Economics 200 Chapter 7 Inputs and Outputs: Marginal Products Production takes place when inputs are combined (typically under the direction of firm owners or managers) to produce some sort of output (purchased by consumers or other firms). Examples of inputs: Capital, Raw Materials, Labor, Land, Energy Definitions: Durable goods ? Those that last for more than one production period. Capital ? Durable goods which are produced specifically to be inputs in the production of other goods. Comprises any resource that generates income. Human capital ? The knowledge and skills we possess that aid in the process of production. Production Function ? A measure of the maximum amount of output that can be produced from a given combination of inputs. Example of Production Function: Assume there are 2 Inputs: Labor (L) and Capital (K) Q = f(L,K) Q = Output Inputs are both complements and substitutes Marginal Product ? Additional output that is created when an additional (one more) unit of input is employed. When we talk of the marginal product of an input, we hold the level of all other inputs constant. Calculus aside: Consider the production function: Q = f(L,K) MPL = ?f(L,K)/?L = fL(L,K) The marginal product of labor is just the partial derivative with respect to labor. MPK = ?f(L,K)/?K = fK(L,K) The marginal product of capital is just the partial derivative with respect to capital. Law of diminishing marginal product As one input of production is added to a fixed amount of other inputs, after some point, the marginal product of the variable input continually diminishes. Table 7-1 Consider a farm in a developing country where there are only 2 inputs, human labor and land. Land is your capital. Q = f(Labor, Land) = f(L,K) The output of the farm is food. Assume the amount of capital or land is fixed at a certain amount so the only input changing is labor. In other words, labor is added to a fixed parcel of land. Number of Laborers Total Product Marginal Product Average Product 1 3 3 3.00 2 8 5 4.00 3 15 7 5.00 4 21 6 5.25 5 26 5 5.20 6 30 4 5.00 7 33 3 4.71 8 35 2 4.38 9 36 1 4.00 10 36 0 3.60 11 35 -1 3.18 We have been discussing marginal product within a firm, this is diminishing returns at the intensive margin as opposed to diminishing returns at the extensive margin. Definitions: Diminishing Returns at the Extensive Margin: As greater final output is demanded, resources less specialized and less efficient than those already used must be utilized to produce the additional output. Examples: Farming (land), mining, professional athletes Diminishing Returns at the Intensive Margin: Diminishing marginal product within a firm. Units of input are of identical efficiency but their application to a fixed amount of other factors makes the addition to output (marginal product) created by the last unit of variable input employed less than the marginal products of the previous units Rents How are rents created? Rents represent the gains in employing resources in their current use versus the next best alternative. Definition: A payment in excess of what is necessary to call forth production of one good or service. Rents occur because of the inability to completely replicate (the most efficient) resources. 7.2 ? Profit Maximization and Factor Demand First, we will look at production where the land is privately owned, so the property owner is the decision-maker. The owner is the residual claimant on all rents generated on the land, and the owner has the incentive to figure out how to maximize the rents (profits). Table 7-2 Definitions: The value of the total product is the amount that consumers pay for the total product. It is the total product of an input multiplied by the per unit output price. The value of the marginal product (marginal product value) is the marginal product of an input multiplied by the per unit output price. Number of Laborers Value of Total Product Value of Marginal Product Value of Average Product Total Labor Cost Rent 1 2 3 4 5 6 7 8 9 10 11 Assume the per unit output price =$10 Assume the wage rate is $40 per worker per day We use the number provide in table 7-1 to solve for the table 7-2. As long as the value of the marginal product exceeds the input price, rents may be increased by using more of the input. Then in equilibrium, it must be the cast that: P * MPi = wi Output Price * Marginal Product of i = Price of Input i Calculus Aside: Max ? = p*Q ? wLL ? wKK Subject to Q = f(L,K) This can be rewritten substituting in the constrain for Q in the profit function. Max ? = p*f(L,K) ? wLL ? wKK Now you can take the derivative with respect to capital and then labor. ? ?/?k = p*? f/?k - wK = 0 then p*MPK = wK ? ?/?L = p*? f/?L ? wL = 0 then p*MPL = wL And we can set the results equal to each other through price giving the following: MPK / wK = MPL / wL and MPK / MPL = wK / wL The Effects of Property Rights on Resource Allocation We will now look at how different types of ownership effect the process of production. Private Ownership Under private ownership the rents are maximized. This occurs where the Value of the marginal product of labor (VMPL) is equal to the wage. How many workers will they hire in our example? At 6 workers, VMPL = Wage = $40 What is the rent? Rent is equal to $60. Common Property Rights or Ownership of the land No person owns the land and no individual has the right to exclude any others from the land. All workers on the land share equally in the output of the land. Or instead of wages, each person receives the value of the average product of the farm. Since there is no way to exclude additional workers, people will continue to join the farm until their share falls to what they could be earning elsewhere. When does this occur in our example? Set Wage = Value of the average product of labor This occurs at 9 workers. Why won?t a 10th worker join? The VAPL would be $36, which is less than the alternative wage of $40. What are the rents under common property? The rents are always completely dissipated under common property or rents equal zero. Socialist Cooperative Workers all have equal ownership shares of the land and split all output equally. Unlike the common property, the workers vote to decide how many workers there will be. So workers can exclude additional workers from joining them to work the land. Each worker would want to maximize their income. This occurs when the value of the average product is maximized. When does this occur in our example? It occurs at 4 workers where the VAPL = $52.50. What are the rents? The rents are equal to $50. Which option is economically efficient or leads to the exhaustion of mutual benefits? The key is the wage. When an individual decides to work on the farm, $40 of output necessarily disappears from somewhere else in the economy. The mutual benefits are exhausted if the total value of output in the combined farm and nonfarm economy is as great as possible. Under common property, what does the 7th, 8th, and 9th worker add to production? $30, $20, and $10 Why is this inefficient? Each of these workers could be producing $40 of output somewhere else. The total loss to society is equal to $60 which is exactly the maximum obtainable rent on the land. Why would this many people work there then? What is the wage if there are 8 workers? Wage = VAPL = $43.75 > Alternative Wage Does Private Ownership lead to the efficient outcome? A private owner seeks to maximize profits, but by doing this he reaches the efficient outcome where the mutual benefits are exhausted. This is a Pareto efficient allocation. The owner will continue to hire as long as the next person produces at least as much as the wage they are paid. What about a socialist cooperative? Why don?t they hire the 5th worker when this worker increases the production by $50 which is $10 more than the wage? Would decrease everyone?s earnings by $2.50. Similar logic follows for the 6th person. Under a socialist cooperative, each of the 4 workers earns $12.50 more than the alternative wage of $40. People who are excluded from the socialist cooperative would devote resources towards being included in a socialist cooperative because they would be able to make more than the alternative wage. Number of Laborers Value of Total Product Value of Marginal Product Value of Average Product Total Labor Cost Rent 1 $30 $30 $30.00 $40 -$10 2 $80 $50 $40.00 $80 $0 3 $150 $70 $50.00 $120 $30 4 $210 $60 $52.50 $160 $50 5 $260 $50 $52.00 $200 $60 6 $300 $40 $50.00 $240 $60 7 $330 $30 $47.14 $280 $50 8 $350 $20 $43.75 $320 $30 9 $360 $10 $40.00 $360 $0 10 $360 $0 $36.00 $400 -$40 11 $350 -$10 $31.82 $440 -$90 PAGE 1 PAGE 1
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