Graphical Evidence of MP < AP on a Concave Function
A visual inspection of a concave production function provides evidence for the relationship between marginal and average product. For instance, at a given point P, the tangent line, representing the marginal product, is visibly less steep than the ray from the origin to P, which represents the average product. This graphical comparison demonstrates that the marginal product is less than the average product when the average product is diminishing.
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Marginal Product of Labour
The Condition for a Diminishing Average Product
Graphical Evidence of MP < AP on a Concave Function
Consider a standard production function graphed with the quantity of an input on the horizontal axis and the quantity of output on the vertical axis. At a specific point 'P' on this function's curve, a straight line (Line 1) is drawn tangent to the curve at that point. A second straight line (Line 2) is drawn from the origin (0,0) directly to point 'P'. Based on the geometric properties of this graph, what do the slopes of these two lines represent?
On a graph of a production function, with input quantity on the horizontal axis and output quantity on the vertical axis, consider a specific point on the curve. If the line tangent to the curve at this point is steeper than the line drawn from the origin (0,0) to this same point, it implies that the addition of one more unit of input will cause the average output per unit of input to increase.
On a standard production function graph, which plots total output (vertical axis) against the quantity of a single variable input (horizontal axis), match each graphical feature with the economic concept it represents.
Analyzing Production Efficiency from a Graph
Analyzing Productivity from Graphical Slopes
Analyzing the Relationship Between Marginal and Average Productivity
A firm is operating at a point on its production function where the output gained from adding one more unit of a variable input is less than the current average output per unit of that input. On a standard graph of this production function (with input on the horizontal axis and output on the vertical axis), what must be true about the geometry at this specific point of operation?
On a graph of a production function where total output is plotted against the quantity of a single variable input, the point where the average output per unit of input is at its maximum occurs where the slope of the line tangent to the curve is equal to the slope of the line drawn from the origin to that same point.
Evolution of Productivity on a Production Curve
A firm's production process is represented by a typical production function graph (with input on the horizontal axis and output on the vertical axis) that is initially convex and then becomes concave. Consider the progression of production as the quantity of the variable input increases. Arrange the following statements, which describe the geometric relationship between the tangent line and the ray from the origin at a point on the curve, into the correct sequence that reflects this progression.
Learn After
Consider a production function graph where output is on the vertical axis and a single variable input is on the horizontal axis. The function is shaped as a curve that starts at the origin and increases at a decreasing rate (i.e., it is concave). At a specific point 'P' on this curve (not at the origin), two lines are drawn: Line A is tangent to the curve at point P, and Line B is a straight line drawn from the origin (0,0) to point P. Based on the geometric properties of these lines on such a curve, what is the relationship between their slopes?
Analyzing Production Efficiency from a Graph
Consider a production function graphed with output on the vertical axis and a single variable input on the horizontal axis. The function starts at the origin and is concave (i.e., it increases at a decreasing rate). For any given level of input (other than zero), the slope of a line drawn from the origin to the corresponding point on the production curve is greater than the slope of the line tangent to the curve at that same point.
Explaining Slopes on a Concave Production Curve
Evaluating a Consultant's Production Advice
Consider a production process where output (on the vertical axis) increases at a decreasing rate as a single input (on the horizontal axis) increases. For any given point on the production curve (other than the origin), match the economic concept to its correct graphical description.
A firm's production process is represented by a concave curve on a graph with output on the vertical axis and a single input on the horizontal axis. At the current operating level, the firm calculates that the output per unit of input is 25 units, and that adding one more unit of input would increase total output by 20 units. This numerical relationship is visually represented on the graph by the fact that the slope of the tangent line at the current operating point is ________ than the slope of the line drawn from the origin to that same point.
Evaluating a Production Decision
An analyst is examining a production process where output (vertical axis) increases at a decreasing rate as a single input (horizontal axis) is added. This is represented by a concave curve starting from the origin. The analyst makes the following claim about a specific operating point, P, on the curve: 'At point P, the contribution to output from the last unit of input added was greater than the average output per unit of input.' Based on the geometric properties of such a curve, evaluate the analyst's claim.
Correcting a Misinterpretation of a Production Graph