# How to derive consumer expenditures in EMEA 14.2.5

I am working on a problem 14.2.5 from EMEA by Sydsaeter, Hammond and Strom.

Consider the consumer demand problem: $$\max_{x,y} U(x,y) = \alpha \ln(x-a) + \beta \ln(y-b) \text{ s.t. } px+qy=m \tag{*}$$ where $$\alpha, \beta, a, b, p, q$$ and $$m$$ are positive constants with $$\alpha + \beta =1$$, and moreover, with $$m>ap+bq$$. Show that if $$x^*, y^*$$ solve problem (*), then expenditure on the two goods is given by the two linear functions: $$px^*= \alpha m+ pa -\alpha(pa+qb), qy^* = \beta m+ qb - \beta (pa + qb)$$

My attempt:

I set up Lagrangian:

$$L= \alpha \ln(x-a) + \beta \ln(y-b) - \lambda (px+qy-m)$$

and get FOCs:

$$\frac{\alpha}{x-1} = p \lambda$$

$$\frac{\beta}{y-1} = q \lambda$$

$$px+qy = m$$

I tried to solve the system above and then my plan was to multiply $$x^*,y^*$$ with $$p$$ and $$q$$ but that gives me different consumption function. Book has a hint that I should solve for $$px$$ and $$qy$$:

$$px = pa + \frac{\alpha}{\lambda}$$

$$qy = qb + \frac{\beta}{\lambda}$$

Then the book hints I should use budget constraint to eliminate lambda so I plug it there:|

$$pa + \frac{\alpha}{\lambda} + qb + \frac{\beta}{\lambda} = m$$

$$\lambda = \frac{\alpha + \beta}{m- pa - qb}$$

When I plug the solution for lambda back to equation for $$px$$ I just get:

$$px^* = pa + \frac{\alpha m - \alpha pa - \alpha bq}{\alpha +\beta}$$

And this is again not the consumption expenditure function the textbook says I should get. I don't get why I can't end with the same consumption expenditure as the book. Does anyone know how to solve this?

• As $\alpha + \beta = 1$, I think these are indeed the same solutions. – tdm May 4 at 18:21
• @tdm I don't get it, can you please explain why? – WilliamT May 4 at 18:21
• If you put $\alpha + \beta = 1$ in your solution, you get $px = pa + \alpha m - \alpha(pa + bq)$ as in the assignment. There is a typo though, because $1 b$ should be $q b$. – tdm May 4 at 18:25
• @tdm ooooo I see now, I am so dumb... if you want you can post it as an answer I will accept it. Yes, that was just typo it was qb – WilliamT May 4 at 19:00
• No problem. By the way, the utility function in you example is called a Stone-Geary utility function wiki. It is very similar to a Cobb-Douglas utilities function, with the addition of subsistence levels $a$ and $b$ that provide the minimal levels of both goods that need to be consumed. – tdm May 5 at 5:06