Introduction to Production PDF

9 Introduction to Production So far in the class we have focused entirely on the demand side of economics. We looked at how consumers make decisions about which goods to buy and how these decisions are based on their preferences, the prices they face, and their income. But wher...

9 Introduction to Production So far in the class we have focused entirely on the demand side of economics. We looked at how consumers make decisions about which goods to buy and how these decisions are based on their preferences, the prices they face, and their income. But where do these goods come from? It is time to turn our attention to the other side of the economy - the supply side. In these notes, we will introduce a simple theory of production. Concepts Covered Production Functions Marginal Product Marginal Rate of Technical Substitution Isoquants 9.1 Production Functions We will assume that each firm in the economy produces a single type of good. In order to produce a certain amount of output, the firm needs to use two inputs: labor and capital. Labor represents all of the workers that the firm employs and capital includes buildings, machines, equipment, and other tools the firm uses in production. The firm combines their capital (K ) and labor (L) to produce a quantity Q according to a function called a production function. Q = F (K, L) Although in some sense we can think of a production function as being similar to the utility function we had in the consumer’s problem, but unlike utility, the quantity that a firm produces has a physical meaning. While utility of 100 has no meaning on its own, producing 100 apples does. In other words, the production function that we assign to a firm is not an ordinal function. We cannot transform it freely as we could utility without changing its meaning. Returns to Scale An important property of production functions is called returns to scale. Imagine that the firm doubled both capital and labor. How much more output would it be able to produce? We could say that the firm has increasing returns to scale if its output more than doubles as a result of this change, constant returns to scale if it exactly doubles, and decreasing returns if it less than doubles (in general we will assume the production function is increasing in both inputs so it will never go down if labor or capital go up). 1  More formally, this relationship needs to hold for scaling up capital and labor by any constant. If F (K, L) = Q, we say a function has Increasing returns to scale if F (zK, zL) > zQ for all z > 1 Constant returns to scale if F (zK, zL) = zQ for all z > 1 Decreasing returns to scale if F (zK, zL) < zQ for all z > 1 Although there are possible arguments for any of these production functions being useful for different situations, constant returns tends to be the most common. The assumption of constant returns makes some things easier mathematically, but it also has intuitive appeal. If a firm just creates an exact copy of itself, it makes sense it would be able to produce twice as much. Examples Imagine a firm has a production function Q = F (K, L) = K 1/2 L1/2 This function should look familiar. It is our Cobb-Douglas function (with α = 1/2) applied to production. Let’s plug in some numbers to see how the production function works. How much would the firm produce if 1. L = 100, K = 100 2. L = 200, K = 200 3. L = 300, K = 300 Based on your answers, is this production function increasing, decreasing, or constant returns to scale? Are the following functions increasing, decreasing or constant returns to scale? 1. F (K, L) = 4K 1/3 L2/3 3. F (K, L) = K 1/4 L1/4 5. F (K, L) = L1/2 + K 1/2 2. F (K, L) = 10K 2 L2 4. F (K, L) = 2L 6. F (K, L) = 3L + 6K (1: Constant, 2: Increasing, 3: Decreasing, 4: Constant, 5: Decreasing, 6: Constant) 9.2 Marginal Product Notice that when we checked for returns to scale above, we scaled both inputs. What if we looked at increasing each input, holding the other one fixed? In mathematical terms, this exercise means taking the partial derivative of the production function. We call these partial derivatives marginal products. In particular, we have ∂F (K,L) Marginal Product of Capital (MPK): ∂K ∂F (K,L) Marginal Product of Labor (MPL): ∂L 2  Heuristically we can think of the marginal product as being how much more we can produce when we hire one more worker (MPL) or use one more unit of capital (MPK). With the Cobb- Douglas example above, we would have ∂F (K, L) 1 MP K = = K −1/2 L1/2 ∂K 2 ∂F (K, L) 1 MP L = = K 1/2 L−1/2 ∂L 2 Let’s analyze the function above. Notice that as we increase capital, MPK falls and as we increase labor, MPK rises. This result means that if you hold labor fixed, adding additional units of capital will have less and less effect on output. To see the intuition, imagine you only have 10 workers. If you give these workers one computer (a type of capital) to aid their work, it will probably greatly increase their output. But if you then add another computer, it probably won’t add as much to production. If you add 1000 more computers, the 1000th probably won’t add very much of anything. The marginal product of capital falls as the amount of capital relative to labor increases. The same logic applies to labor. The marginal product of labor decreases as more workers are added, but increases as more capital is added. Here the intuition is that if you only have 10 computers, the first 10 workers will be very productive. But as more workers are added, they won’t be able to effectively make use of the small amount of capital, so the productivity of each additional worker falls as more are added. To summarize, even though the original function has constant returns to scale, it has dimin- ishing returns to each input when we hold the other one fixed. If we increase both capital and labor at the same time, production scales up at a constant rate. If we only increase capital, or only increase labor, the effects on production will be diminishing as we increase that factor. The graphs below show production as a function of capital (holding labor fixed) and the marginal product of capital. A similar graph would apply if we replaced K with L. 5 1 4 0.8 3 0.6 Q MP K 2 0.4 1 0.2 0 0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 K K 3 Marginal Rate of Technical Substitution In our study of consumer theory, one of the most important concepts was the marginal rate of substitution (MRS). In producer theory, we have a related concept called the marginal rate of technical substitution (MRTS). You might be able to guess how we define this concept. We will say that the MRTS of L for K is given by ∂F (K,L) MP L ∂L M RT S = = ∂F (K,L) MP K ∂K Like the MRS, which told us the relative benefits of two goods, the MRTS tells us the relative benefit of our two inputs, capital and labor. Let’s take an example where the MRTS is constant. Assume our production function is Q = F (K, L) = 2L + K Using the definition of MRTS above, we get ∂F (K,L) ∂L M RT S = ∂F (K,L) =2 ∂K What does this value for MRTS mean? It tells us that labor is twice as productive as capital. In other words, if we wanted to keep production constant, for every unit of labor we lose, we would have to add 2 units of capital. Alternatively for every unit of capital we lose, we would have to add 1/2 a unit of labor. Isoquants Just as we had an analogue to the MRS, we will also have a closely related concept to an indifference curve for the producer side. We will call this concept an isoquant, but the idea is almost exactly the same as an indifference curve. In the consumer choice version, we found all combinations of x and y that gave the same utility. Here we will find all combinations of K and L that give the same output Q. Using our example above, let’s fix a level of output for Q, Q = Q̄, so that we have Q̄ = 2L + K Now solve for K to get K = Q̄ − 2L We can then plug in a value of Q̄ and plot the curve of L against K. Just as the slope of an indifference curve was the (negative) MRS, the slope of an isoquant is the (negative) MRTS. 4  The figure below plots a few isoquants for this utility function. 10 8 6 K 4 2 0 0 2 4 6 8 10 12 14 16 L In this case, isoquants were linear, but in general they can have any shape just like indifference curves. Using the Cobb-Douglas example, Q̄2 Q̄ = K 1/2 L1/2 =⇒ K = L And plotting this function for difference values gives us a familiar looking picture. 10 8 6 K 4 Q̄ = 4 2 Q̄ = 3 Q̄ = 2 0 Q̄ = 1 0 1 2 3 4 5 L One final point is that we have the same condition on the convexity of isoquants as we had for indifference curves. If the MRTS is decreasing in the ratio L/K then the isoquants will be convex (as they are above). In the next notes, we will use this framework to discuss a firm’s optimal choice of K and L. 5