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5

Independence of irrelevant alternatives fails. Your conclusion that removing $X$ will have no effect on the preference between $Y$ and $Z$ is wrong. If one person has preference $Z\gt Y\gt X$ and everyone else has preference $X\gt Y\gt Z$, then $Z$ is socially preferred to $Y$ if $X$ is in the race, whereas otherwise $Y$ is socially preferred to $Z$. (I'm ...


2

Putting $\beta=\frac{2}{\gamma+\kappa}$ and $\alpha=\beta\gamma$ we have \begin{align} I&=\int_0^x \frac{2(\mathrm e^{\gamma u}-1)}{(\gamma+\kappa)(\mathrm e^{\gamma u}-1)+2\gamma} \mathrm du=\beta\int_0^x \frac{(\mathrm e^{\gamma u}-1)}{(\mathrm e^{\gamma u}-1)+\alpha} \mathrm du\\&=\beta\int_0^x \frac{(\mathrm e^{\gamma ...


2

Dividing the numerator and denominator of the integral by $\gamma + \kappa$ gives $$\int_{0}^{x}\frac{a(e^{\gamma u}-1)du}{e^{\gamma u}-1+a\gamma}$$ Where $a=\frac2{\gamma + \kappa}.$ Breaking this into $2$ integrals, $$=\frac a\gamma\int_{0}^x\frac{\gamma e^{\gamma u}du}{e^{\gamma u}-1+a\gamma}-a\int_0^x\frac{du}{e^{\gamma u}-1+a\gamma}$$ For the first, ...


1

A fixed-effects model may be useful here. Let $t_{pcrn}$ be completion time for a given person $p$ driving in car $c$ on track $r$ for run $n$ (assuming there can be multiples of each triplet $pcr$. We could model $t_{pcrn}$ using a three-factor, additive, fixed-effects model: $$t_{pcrn} = \mu + \phi_p+\alpha_c + \beta_r + \epsilon_{pcrn}$$ Where: $\mu$ ...


1

You need to produce seven units of output, therefore: $7 = 4L^{0.5}M^{0.5}$. Squaring, you find that are restricting yourself to the curve $LM = 49/16$ in the first quadrant as $L,M>0$. The cost function is $C(L,M) = 100L + 16M$. You want to minimize this subject to the constraint $LM = 49/16$. We can do this more generally, but in this case, it is ...


1

For the new machine, consider the following: PERIOD COST UPDATED COST 0 -75000 `=-75000/(1,12)^0 -75000,00 1 -5500 `= -5500/(1,12)^1 -4910,71 2 -5500 `= -5500/(1,12)^2 -4384,57 3 -5500 `= -5500/(1,12)^3 -3914,79 4 -5500 `= -5500/(1,12)^4 ...



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