Question

hi there! how did they get thst equation x1=am/p1? i dont know understand this paragraph. if someone could help that would be great!

Cobb-Douglas Preferences For the case of Cobb-Douglas preferences it is easier to look at the algebraic! form of the demand functions to see what the graphs will look like. If u(11,12) = rm-9, the Cobb-Douglas demand for good 1 has the form 2 = am/pi. For a fixed value of Pı, this is a linear function of m. Thus doubling m will double demand, tripling m will triple demand, and so on. In fact, multiplying m by any positive number t will just multiply demand by the same amount. The demand for good 2 is 12 = (1-a)m/P2, and this is also clearly linear. The fact that the demand functions for both goods are linear functions of income means that the income expansion paths will be straight lines through the origin, as depicted in Figure 6.6A. The Engel curve for good 1 will be a straight line with a slope of pi/a, as depicted in Figure 6.6B.

Answer 1

Use the fact that optimal choice has MRS = p1/p2

MRS = MUx1/MUx2 = αx1^(α-1)x2^(1-α) / (1-α)x1^(α)x2^(-α)

= (α/(1-α)) * x2/x1

Now we have MRS = p1/p2

(α/(1-α)) * x2/x1 = p1/p2

x2 = (p1/p2)*((1-α)/α)x1

Budget equation is M = x1p1 + x2p2

M = x1p1 + p2 * ((p1/p2)*((1-α)/α)x1)

M = x1p1 + p1(x1/α - x1)

M = x1p1 + p1x1/α - p1x1

M = p1x1/α

Hence x1 = αM/p1

This is the demand function for x1

Now x2 = (p1/p2)*((1-α)/α)x1

x2 = (p1/p2)*((1-α)/α) *  (αM/p1)

= (1-α)M/p2

This the demand function for x2

Both functions are linear for income which means doubling the income will double the demand for the two goods.