Economics
301


Last updated: January 25, 2000 Note: These notes are preliminary and incomplete and they are not guaranteed to be free of errors. Please let me know if you find typos or other errors. The Production FunctionThe aggregate production function for the economy is assumed to take the form Y = A·F(K, N)where
Assumptions concerning the shape of F(K, N)
The graph above shows the production
function as a function of N holding A and K fixed.
The graph above shows the production function as a function of K holding A and N fixed. Example: CobbDouglas constant returns to scale production functionY = A·K^{a}N^{1a} , 0 < a < 1 where
Note: Constant returns to scale means that if we double all inputs to the production function then output exactly doubles. To see that the CobbDouglas exhibits this property, note that A·(2K)^{a}(2N)^{1a} = A·2^{a}2^{1a}K^{a}N^{1a} = 2A·K^{a}N^{1a} = 2Y For US data, from the national income and product accounts, we know that labor's share of income is about 0.7 which implies that for the US 1a = 0.7 and a = 0.3. A Mediumrun AssumptionThe production function shows how real GDP, Y, is determined over some fixed time period like a quarter or a year. Over this time period it is assumed that capital stock, K, and the level of overall productivity, A, is held fixed whereas it is assumed that the amount of labor can vary. That is, over the assumed time period, firms in the economy cannot easily alter the amount of capital they use (i.e. buildings, equipment etc.) but they can easily vary the amount of labor that they hire. Similarly, it is assumed that productivity does not change very quickly. Given, these assumptions we can represent the production function as Y = A_{0}F(K_{0}, N) where the subscript "0" indicates that the level of productivity is fixed at the amount A_{0} and the level of capital is fixed at the amount K_{0}. The unsubscripted variable, N, is allowed to vary. Moving Along vs. Shifting CurvesMoving along the production functionAs N increases from N_{0} to N_{1} output increases from Y_{0} to Y_{1} as we move along the production function. Shifting the production function: An increase in productivityWhen the index of productivity increases from A_{0} to A_{1}, holding everything else fixed, the production function shifts up. Then for a given amount of labor, N_{0}, the amount of output produced in the economy increases from Y_{0} to Y_{1}. Shifting the production function: An increase in the stock of capitalWhen the capital stock increases from K_{0} to K_{1}, holding everything else fixed, the production function shifts up. Then for a given amount of labor, N_{0}, the amount of output produced in the economy increases from Y_{0} to Y_{1}.
