Types of Relationships between the Input and Output
Analysis of the input–output relationship  is less computationally expensive, but it .. This model considers the interaction between the microbial population, .. cases where this upper limit on the number of coefficients required is not known. Relationship between input and output of cells in motor and somatosensory type of organization, with identical thresholds, existed in the so-called "Motor" and. These are called registers. For Input/Output, usually, the goal is to move information between the I/O This is called Memory Mapped I/O.
However, the average product of fixed inputs not shown is still rising, because output is rising while fixed input usage is constant. In this stage, the employment of additional variable inputs increases the output per unit of fixed input but decreases the output per unit of the variable input. In Stage 3, too much variable input is being used relative to the available fixed inputs: The output per unit of both the fixed and the variable input declines throughout this stage.
At the boundary between stage 2 and stage 3, the highest possible output is being obtained from the fixed input. Shifting a production function[ edit ] By definition, in the long run the firm can change its scale of operations by adjusting the level of inputs that are fixed in the short run, thereby shifting the production function upward as plotted against the variable input.
If fixed inputs are lumpy, adjustments to the scale of operations may be more significant than what is required to merely balance production capacity with demand.
- Input–Output Analysis
- Production Function: Relation between Physical Inputs and Output of a Good
- Types of Relationships between the Input and Output
For example, you may only need to increase production by million units per year to keep up with demand, but the production equipment upgrades that are available may involve increasing productive capacity by 2 million units per year. Shifting a production function If a firm is operating at a profit-maximizing level in stage one, it might, in the long run, choose to reduce its scale of operations by selling capital equipment.
By reducing the amount of fixed capital inputs, the production function will shift down. The beginning of stage 2 shifts from B1 to B2. The unchanged profit-maximizing output level will now be in stage 2. Homogeneous and homothetic production functions[ edit ] There are two special classes of production functions that are often analyzed. The V; usually consist of wages, interest on capital and entrepreneurial revenues credited to households, taxes paid to the government, and so on.
The solution of the price equations 9 permits the determination of prices of all products from given values added by each sector. The solution can be written The constant, Ai j, measures the dependence of the price of the product of sector j on the value added by sector i. The coefficients ai j appearing in each row of the output equations 3 make up the corresponding column of coefficients appearing in the price equations 9 ; the coefficients Aij appearing in each row of the output solution 4 make up the corresponding column of coefficients in the price solution Only if all the Aij in the price solution are non-negative will there necessarily exist positive prices enabling each sector to balance exactly its input-output accounts in value terms for any given set of positive values added.
Since Aji in the price solution equals A;, in the output solution, this condition is the same as that needed to assure positive outputs for any given set of final demands. Inserting into 10 the inverse computed for the example used above, we have: The internal consistency of the price and quantity relationships within an open input-output system is confirmed by the following identity derived from equations 4 and 9: On the left-hand side is the sum of the values added paid out by the endogenous sectors to the exogenous sectors of the system; on the right-hand side is the sum of the values quantities times prices of products delivered by all endogenous sectors to the final exogenous demand sector.
This identity confirms, in other words, the accounting identity between the national income received and the national income spent, as shown in Table 2. Theory of dynamic input-output systems Dynamic input-output theory grows out of the static theory through consideration of intersectoral dependences involving lags or rates of change of variables over time.
Structural relations between stocks and flows of goods constitute the theoretical basis for the input-output approach to empirical analysis of the accumulation process and of developmental planning.
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The stock of goods produced by sector i that sector j must hold per unit of its full capacity output is called the capital coefficient of good i in sector j and is usually designated by bi j. The matrix describes the real capital structure of a national economy as a whole. The current inputs and capital stocks required to produce the output of a particular industry might have to be utilized during the period in which the output is produced, or they might have to be made available and used, at least in part, one or more periods before that.
An analytically general and at the same time realistic description of dynamic input-output relationships can be given if separate variables are used to designate the flows of inputs and of outputs absorbed or produced by the same industry in different years. The balance between the output and the available capacity of sector i in a particular year t can, for example, be described by a linear differential equation involving structural interrelationships between the inputs and the outputs of the various sectors and the rates of change of the inputs and outputs.
The equation is where xi t is the output and xi t the rate of change of output of sector i at time t. If the time path of all final demands and the levels of all outputs at an initial point of time are assumed to be given, a system of n such linear differential equations, one for each sector, can be solved for all the n outputs.
The solution gives the level of each output, Xi tat any point of time—that is, for any t. Although this approach to the study of dynamic input-output relationships offers certain theoretical advantages, most empirical work in the field is conducted in terms of discrete period analysis based on systems of difference equations of the following kind: Superscripts indicate the time period to which the variables refer. The next n terms represent the deliveries from sector i to itself and to all other sectors in response to needs for additional productive capacities, which in turn depend on the differences between current and future outputs.
These changes in outputs multiplied by the appropriate capital coefficients, i. In a static formulation investment in additional productive capacity is treated as a component of the given final demand, but in a dynamic analysis investment must be explained and cannot be considered as having been fixed beforehand.
Hence the final demand for the product of industry i in period t, y t 1 now comprises deliveries to households, government, and so on, but no additions to the stock of productive capital.
Equation 14 is a basic building block that can be used to construct a system describing inter-temporal input-output relationships between the different sectors of a particular economy over an interval of time containing any number of years. The set of six equations 15 spans the intersectoral relationships within a three-sector economy, of which only two are endogenous, over a period of three years. In the last two equations, which describe the input-output balance of both industries in the third year, the amounts allocated to investment are seen to depend on the output levels of the next, that is, the fourth year.
Production Function: Relation between Physical Inputs and Output of a Good
Thus, for example, if the outputs of both sectors in the first year, i. Instead of being anchored in the first year and solved for the next three years, the system can be used in reverse; that is, after having fixed the output of both endogenous sectors in the last fourth year, the system can be solved so as to display the dependence of production on the final consumption levels over the period of the first three years.
The numerical example of a three-sector economy presented above can now be extended to demonsrate the solution of a dynamic input-output system. The flow coefficients shown in Table 3 must first be supplemented by a corresponding matrix of capital coefficients. Let it be The entries in the first column show that 0. The two figures in the second column supply analogous information on the capital structure of manufacturing.
The terms containing and are transferred to the right-hand side in the last two equations, because in the general solution shown below these outputs will be considered given.
The solution for the remaining unknown outputs is given in equations Equations 17 represent a general numerical solution of the dynamic input-output system 16 in the same sense in which the inversion of the flow coefficient matrix incorporated in 5 yields a general solution of the original static system.
These six equations describe explicitly the dependence of the total outputs of both industries in the first, second, and third years on the levels of final deliveries of both products in the first, second, and third years.
As a simple check on the internal consistency of this general solution, the amounts of 55 bushels of wheat and 30 yards of cloth actually allocated to households in Table 1 can be substituted, respectively, for y1 and y2 in each of the six equations. After the performance of appropriate multiplications and additions the result would show that in this particular case the total output of wheat would be maintained at a constant level of bushels and the total annual output of manufactured products at a constant level of 50 yards of cloth throughout the entire period.
Since from the first year on nothing would have been added to, or subtracted from, its productive stocks, the economy would in this particular case maintain itself without expansion or contraction in either sector. The same analytical procedure can be used to construct and solve an open dynamic input-output system incorporating structural change. Since no outputs can be negative, only those sequences and combinations of final deliveries that turn out to require nonnegative total outputs in all sectors for all years can in fact be realized within the framework of a particular dynamic structure.
The presence of many negative constants in the general solution of the type presented above indicates how narrow the range of alternative developmental paths open to a particular economy might actually be. The major deficiency of the simple input-output approach to the description of dynamic processes presented here is its inability to handle situations in which one or more industries operate over significantly long periods of time under conditions of excess capacity.
Stocks of fixed capital invested in one sector cannot as a rule be dismantled and shifted to use in another sector.