ECO2003P – Production PDF

ECO2003 – Intermediate Microeconomics Production in the short run Geraint Van der Rede [email protected] 1  An Introduction to Production We generally think about production as “raw materials being turned into final goods”...

ECO2003 – Intermediate Microeconomics Production in the short run Geraint Van der Rede [email protected] 1  An Introduction to Production We generally think about production as “raw materials being turned into final goods” We see production as a tangible thing, but it is so much more than this! Production in economics is defined as “any activity that creates present or future utility” or “the process of transforming inputs into outputs” So, production could be anything from actually making goods, to telling a joke, to teaching. 2 The Production Function It makes more sense to think of the production function in the context of the second definition of production. The production function is the relationship by which inputs are combined to produce outputs. Inputs = generally factors of production e.g. land, labour, capital, knowledge, technology Outputs = the objects that provide us with utility 3 The Production Function Think of this like a recipe … but instead of eggs and flour and butter, we use capital and labour Obviously, capital and labour by themselves are not enough to make output – you need raw materials as well. You can’t just make bread out of bakers and ovens – you need flour, yeast, salt, etc. But for simplicity of analysis, we will not model raw materials into our analysis separately, but there is no reason why you can’t include these intermediate products in the model. 4 The Production Function 2(2) (2)=8 2(3) (3)=18 As an example, consider the production function given by: Q=2KL K 1 2 3 4 How much output do you get 1 2 4 6 8 when you combine 4 machine- hours with 5 person-hours? 2 4 8 12 16 L How many machine-hours would 3 6 12 18 24 you need to create 60 units of output with 6 person-hours? 4 8 16 24 32 5  Fixed and Variable Inputs Different factors take different lengths of time to change. For example, does it take the same amount of time to buy and set up a new factory as it does to hire a new worker? This introduces the idea of the short run and the long run VS 6 Fixed and Variable Inputs Short run: The longest period of time during which at least one of the inputs cannot be varied Long run: The shortest period of time required to alter the amounts of all inputs in the process If the input can be changed in the short run, then it is a variable input. If the input cannot be changed in the short run (or if it can, it costs an enormous amount of money), then it is considered a fixed input. In the long run, all inputs become variable. In the production of good grades for this course, what would the inputs be? Menti.com - 530646 7  Production in the Short Run In the short run we assume that capital is a fixed input. In other words, we say that K is fixed at some value of K0. Consider the example from earlier: Q = 2KL In the short run, we might only have 1 machine, so K=1 and the production function simplifies to Q = 2(1)L = 2L Similarly, if another company has 3 machines, then their production function becomes Q = 2(3)L = 6L 8 Production in the Short Run Let’s do an example: Draw the short run production function that corresponds to where capital is fixed at K0 = 4. 𝐹 ( 4 , 𝐿 )= √ 4 𝐿=2 √ 𝐿 Output Labour 9 Production in the Short Run As you saw in the example, we aren’t limited to straight line production functions – in fact, one of the most common production function shapes is non- linear. It still represents 3 general properties of short run production functions: 1. The production function should pass through the origin 2. Initially, adding extra units of variable input should increase output at an increasing rate 3. After some point, we should see extra units of the variable input increasing output by smaller and smaller increments 10  Production in the Short Run Note how each extra unit of labour gives us less and less extra output after L=4, and output decreases after L=8. This is the law of diminishing marginal returns: If other inputs are fixed, the increase in output from an increase in the variable inputs must eventually decline 11 Production in the Short Run Some things to think about: Why might you start off by seeing increasing levels of the variable input increasing output at an increasing rate? Maybe there is something to do with benefits from division of tasks – when you’re making sandwiches, maybe having two people instead of one person means you get specialisation. Does the concept of diminishing marginal returns make sense in reality? Do we see it in practice? It does seem to make sense. Think about our group of sandwich makers. Does adding a third person really make such a difference to our total production? What about a fourth person? At what point do we see everyone getting distracted and sitting around drinking tea? 12  Technology in the Production Function In 1798, Thomas Malthus said that the law of diminishing marginal returns basically meant that because land is fixed, having more farmworkers would ultimately add less and less to food production and we would all starve to death. He predicted that average food consumption per capita would be driven down to starvation level. But he didn’t account for technological advances, which might not spell doom and gloom for the human race. BUT we still saw diminishing returns – it’s just that the growth in technology outstripped the effect of diminishing returns. Unfortunately, the logic of Malthus’ prediction still holds: if the population continues to grow, the limits on the amount of land we have will spell food shortages. Technology is embedded in the construction of the production function. How does technological growth impact the production function? 13 Technology in the Production Function Technological advancement will shift our production function upwards. Why? Basically, at the same level of variable input, we can produce more because of technological advances. Think about how much 10 workers could harvest by hand vs. with combine harvesters. 14 Total, Marginal and Average Products We are now moving on to define some important concepts in the theory of production: Total product curve: A curve showing the total amount of output as a function of the amount of variable input (think about this as the graph of the production function) Average product: Total product divided by the number of units of the variable input used in the production process. Marginal product: Change in total product due to a one-unit change in the variable input. 15  Marginal Product Geometrically, marginal product is the slope of the total product curve. Algebraically, we can calculate the slope in one of two ways: Discrete MP: Continuous MP: Marginal Product reaches its maximum at the point of inflection of the TP curve. 16 Marginal Product Q=TP Continuous MP – the slope of the tangent line at any given point Discrete MP – the slope of the line joining two points on the TP curve L 17 Average Product Graphically, Average Product Q=TP (AP) is the slope of the ray from the origin to any point on the TP curve. Algebraically: We can see that AP begins by increasing, and then starts decreasing L 18 Average Product and Marginal Product The basic idea is this: If you have MP>AP, then your AP curve is increasing. If you have MPAP For L > 6, the slope of the AP line is greater than the slope of the TP curve  AP>MP At small values of L, slope of ray and slope of TP are indistinguishable, and flat  MP=AP=0 At L = 6, the slope of AP must be the greatest it can be. It also happens to be tangent to TP at that point  AP=MP at its maximum 20 Practical Considerations of MP and AP Suppose you have a fleet of fishing boats and you have 2 boats on each end of a lake. You get 100kg of fish from each boat sent to the East end of the lake, and 120kg of fish from each boat sent to the West side of the lake. Should you change your allocation of boats? Not necessarily – it depends on what gains you get from sending the third boat to the West side. The question hinges on marginal product, but most people seem to answer this question by thinking about average product. 21 Practical Considerations of MP and AP Number East End West End of Boats TP AP MP TP AP MP 0 0 0 0 0 1 100 100 100 130 130 130 2 200 100 100 240 120 110 3 300 100 100 330 110 90 4 400 100 100 400 100 70 Here, it makes no sense to send the third boat to the West End as the marginal gain from doing so is not worth it. 22 Practical Considerations of MP and AP Number East End West End of Boats TP AP MP TP AP MP 0 0 0 0 0 1 100 100 100 120 120 120 2 200 100 100 240 120 120 3 300 100 100 360 120 120 4 400 100 100 480 120 120 Here, it does make sense to send the third (and fourth) boat to the West End as the marginal gain from doing so outstrips what they could earn you in the East End 23 Practical Considerations of MP and AP Basically, you have to consider how to best allocate your resources – how do you do it? Economically, the most efficient way of allocating your resource is to allocate the next unit of input to the process in which its marginal product is highest. If you have perfectly divisible inputs, and for processes where the MP of one isn’t always higher than the MP of the other, you should allocate this input between production processes until the marginal product is equal across both processes. 24  Example Given the following function for baking loaves of bread To maximise output, we need to hire 7 labourers 2. 215 Where L= hours spent baking loaves The maximum output produced when the firm hires 7 labourers 1. Where is Q maximised? 3. 2. What is this maximum value of This can be used to find the marginal product of each Q? respective worker. To find the marginal product of the 5 worker, th simply substitute 5 in for L. 3. Write an expression for MPL. 4. Write an expression for APL. 4. 5. What is the MPL and APL when We could use this function to find the average product L=2? of labour for any number of workers 5. 25  ECO2003 – Intermediate Microeconomics Production in the long run Geraint Van der Rede [email protected] 26 Production in the Long Run Recall: the long run is the shortest period of time it takes for all inputs to become variable. So … Now, everything can change. We no longer assume K to be fixed, so we can vary the level of K. So, how do you draw the production function Q = 2KL ? 27 Production in the Long Run How do we graph production in the long run? We draw what are called isoquants Isoquant – “iso” means same; “Quant” for quantity. So, we choose a value for Q, say Q0, and plot our graph in L-K space. Does this remind you of something? Maybe how we represent 3D utility functions with indifference curves. 28  Production in the Long Run So, for Q = 2KL: Assume Q=Q0=16  Assume Q=Q1=32  Assume Q=2C (Arbitrary constant)  This is a rectangular hyperbola defined for positive K, L and C. So, you choose an output level, and then solve for K as a function of L. Remember that numerical labels on isoquants have meaning here. 29  Marginal Rate of Technical Substitution In consumer theory, we asked ourselves “how much of x would we substitute for y to remain at the same level of utility?” In the world of the firm, why don’t we ask an analogous question: “How much labour could we substitute for capital in order to continue producing the level of output we were before?” The answer to this question is the Marginal Rate of Technical Substitution (MRTS) 30  Marginal Rate of Technical Substitution If we were to give up of capital, then to remain on the same isoquant, we need to hire units of labour. So, MRTS is the absolute value (positive value) of the slope of the isoquant. We can define this at the point A as If MRTS=5, this means I am substituting 5 units of capital for 1 unit of labour. 31  Marginal Rate of Technical Substitution So, we know that if we give up some of one input (K), we have to increase the amount of the other input (L) to stay on the same isoquant. Also, we can work out that the loss from the decreased K must offset the gain from the increased L, in production terms. How much do we lose from decreasing K? How much do we gain from increasing L? So, we know these terms are equal: Rearranging, we get: Thus, we can write 32  Marginal Rate of Technical Substitution The MRTS effectively tells us how a firm is willing to substitute between inputs (for the purposes of this course between K and L) Note that if you have a lot of a particular input, you would be willing to give up a lot of it for a little bit of the other one. Note how this supports the diminishing marginal returns assumption from earlier. 33 Returns to Scale Returns to scale are a long run phenomenon This is because they require a change in all of the inputs (this means all the inputs need to be variable  long run) Returns to scale have to do with how production responds to a proportional increase in all inputs. Increasing Returns to Scale – a proportional increase in all inputs leads to a more than proportional increase in output Constant Returns to Scale - proportional increase in all inputs leads to an equally proportional increase in output Decreasing Returns to Scale – a proportional increase in all inputs leads to a less than proportional increase in output 34 Returns to Scale (Mathematically) Assume that we increase all of our inputs by a factor of c. Then, the three types of returns are characterised by: Increasing Returns to Scale Constant Returns to Scale Decreasing Returns to Scale 35 Diminishing vs. Decreasing Returns It is very important to remember the distinction between diminishing returns and decreasing returns: Diminishing marginal returns: a short run concept which talks about how output changes due to an increase in a specific factor. Decreasing returns to scale: a long run concept which talks about how an increase in every factor affects total production. Firms with increasing, decreasing or constant returns to scale will probably still exhibit diminishing returns. 36  Returns to Scale vs. Diminishing Returns Your returns to scale depend crucially on the production function you are dealing with. It is possible for one production function to exhibit increasing returns to scale at some point, constant returns to scale at others, and even decreasing returns to scale. BUT NOTE: The function shows diminishing returns at all levels of production – this can be seen from the curved isoquants In fact, diminishing returns is a pretty universal characteristic of production functions 37 Types of Production Functions There are 3 basic types of production functions you will come across: 1.Cobb-Douglas function (this is the most commonly used function) Also, If as well, then diminishing marginal returns to that factor. If , then increasing marginal returns to that factor. 2.Leontief (or fixed proportions) production function This is the function used when the inputs are perfect complements 3.Straight line production function This is the function used when the inputs are perfect substitutes 38 The Leontief (Fixed Proportions) Production Function Example: MRTS does not exist Assume K=1 and L=1. MRTS=/undefined Thus, Q = min{2(1), 3(1)} = min{2, 3} = 2 Does it help us at all if K increases and L doesn’t? We need K and L to increase in the same proportion to increase output. These inputs are thus perfect complements. MRTS=0 The line of efficient combinations of inputs lies along the line where 2K=3L. Solving out gives that 39 The Straight Line Production Function Example Shell K and L are related in a linear way, and when you solve for an isoquant, you will get the equation of a straight line: This is a relationship between perfect substitutes. The MRTS is constant at all levels of inputs – exactly because of the straight line relationship. Examples: Whether you get petrol from Shell Caltex or Caltex, you’ll still be able to drive the same distance 40 The Cobb-Douglas Production Function Let’s look at this function in terms of returns 1 1 to scale. 2 2 𝑄= 𝐾 𝐿 Let’s increase all of our inputs by a factor of c: So, our different types of returns depend on the parameters and If then increasing returns to scale f then constant returns to scale f then decreasing returns to scale 41 Cobb-Douglas and MRTS How do we calculate the MRTS for a Cobb-Douglas production function? Well, we make use of the formula So, now assume we have a Cobb-Douglas function in its most general form: To derive the MRTS, we need MPL and MPK: By dividing these out, we get: 42