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I am studying Essential Mathematics for Economic Analysis and in chapter 12 problem 10 there is a problem that I can solve but I do not understand why the solution works.

The problem asks us to differentiate the following system from consumer demand theory:

$$U'_1(x_1,x_2) = \lambda p_1 \\ U'_2(x_1,x_2) = \lambda p_2 \\ p_1 x_1 + p_2x_2=m $$

where system defines $x_1, x_2$ and $\lambda$ as differentiable functions functions of $p_1,p_2$ and $m$. We are asked afterwards find expression for $\partial x_1 / \partial p_1$.

Now I have no problem with the first step - the differentiated system will be:

$$U^{''}_{11}(x_1,x_2)dx_1 + U^{''}_{12}(x_1,x_2)dx_2 = \lambda dp_1 + p_1 d\lambda \\ U{''}_{21}(x_1,x_2)dx_1 + U{''}_{22}(x_1,x_2)dx_2 = \lambda dp_2 + p_2 d\lambda \\ p_1 dx_1 + x_1 dp_1 + p_2 d x_2 + x_2 dp_2=dm $$

Now my first attempt was to solve 2nd equation for $dx_2$ substitute it into the first and solve for $d x_1 /dp_1$ but I got wrong result. Afterwards in the solution in the back there is a hint that says that to solve this we should put $dp_2=dm=0$ after I follow this hint I get the correct solution:

$$\partial x_1 / \partial p_1 = \frac{[\lambda p_2^2 + x_1(p_2 U^{''}_{12}-p_1U^{''}_{21})]}{(p_1^2 U^{''}_{22} -2p_1p_2U^{''}_{12} + p_2^2 U^{''}_{11})}$$

But I don't get why we can simply assume $d p_2 = dm = 0$ that seems so arbitrary, how come that is necessary to find the correct solution?

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  • $\begingroup$ Are they specifycally trying to derive an expression that says something about what happens as a result of changes in price $p_1$? $\endgroup$ – Jesper Hybel Dec 16 '20 at 21:59
  • $\begingroup$ @JesperHybel idk, they just state find $\partial x_1 / \partial p_1$, they don't ask for much else or even say why that should be done. It is a review problem at the end of the chapter so they don't explain it. I just don't understand why solution to this one exercise requires $d p_2=dm=0$ I can solve all other exercises and in the main text of a chapter there is no example where arbitrarily differentials are set to zero for no reason. I guess there might be some economic reason for it but idk... $\endgroup$ – WilliamT Dec 16 '20 at 22:54
  • $\begingroup$ I must admit I'm not so strong when it comes to differentials. My intuitive understanding of why it is ok is that $(p_1,p_2,I)$ are parameters of the optimization problem and as such they are simply given. The solution $(x_1,x_2)$ is a function of chosen parameters $(p_1,p_2,I)$ - assuming it is unique - so $(p_1,p_2,I) \rightarrow (x_1,x_2)$. So if you were interested in how solution change when $p_1$ change you would keep $dp_2=dm=0$. $\endgroup$ – Jesper Hybel Dec 16 '20 at 23:02
  • $\begingroup$ I guess in your initial approach you ended up finding the total derivative $d p_1$. However, you are interested in the partial derivative $\partial p_1$, keeping the other variables constant and only changing $p_1$. $\endgroup$ – Bayesian Dec 17 '20 at 9:24
  • $\begingroup$ @Bayesian but then why not set $d \lambda$ to zero as well? $\endgroup$ – WilliamT Dec 17 '20 at 17:40
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The assumption $dm =0$ says that we examine the behavior of the consumer under a fixed nominal income, and this is something interesting to study, because it aligns to a large degree with the observed reality of many people that have approximately constant income.

The assumption $dp_2=0$ assumes away general equilibrium effects, since we are looking at changes in $p_1$ which is price in one market, while $p_2$ is the price in another market, presumably of a different good. So while formally the justification here is that "we do partial equilibrium analysis", the vindication of partial equilibrium analysis is that the effects from changes in one goods market to another goods market take some time to materialize, so partial equilibrium analysis has relevance to the real world.

So we see that the assumption $dm=dp_2=0$ is not "necessary to find the correct solution" (there is not one correct solution, but solutions each consistent with the corresponding assumptions made). But it is neither an "arbitrary" set of assumptions, on the contrary, it has a specific logic and purpose for which it is imposed.

The OP asked in the comments

"Then why not also assume that $d\lambda=0$"?

But the optimal value of lambda is endogenous to the model, and it summarizes aspects of the consumer's optimal behavior. So whether it changes or stays constant is for the model to tell us.

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This dx2=dm=0 assumption is necessary because it is the application of theory of partial derivative mathematics. If you don't follow this rule as you find dx1/dp1 you are not following partial derivative rules and you will get a wrong answer. So the assumption is not arbitrary but based on the rules of partial differentiation. dx2=dm=0 assumption is partial differentiation rule.

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  • $\begingroup$ but then I still do not understand why we do not set $d \lambda$ to zero as well $\endgroup$ – WilliamT Dec 17 '20 at 17:38
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While the x’s are functions of price, both P_2 and m are constants. Thus the zero-valued differential.

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  • $\begingroup$ I made the same comment under the other answer, this makes sense but then raises another question about why we do not set $d \lambda = 0$ too $\endgroup$ – WilliamT Dec 17 '20 at 17:39
  • $\begingroup$ Lambda, which ends up being the monetary value of loosening the constraint, is solved for as a function of prices as well. $\endgroup$ – philE Dec 19 '20 at 4:06
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Del lambda is also 0 when you find dx1/dp1 as seen in the equation. Why are you getting confused? When you find dx1/dp1 you set all the other partial derivatives =0. This is the rule of partial differentiation. Similarly when you find dx1/dm you set all the other partial derivatives =0. Hope u understand now.

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