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use the general form of the Cobb Douglas utility function $U(x,y)= (x^a)(y^b)$ and the budget constraint in the form $B=p_{x}X + p_{y}Y$ to find the demand functions for good x and good y.

Is this even possible? I took the derivative with respect to x and y and set them equal to $\frac{p_x}{p_y}$. I can’t get past $\frac{ay}{bx}= \frac{p_x}{p_y}$

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    $\begingroup$ Hint: With ay/bx= px/py and B=pxX + pyY, you have two equations and two unknowns. So you should be able to solve for x and y, the quantity demanded for each good. $\endgroup$ – Herr K. Jul 30 at 18:26
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its now been a couple days without response to @Herr so I will show you the general solution. Usually you will be given values for $a$ and $b$ and the respective prices, but you can always solve this symbolically treating them as unknowns. The solution follows: $$max\;U(X,Y)=X^{a}Y^{b}$$ $$s.t.\; B=P_{x}X+P_{y}Y$$ Taking first order conditions of the utility function, we get: $$FOC_{x}=aX^{a-1}Y^{b}$$ $$FOC_{y}=bX^{a}Y^{b-1}$$ Setting these equal to the price ratio as you suggested, it simplifies to $$\frac{aY}{bX}=\frac{P_{x}}{P_{y}}$$ As you found. Here you have this equation and the equation for the budget line: two equations and two unknowns, algebra is all that is needed now. Rearrange the tangency condition ($MRS=\frac{P_x}{P_y}$) and we get: $$\frac{aY}{bX}=\frac{P_{x}}{P_{y}}$$ $$aY=\frac{P_{x}}{P_{y}}bX$$ $$Y=\frac{P_{x}}{P_{y}}\frac{b}{a}X$$ then we can substitute this demand for Y (as a function of X) into the budget line: $$B=P_{x}X+P_{y}(\frac{P_{x}}{P_{y}}\frac{b}{a}X)$$ which leaves us with an equation of only $X$! Continue solving: $$B=P_{x}X+P_{x}\frac{b}{a}X$$ $$B=P_{x}X(1+\frac{b}{a})$$ $$\frac{B}{1+\frac{b}{a}}=P_{x}X$$ $$X=\frac{B}{P_{x}(1+\frac{b}{a})}$$ $$X=\frac{a}{a+b}\frac{B}{P_x}$$ where the RHS are all values of parameters, hence the demand of $X$ depends on the prices and level of income. Then you use the original $Y=\frac{P_{x}}{P_{y}}\frac{b}{a}X$ derived from the tangency condition and substitute $X$ in: $$Y=\frac{P_{x}}{P_{y}}\frac{b}{a}\frac{B}{P_{x}(1+\frac{b}{a})}$$ $$Y=\frac{b}{a}\frac{B}{P_{y}(1+\frac{b}{a})}$$ $$Y=\frac{a}{a+b}\frac{b}{a}\frac{B}{P_{y}}$$ $$Y=\frac{b}{a+b}\frac{B}{P_y}$$

Note the symmetry between the demand functions of $X$ and $Y$. While these questions are much easier with actual values for the unknowns, you come up with a rather intuitive answer when working with the parameters instead! The relationship still holds true and if you plug the values given into $X$ and $Y$ you will get the correct answer.

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    $\begingroup$ Note that $(1+\frac{b}{a})^{-1}$ is sometimes written as $\frac{a}{a+b}$ for easier memorization. $\endgroup$ – Herr K. Jul 31 at 17:40
  • $\begingroup$ That's the step I was forgetting, thank you! $\endgroup$ – Brennan Jul 31 at 19:03
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You need to maximize the utility function subject to the budget constraint. I invite you to do the full procedure with the Lagrangian and the first order conditions. However the shortcut as you mention consist of equalizing the Marginal Rate of Substitution to the ratio of prices (Since it is the condition of optimality). Then, from this expression you need to solve for x as a function of y (or the reverse) and plug the result in the budget constraint. Doing so you will be able to solve for the demand for good y. Then plug the demand function of y into the expression of x and you get the demand function for x.

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