# Cobb Douglas, Budget Line, Demand function question

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 $$\frac{ay}{bx}= \frac{p_x}{p_y}$$

• 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. – Herr K. Jul 30 at 18:26

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.
• Note that $(1+\frac{b}{a})^{-1}$ is sometimes written as $\frac{a}{a+b}$ for easier memorization. – Herr K. Jul 31 at 17:40