RETURNS TO SCALE. Right.

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1 CHAPTER 7, PART C, KWAN CHOI RETURNS TO SCALE When the price of a good rises, a producer wants to increase output. In the SR, only the variable input can be increased. In the long run, even the quantity of the SR fixed inputs can be changed. Why study returns to scale? For instance, in response to a decline in demand, GM and Toyota are forced to reduce their outputs. They will have to lay off many workers. Of course, they will have to move along their expansion path. If the expansion path is a straight ray from the origin, what does it really mean? In response to price changes, a firm needs to increase or decrease output. It needs to know inputs should increase or decrease proportionately. That would be a short run response. In the SR, a firm can change the output level by adjusting the variable input. However, in the LR, all inputs are variable inputs, and firms need to know how it should vary the input levels. The expansion path may have an odd shape, but for many industries with standardized production process, the expansion path may be a straight line, or a ray from the origin. As output increases, the firm does not change the input mix or the capital intensity (K/L). Otherwise, as demand increases, the firm increases output, but the input mix will change. If a small hospital uses on MRI scanner, as demand doubles, it buys 2 scanners and double the staff. 1

2 Even if the expansion path is a ray from the origin, it does not mean that output will rise proportionately as input levels increase. Thus, it is important to understand the returns to scale. In other words, how much will output increase if all inputs increase proportionately? Let the production function be y= F( K, L). If all inputs double, then output will be F(2 K,2 L ). The question is then whether the output will more than double, exactly double or less than double. If the output exactly doubles, we say that the production function exhibits constant returns to scale(crs). What if the output does not exactly double? We can formalize the case by choosing a scalar multiple λ (which could be 2). Specifically, the production function exhibits CRS if F( λl, λk) = λf( K, L). It exhibits decreasing returns to scale (DRS) if F( λl, λk) < λf( K, L), and increasing returns to scale (IRS) if F( λl, λk) > λf( K, L). If all inputs double, and the output less than doubles, the industry shows decreasing returns to scale (DRS). If the output more than doubles, it exhibits increasing returns to scale (IRS). 2

3 Sure. Can you give me a practice problem? Consider the Cobb-Douglas production function, y = AL α K β, α + β = 1. (1) If all inputs were to double, then the new output will be α y' = A(2 L) (2 K) α β α β = 2 + ial K = 2 y. β (2) Thus, with this Cobb-Douglas production function, when all inputs double, the output exactly doubles, and hence it exhibits CRS. Jawohl! In this case, we also say that the production function is homogenous of degree 1, or linearly homogeneous. More generally, a production function is homogeneous of degree r, if r F( λl, λk) = λ if( L, K). (3) For instance, if λ = 2, r F(2 L,2 K) = 2 i F( L, K). Exactly. The parameter r represents the degree of homogeneity. 3

4 Practice Problem A company produces yo-yos at its factory in Shenzhen (See Fantasia 2000). Wages are $12 per hour and the rent for yo-yo making machine is for $4 per hour. The production function is = 40. (4) q K L where q = boxes of yo-yos per week, K = hours of machine use, and L = hours of labor. Therefore, L = 30, and MP K L K = 10. (5) MP K L Determine the cost minimizing capital-labor ratio (expansion path). To minimize cost, the firm chooses K and L so that w / r = MP / MP. Thus, K L K = or 3= 3, or K = L K L L Thus, the expansion path is given by K = L. If the wage rate were $16 per hour, then the 4 wage/rent ration would be 16/12 = 3K/L, and the expansion path would be K = L. Thus, the 3 expansion path becomes steeper as the wage rate increases. This indicates that as the wage rate rises, the firm uses the cheaper input, capital, more intensively. L K Assume that the desired output level is 640. How much K and L should be used? Substitute K = L into the production function to get q K K K = 40 = 40, or K = 16 = L. What is the minimized cost? $ $4 16 = $256. What is the unit cost? $256 $ = 4

5 Does the industry have increasing, constant, or decreasing returns to scale? When the exponents in a Cobb-Douglas production function add up to 1, the production function exhibits CRS. If the exponents add up to less than one, it has decreasing returns to scale. In this case, α + β = 1. Does the unit cost change as output increases? No, because of CRS, unit cost is constant and does not depend on the output level. In the short run, the firm cannot change capital but can vary only labor. Suppose the firm desires to double its production of yo-yos. How much labor is required? (16) L 40 2 L, 3/4 = = i or 16 = L. 4/3 Thus16 = L. April 29, 2009, BW Why Indians Are So Thrilled by Tata's Nano Car ownership in India brings feelings of pride and freedom, which is why 350,000 people have entered a lottery to buy a Tata Nano 5