Re: Newbie with simple questions (take 2)

• To: mathgroup at smc.vnet.net
• Subject: [mg62325] Re: Newbie with simple questions (take 2)
• From: misha <iamisha1 at comcast.net>
• Date: Sun, 20 Nov 2005 04:50:29 -0500 (EST)
• References: <dlp37m\$1hp\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```I've narrowed my question down quite a bit...

Why can't I use superscripts?

I initially tried q(superscript 2, subscript h) and q(superscript h,
subscript l) and got error messages.  Then I tried q(superscript h,
subscript 2) and q(superscript l, subscript 2), but this didn't seem to
help.  It seems that Mathematica interprets superscripts as exponents.
Is there a way around this?

In any case, I did the following and got some satisfying results:
Solve[{D[(a-x-z-c)x==0, x], D[(a-y-z-d)y==0, y], D[t(a-x-z-g)z +
(1-t)(a-y-z-g)z==0, z]}, {x,y,z}]

So the question regarding solving FOCs and simultaneous equations no
longer needs answering, but I am very open to suggestions regarding such
problems!

In short, my question boils down to naming variables and using
superscripts in doing so.

Thank you!

Misha

misha wrote:

> I am a new user (errr..purchaser) of Mathematica, but I have not been
> able to find answers to these (probably) simple questions with
> Mathematica?s help browser.  I am trying to use Mathematica to solve a
> simple system of simultaneous equations.  I suppose I could use it to
> solve the first order conditions (FOCs), but I?m having enough problems
> as it is.  I have more ambitious goals than this, but I thought this
> would be an easy place to start.
>
> By the way, can anyone recommend a book heavy in examples for a
> beginning user such as myself?
>
> Here is the complete problem:
>
> P(Q) = a - Q (inverse demand curve)
>
> Q = q1 + q2 (Cournot) duopoly
>
> C1(q1) = c*q1 (firm 1?s commonly known cost function, with constant
> marginal cost, c)
>
> C2 =  cL*q2 with probability t
>        cH*q2 with probability 1 - t
>
> (firm 2?s cost functions for constant marginal costs cL < cH, known to
> firm 2 but unknown with certainty to firm 1)
>
> If Firm 2 has constant marginal cost cH, firm 2 chooses q2 to solve
>
> max{[(a - q1* - q2) - cH]*q2}
>
> If Firm 2 has constant marginal cost cL, then firm 2 chooses q2 to solve
>
> max{[(a - q1* - q2) - cL]*q2}
>
> The resulting FOCs are:
>
> <<the asterisk denotes ?optimal? and the cH in parentheses denotes that
> it is a function of cH>>
>
> q2*(cH) = (1/2)*(a - q1* - cH)
>
> q2*(cL) = (1/2)*(a - q1* - cL)
>
> Similarly, Firm 1 chooses q1 to solve
>
> max{t[(a - q1 - q2*(cH)) - c]*q1 + (1 - t)[(a - q1 - q2*(cL)) - c]*q1},
>
> which yields FOC:
>
> q1* = (1/2)*[t(a - q2*(cH) - c) + (1 - t)(a - q2*(cL) - c)]
>
> So I want to use Mathematica to do the tedious algebra to get me the
> following:
>
> q2*(cH) = (a - 2cH + c)/3 + (1 - t)(cH - cL)/6
>
> q2*(cL) = (a - 2cL + c)/3 - t(cH - cL)/6
>
> q1* = (a - 2c + tcH + (1 - t)cL)/3
>

```

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