Question

1. Miguel and Jake run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's long-run production function is Q = 4K 0.75 0.25, where a is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. The weekly cost function for the paper plant is C = 10K + 2L, where C is the total weekly cost. a. What is the marginal product of labor? b. What is the marginal product of capital? c. Are there diminishing returns to capital? To Labor? What returns to scale exist for capital and labor combined (when both are allowed to change)? How did you determine your answer? d. What ratio of capital to labor minimizes Miguel and Jake's total costs? e. How much capital and labor will Miguel and Jake need to rent and hire in order to produce 1,000 reams of paper each week (based on the original cost function C= 10K + 2L)? f. How would your answer change to the previous question if the price of labor increased to $10 due to unions demanding higher wages and capital was subsidized leading, in turn, to its price decreasing to $2? g. How much will hiring these inputs cost them (based on your answerto letter e)?

Answer 1

0.25 0.25 to Thus, .75 0.75 0.2 0.25 0.25 uk D- 4*0.75 0.25 (a) MP od 10.75 0.25-1 0.75,-0.75 b) MPK = 00 4 (0.75) R0.75 -1 ,0.25 = OK 36-.25 L (a) from b , we can see that as k increases, MPK wro decrease. Thus, there are dininishing utuns to Capital rom a, we can see that as increases. MP will decrease ereare. there are diminishing returns to L. Q = 460.75 ,0.25 0 Cet katk and L=th where tol. 0.21 0.25 0.25 +0.1 So, d'= u(tk 20.75 (th,0.25 u "*°ns 2020:25 = 4º. 0.75 - = d'= tg Thus, there are constant returns to scale - as power of t ist is MRTS = (d) w. that where Ceast cost combination 0.75 +0.25 to.75 -0.75 - MRTS = MPL . O MPK 32-0.25 0.25 3,0.25 +0.75 37 W=2 ; 2=10 (from wst function) =* - }| E k with a hoone (Note Camslostnery subparts at a time. K so, a ž