1) Calculating Returns to Scale

Transcription

1 CLAS Handout Returns to Scale Practice/Intuition for Profit Maximization This handout includes 3 sections: calculating returns to scale, visualizing returns to scale, and importance/application. The goal of the handout is to understand how to calculate RTS, what it means for productivity, and how it effects our demand for L and K. If you feel proficient in one section, feel free to skip it. I recommend the last section regardless of your proficiency. 1) Calculating Returns to Scale Compare to mf(l,k) Which is the same as comparing doubling 1 to doubling outputs What are the returns to scale for the following? A) F(L,K) = L 1/2 K 1/2 B) F(L,K) = LK C) F(L,K) = Min (2L, K) D) F(L,K) = Min (L 1/2, K) E) What is the exponent rule for returns to scale with Cobb-Douglas? (check the slides) 1 If we increase inputs by m, what happens to output? If m is 2, we are doubling our inputs. However, it could easily be 3, 4, etc. I say double (2 for m) throughout this handout because it is the easiest to talk about and explain, but it could easily be tripling, quadrupling, etc.

2 2) Visualizing Returns to Scale Our profit max equation is as follows Max π = P*Q -wl rk Where P*Q is our, wl and rk are our, L and K are our, and Q is our. A) Increasing returns to scale (IRTS) F(L,K)= L 2 P = 1 w= 6 r= 4 L=3, K= L=6, K=0 L=12, K=0 When I double, increase by than double. Therefore, when I double inputs, increase by double, while increase by more than double. Even if I make a loss on the first few units of labor, eventually I will make a profit. This is because revenue (P*Q) is increasing at a rate, while costs (wl + rk) are increasing at a rate. I. If there is no limit to how much Labor you can hire, and there is an endless demand for your output, how many workers would you hire? B) Constant returns to scale (CRTS) F(L,K)= 2k P = 2 w= 4 r= 3 L=0, K=2 L=0, K=4 L=0, K=8 When I double, increase by double. Therefore, when I double inputs, increase by double, and increases by double. The profit I made on the first unit will every time I double inputs. This is because revenue (P*Q) and costs (wl + rk) are increasing at a rate, so every time revenue and costs are doubled, are doubled.

3 I. If there is no limit on how much capital you can purchase, and there is an endless demand for your output, how much capital would you purchase? II. What would the profit/loss be if the P=1? How much capital would you want to purchase? Justify your answer. C) Decreasing Returns to Scale (DRTS) F(L,K)= L 1/2 P = 4 w= 2 r= 3 L=2, K=0 L=4, K=0 L=8, K=0 When I double, increase by than double. Therefore, when I double inputs, increase by double, and increases by less than double. If I continue to hire workers, eventually I will make a. This is because revenue is increasing at a rate while costs are increasing at a rate. I. Explain what is happening in the chart by using the classic too many cooks in the kitchen example. 3) Importance and Application In profit maximization, we are trying to determine the how much labor and capital will maximize profit. The majority of the time, DRTS means we will have a finite answer for (L,K). On the other hand, IRTS/CRTS usually means we will have an answer for (L,K) that is infinite or 0. There are a couple of reasons why DRTS does not guarantee a finite answer, and IRTS/CRTS does not guarantee an infinite or 0 answer. Short run problems Fun twists on certain functions

4 For the following, determine whether demand for L and K will be infinite, zero, or finite. You do not need to do any math for the following problems, but you should consider 4 things: 1. Returns to scale 2. Type of function 3. Restriction on K due to short run 4. The relationship between price, wage, and rate (especially in CRTS functions) A) Long run P=1 w=10 r=6 F(L,K) = L 2 K B) Long run P=10 w=4 r=6 F(L,K) = L 1/4 K 1/2 C) Short Run P=8 w=4 r=2 K=4 F(L,K) = Min (2L, K) L= D) Long run P=12 w=4 r=2 F(L,K) = Min (L, K 1/2 ) E) Long run P=3 w=10 r=2 F(L,K) = L 2 + K F) Short run P=2 w=6 r=15 K=3 F(L,K) = 2L + 10K L= G) Long run P=7 w=1 r=2 F(L,K) = Max (L 1/2, K) H) Long run P=4 w=3 r=1 F(L,K) = L + K 1/2 (Answer key on the next page)

5 A) L= Infinite, K= infinite B) finite, finite C) finite D) finite, finite E) infinite, 0 F) 0 G) 0, infinite H) infinite, finite