Question

Bill sells t-shirts. He has 10 in stock and is currently cashless. His preference as given by u(x1,x2)=10ln x1+x2 which implies that his mrs=10/x1 Find an expression for Bill's willingness to supply shirts at a given price, p. (Be sure it is an expression for how many shirts Bill sells at price p and not an expression for how many he keeps.)

Answer 1

Bill has 10 t-shirts in his stock.

His utility function is given as

u(x1, x2) = 10.ln(x1) + x2

Marginal Rate of Substitution is

MRS = 10/x1

Now, price of t-shirts (x1) is p1 = p and price of all other goods (x2) is p2 = 1.

When Bill maximizes utility, then,

MRS = p1/p2

or, 10/x1 = p/1

or, x1* = 10/p........(1)

This is Bill's demand function for t-shirts.

Now, Bill has 10 t-shirts in stock. Hence, he will consume (10/p) t-shirts to maximize his utility and then he will supply the rest of the t-shirts.

Hence, Bill's supply of t-shirts is

S(x1) = 10 - x1* = 10 - 10/p

or, S(x1) = 10.(p - 1)/p

This is Bill's willingness to supply function.

Hope the solution is clear to you my friend.