Topic I. Consumer Theory (3) (3) e(p,u) is concave in p, that is, for 0 te(p,u) + (1-t)e(p',u). Proof: Let (p,x) and (p',x') be two expenditure-minimizing price- consumption combinations for a given utility level u. Let p" = tp+ (1-t)p' for te [0,1]. e(p",u) = p" . x" = tp. x" + (1-t)p'.x" Since x is the expenditure-minimizing consumption bundle at p, and x' is the expenditure-minimizing consumption bundle at p' , p . x" 2 p . x and p' . x" 2 p' . x' . Thus, p" . x" = tp . x" + (1 -t)p' .x" > tp . x + (1 -t)p' . x' = te(p, u) + (1 - t)e(p', u) e(p,u) is concave in p. 7