We are in two good life, spinach and sprouts, spinach on x axis, sprouts on y axis. If you consume 10 or less servings of spinach, you pay \$5 for each. Above for a quantity above 10, the price for EACH of them becomes equal \$10. So, for example, buying 9 servings is \$45, buying 11 is \$110. Assume continuity of distribution (so you can consume 9.5, etc.) The same deal applies to sprouts -- \$5 before 10, \$10 for EACH starting at quantity 10 and up. Our budget for the time period is \$150. I am at loss how to graph this, even using more than 1 line... Any help appreciated.

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What is it you are trying to graph, if spinach is in one axis and sprout on the other? You could plot the largest number of servings of sprouts that you can afford if you buy a given number of servings of spinach (thus making "number of servings of sprouts" a function of "number of servings of Spinach"). Is that what you want? Or are you trying to do a 3-dimensional "graph" where the $z$-axis indicates total cost? – Arturo Magidin Oct 6 '11 at 5:08
@ArturoMagidin: it's always assumed that expenditure is at \$150, I am trying to graph one against the other, and various jumps and inconsistencies arise. Here's a typical example of a budget constraint with normal prices: -- as you can see, bread is 4 times more expensive than wine. I can't do such a straightforward graph here because of jumps. – keynist Oct 6 '11 at 5:15
I'm not sure what the red and blue line are supposed to represent in that graph, nor do I know what original conditions that plot was based on, so it's not much help. But it looks like you are trying to draw what I describe first: find the maximum number of sprouts that you can buy, as a function of how much spinach you bought. – Arturo Magidin Oct 6 '11 at 5:18
@ArturoMagidin: Sorry, the red line is our budget constraint. So we are trying to spend some amount of dollars on both of the goods, given prices. In the context of the picture, let the budget be 80 dollars, price_wine = 1, price_bread = 4. Then the read line is the budget constraint. I am trying to make an analogous graph for the problem I posted above. – keynist Oct 6 '11 at 5:21
up vote 1 down vote accepted

The amount spent on $n$ servings is given by: $$\begin{align*} s(n) &= \left\{\begin{array}{ll} 5n &\text{if }0\leq n\leq 10;\\ 50+10(n-10) &\text{if }10\lt n \end{array}\right.\\ &=\left\{\begin{array}{ll} 5n &\text{if }0\leq n\leq 10;\\ 10n - 50 &\text{if }10\lt n \end{array}\right.\end{align*}$$

Now, start with $0$ servings of Spinach: if you buy no Spinach, you can buy 20 servings of Sprouts (the first 10 cost you \$50, the next 10 cost you \$100).

Each serving of sprout that you drop betweeen 20 and 10 will "free up" ten dollars. For the first 5 servings, you can buy 2 servings of Spinach of every serving of Sprouts you drop.

So from $x=0$ to $x=10$, your budget constraint graph is just the line segment that joins $(0,20)$ (no spinach, 20 servings of sprouts) and $(10,15)$ (ten servings of spinach, which will cost you \$50, and fifteen servings of sprouts, which will cost you \$50 for the first ten servings, and another \$50 for the next five, for a total of \$150).

At this point, if you drop one serving of Sprouts you have \$10 more available, with which you can buy one more serving of spinach. This will happen until you drop to 10 servings of sprouts, at which point you will be able to afford 15 servings of Spinach (the ten from before, plus five more here). So the budget constraint graph is now a line that joins $(10,15)$ with $(15,10)$.

And now, for every serving of sprout you drop you get \$5 back, but you need \$10 to buy one further serving of Spinach. So you need to drop two servings of sprouts in order to be able to buy one serving of spinach. The budget constrain graph at this point is a line segment joining $(15,10)$ to $(20,0)$

You'll see that there is a certain symmetry to the graph about the line $x=y$ (which makes sense, since the sprouts and spinach are really playing symmetric roles).

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Awesome, thanks!! – keynist Oct 6 '11 at 5:37
@keynist: Here's another way to view this: paint the vertical band above $[0,10]$ pale yellow, and the horizontal band to the right of $[0,10]$ pale blue; that corresponds to the locations where the spinach costs \$5 per serving, and where the parsley costs \$5 per serving, respectively. If you are in the pale-green section (the intersection), parsley and spinach servings cost the same, so any budget line will have slope $-1$. If you are in the pale-blue section, spinach costs twice as much, so lines will have slope $-\frac{1}{2}$; in the pale-yellow, slope $-2$; in the white, slope $1$. – Arturo Magidin Oct 6 '11 at 16:47

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