Question

8.13. A firm produces a product with labor and capital. Its production function is described by Q = L + K. The marginal products associated with this production function are MPL = 1 and MPK = 1. Let w= 1 and r = 1 be the prices of labor and capital, respectively. a) Find the equation for the firm's long-run total cost curve as a function of quantity Q when the prices labor and capital are w = 1 and r = 1. b) Find the solution to the firm's short-run cost minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5), and w= 1 and r = 1. Derive the equation for the firm's short-run total cost curve as a function of quantity Q and graph it together with the long-run total cost curve. c) How do the graphs of the short-run and long-run total cost curves change when w = 1 and r = 2? d) How do the graphs of the short-run and long-run total cost curves change when w = 2 and r = 1?

Answer 1

production function : Q=L+k ; w=r=1 a) At optimal : meln 11-111 1. So, any combination of 24k can be used for produchun of lo such that it selfies production function, So, for producing a output eithe [Q=44k] Cost = wht rk = Ltk [.." width=1] wst = al b) K = 5 (capital fixed at 5 units) ducing a >5 a= Ltk a-K= 2 (&-5-1) So; cost = oltrñ=1(4-5)+5 1. [Cost = ca ] when ass - cost=s for a=s lost AB = short yun Total cost Long run I total cost O A z

w 1 4 r=a The meal > I> So, only Labor will be employed ; Kro L L=0 I cost = ww L=10=07 Longrun cost 1 [cost=a R = 5 fixed capital) fo., for a>5 only labor will be used . a=lta Q-K= L - LQ-SEL] 1 lost Short am lost = whtrk=1(0-5)+2 (5) [ short non cost eats i for ass - cost for this Br /A LOA2 Long run cost | BC= short run cost / a Coutout)

d w=2 & 8=1 The Top 1 mpk 1. 114/21 to only K will be used for production. L=0 ſkace] So, long run cost = rk= 1.= k=5 (fixed in short ron) a= ktk for ass a-kak (a-5=k for a75 lost a ok t* )=r(a-sts) Short I cost=1- a 7 for a35 Tun so. figure will be similar to part 1b)