Microeconomics

What is Microeconomics

Budget Constraint, Cost Line

The process of creating an idiffrerence map

1. Create a budget line.
   
2. Create the indifference curve.
   
3. Finding which indifference curve is tangent to the budget line.

Example: There is a restraint of $50.00. The restraint can be a budget, income, production, etc. The $50.00 dollars must be spend on two goods, Good \(x\) and good \(y\). The two goods have a price. The price for good \(x\) is 3 dollars: \(Px\) = 3.00. The price of \(y\) = 1.50: \(Py\) = 1.50.
The \(U\) function is \(u(x,y) = x^{2}0.25y\) and is given. The restraint line: \(50.00 = x3.00 + y1.50\).

Creating a budget line. A budget line can be seen as a contraint. The total amount of possible income espend. The equation of a line can be use to create the budget line: \(y = mx + b\) where \(b\) is the y intercept.

B = 50 # Total cost. The linear functionrepresenting Incocme, Budget. This is the total income. 
px = 3 # This is the price of good x.
py = 1.5 # This is the price of good y.
n = B/px # Total amount of good x consumption.
n

## [1] 16.66667

n2 = B/py # Total amount of good y comsumption. 
n2

## [1] 33.33333

slope = -n2/n # MRS
slope

## [1] -2

This would be the budget line created for the r code above: \(y = -2b + 33.33\).

budget <- function(x) (slope * x) + n2
ggplot() +
  # This draws the budget constraint. 
  stat_function(data = tibble(x = 0:15), aes(x = x), fun = budget, color = "red", size = 1.5) +
  # This adds the label. 
  annotate(geom = "label", x = 2.5, y = budget(2.5), label = "Budget", color = "blue") +
  labs(x = "x", y = "y")

Any point on this line is the budget constraint.

Create & Plot the Utility indifference curve \(u(x,y) = x^{2}0.25y\)

library(Ryacas)
utility.u = function(x, y, U) x^2*(0.25 * y)
# Test with adding values for x and y. 5 and 5
utility.u(5,5) # This wold tell you his number of utils at each combination of pizza and soda. 

## [1] 31.25

# We then cahnge the equation to solve for y
utility.solved = function(x,y,U) U / (0.25 * x^2)

eval(utility.solved, list(x=5, U = 10))

## function(x,y,U) U / (0.25 * x^2)

n <- 11
foo <- outer(X=seq_len(n), Y=seq_len(n), function(x, y) pmin(3*x, 9*y))

Preferences/Consumer Theory

Utility

Cobb-Douglas Utility

A typical micro-economic problem is to find the optimal level of consumption of two goods. Set to some restrictions like prices, budget constraint, and preferences.

Deriving the utility function: \(U = XY\), Budget constraint: \(M = PxX + PyY\).
\(MUx\) = \(\frac{\partial{U}}{\partial{X}} = Y\). \(MUy\) = \(\frac{\partial{U}}{\partial{y}}\) = \(X\)
\(\frac{MUx}{Px}\) = \(\frac{MUy}{Py}\): Maximizing his/her utility. Solve for \(Y\). Substituted your \(MUx\) and \(MUy\) into the utility funceion: \(U = XY\).

Choice

Demand

Finding the demand function: a demand function is a linear function which could be interpertec as y=mx+b. m represents the slope and b represents the y intercept. This lead to Qd = mP + b. Qd is quantity demanded P is price. You then find two order pairs of price and Quantity. It can be represented as (x_1,y_1) & (x_2,y_2). 2 slices of pizza = $2 in time one. lets increase price in time two. find the slope which is the marginal rate of substitution. with all these three information you cal clulate the y intercept.
Typical Demand Function for \(y\): \(y = \frac{I}{x}\). Where \(y\) is total demand is equal to Total income diided by the good \(x\).

Income and Substitution Effect

There are two major most common ways of deriving the income and substitution effect. One is the Slutsky and the other one is Hickman.
Hicksians is the demand curve created by Hicks, demand for consumption bundles whet a utility is fixed for the consumers as the expenditure I reduces. Hicks separates the demand curve into the price effect and the substitution effect. through the compensating income variation caused by the changing i the relative price of a good, keeping the real income constant.
Slutsky expresses the demand for consumption bundles due to a price change while the utility is fixed.

Slutsky Equation
States that total change in demand is composed of an income and substitution effect and that the two effects together must equal the total change in demand.
Compensating Variation The amount of money to give to the consumer to completely offset the price increase, so the consumer’s utility is unchanged.

Equivalent Variation is the measure of economic welfare changes associated with changes in price.

Example of finding the Demand Function from the Utility function Utility function: \(U(X,Y) = XY\)
Demand functions of good x & y: \(X(M,p_x) = \frac{M}{2P_{x}}\). \(Y(M,p_y) = \frac{M}{2P_{y}}\).

Substitution effect: The change in demand due to the change in the ate of exchange between two goods. The change in the consumption of one good and buying more of the other good.
Income Effect: The change in demand due to having more purchasing power parity. The change in consumption of one good because the buying power of the agent has either fallen or increase.
There are two reason the consumption either increase or decrease when the price of one of the goods increase or decrease. The first source of the decrease is called the substitution effect. The second is call the income effect.

Function: \(U = x^{\alpha}y^{1-\alpha}\). \({\alpha}\) is a parameter given. K and L are unknown.
The Budget constraint is: \(x + py = w\). \(w\) is the income.

The way to find out the substitution effect and income effect is to first let relative prices change and adjust money income so as to hold purchasing power constant.
2. let purchasing power adjust while holding the relative prices constant.
In a graph the first step is to pivot the movement where the slope of the budget line changes while its purchasing power stays constant. The second step is a movement where the slope stays constant and the purchasing power changes.

Determining the substitution and income effect.

There are two ways of determining the substitution and income effect. One is Hicksian Method and the other is Slovaki method. Substitution effect: The change in demand due to the change in their ate of exchange between two goods.
Income Effect: The change in demand due to having more purchasing power parity.

Engel Curve: Shows the relationship between optimal choice of a good and income. The curve would slope upward for normal good and down ward for an inferior good. Normal good are categorize into luxury goods and necessary good. If the case the

Production Technologies

Profit = Total Revenue - Total Cost
\[\Pi = pQ - (wl-rk)\] Isoquant: The set of all possible combinations of inputs that are just sufficient to produce a given amount of output.
Production function is written as \[f(x_1,x_2) = min[x_1,x_2]\]
Perfect substitute \(f(x_1,x_2) = x_1 + x_2\).
Cobb-Douglas Production function: \(f(x_1,x_2) = Ax_1^ax_2^b\). \(A\) is the numerical magnitude of the utility function. \(a + b =1\) \(A\) measures, roughly speaking, the scale of production, how much output we would get if we used one unit of each input. \(a,b\) measures how the amount of output responds to.
We assume technologies are monotonic: if you increase the amount of at least one of the inputs, it should be possible to produce at least as much output as you were producing originally.
convex: It means that if you have two ways to produce \(y\) units of output, \((x_1,x_2)\) and \((z_1,z_2)\), then their weighted average will reduce at least y units of output. Marginal Product: The extra amount of output per unit of extra input.
Technical rate of substitution: the trade off between two inputs in production . It measures the rate at which the firm will have to substitute one input for another in order to keep output constant.
\[TRS(x_1,x_2) = \frac{\delta x_2}{\delta x_1} = - \frac{MP_1(x_1,x_2)}{MP_2(x_1,x_2)}\]
Diminishing marginal product: Output will go up at a decreasing rate. It applies only when all other inputs are being held fixed.
Diminishing technical rate of substitution As we increase the amount of factor 1, and adjust factor 2 so as to stay on the same isoquant, the technical rate of substitution declines. It means the slope of an isoquant must decrease in absolute value as we move along the isoquant in the direction of increasing \(x_1\) and it must increase as we move in the direction if increasing \(x_2\).
Diminishing marginal product is an assumption about how the marginal product changes as we increase the amount of one factor, holding the other factor fixed. Diminishing TRS is about how the ratio of the marginal product, the slope of the isoquant, changes as we increase the amount of one factor and reduce the amount of the other factor so as to stay on the same isoquant.

Short Run & Long Run

Short run there will be some factors of production that are fixed at predetermined levels. \(f(x_1, \bar{x}_2)\). Bar denotes fixed variable.
Long Run All factors of production can be varied.

Return to Scale

Constant Return to Scale: It means that two times as much of each input gives two times as much output. For example, an investmet or 100 would give out a return of 100. In math terms, it could be represent as this: \[2f(x_1,x_2) = f(2x_1,2x_2)\].

If we scale all o f the inputs up by some amount \(t\), constant return to scale implies that we should get \(t\) times as much output: \(tf(x_1,x_2) = f(tx_1,tx_2)\).

Returns to scale describes what happens when you increase all inputs.
Diminishing marginal product describes what happens when you increase one of the inputs and hold the other fixed.

Increasing returns to scale: if we scale up inputs by some factor \(t\) and we get more than \(t\) times as much output. \(f(tx_1,tx_2) > tf(x_1,x_2)\).

Decreasing returns to scale \(f(tx_1,tx_2) < tf(x_1,x_2)\). If we get less than twice as much output from having twice as much of each input.

Summary

1. The technological constraints of the firm are described by the production set, which depicts all the technologically feasible combinations of inputs and outputs, and by the production function, which gives the maximum amount of output associated with a given amount of the inputs.
   
2. Another way to describe the technological constraints facing a firm is through the use of isoquants curves that indicate all the combinations of inputs capable of producing a given level of output.
   
3. We generally assume that isoquants are convex and monotonic, just like well behaved preferences.
   
4. The marginal product measures the extra output per extra unit of an input, holding all other inputs fixed. We typically assume that the marginal product of an input diminishes as we use more and more of that input.
   
5. The technical rate of substitution TRS measures the slope of an isoquant. We generally assume that the TRS diminishes as we move out along an isoquant which is another way of saying that the isoquant has a convex shape.
   
6. IN he short run some inputs are fixed, while in the long run all inputs are variable.
   
7. Return to scale refers to the way that output changes as we change the scale of production. If we scale all inputs up by some amount \(t\) and output goes up by the same factor, then we have constant returns to scale. If output scales up by more than \(t\), we have increasing returns to scale; and if it scales up by less than \(t\), we have decreasing returns to scale.

Profit Maximization

Profits is equal to revenues - cost.
Suppose the firm produces \(n\) outputs \((y_1,...,y_n)\) and uses \(m\) inputs \((x_1,...,x_m)\). Prices of goods are \((p_1,...,p_n)\), and prices of inputs \((w_1,...,w_m)\). There for profits can be written as \[\pi = \Sigma_{i=1}^{n} p_{i}y_{i} - \Sigma_{i=1}^{m} w_{i}x_{i}\]

The first term (\(\Sigma_{i=1}^{n} p_{i}y_{i}\)) is revenue and the second term (\(\Sigma_{i=1}^{m} w_{i}x_{i}\))is cost. The first term represents total output times the price. The second term represents total wages (cost) times the amount of output produce.

Fixed and Variable Factors

It may be difficult for firms to adjust some inputs. For example, capital. A demolishing and changing a building would be difficult for a firm to do. The factors of production that is in a fixed amount are called fixed factors. The short run is defined as the period of time in which there are some fixed factors. The firm is obligated to employ some factors, even if it decides to produce zero output. Therefore, it is possible for the firm to make negative profits in the short run. Fixed factors of production, must be paid even if there is no production. Some examples are: machines, factory buildings, plants, permanent employees.

Opportunity cost The forgone opportunity of using capital elsewhere.

Fixed and Variable Factors

If the factor can be used in different amounts, we refer to it as Variable factors. In the long run all factors are variable factors.

Quasi-fixed factors Factors of production that must be used in a fixed amount, independent of the output of the firm, as long as the output is positive.
Examples: Costs of hiring new workers, Costs of training new workers, Social insurance programs (e.g., social security, worker’s compensation insurance.

Short run profit maximization

Short run profit maximization problem when input 2 is fixed at some level \(\bar{x}_2\). \(f(x_1, x_2)\) is the production function. \(p\) is the price of output. \(w_1\) and \(w_2\) are the price of inputs. Therefore the profit maximization is:
\[max_{x_1}pf(x_1, \bar{x}_2)-w_1x_1-w_2\bar{x}_2\] The condition for the optimal choice of factor 1. If \(x_1^*\) is the profit maximizing choice of factor 1, then the output price times the marginal product of factor 1 should equal the price of factor 1. \[pMP_1(x_1^*,\bar{x}_2) = w_1\]
The marginal product of a factor should equal its price. The rule comes from the first order condition for the maximization problem:
\[p\frac{\partial f(x_1^*, \bar{x}_2)}{\partial x_1} - w_1 = 0\] Using \(y\) to denote the output of the firm, profits are given by:
\[\pi = py - w_1x_1 - w_2\bar{x}_2\]
The above expression can be solve for \(y\). This equation is the Isoprofit lines:
\[y = \frac{\pi}{p} + \frac{w_2}{p}\bar{x}_2 + \frac{w_1}{p}x_1\]
\(\frac{w_1}{p}\): is the slope of the lines. \(\frac{\pi}{p} + \frac{w_2}{p}\bar{x}_2\) is the vertical intercept and it measures the profits plus the fixed costs of the firm. Higher levels of profit will be associated with isoprofits lines with higher vertical intercepts. It is use to find the point on the production function that has the highest associated isoprifit line.

To find the optimum, the slope of the production function should equal the slope of the isoprofit line. The slope of the production is the marginal product and the slope of the isoprofit line is \(\frac{w_1}{p}\). The optimum condition can be written as:
\[MP_1=\frac{w_1}{p}\]
Increasing \(w_1\) will make the isoprofit line steeper. If is stepper then the tangentcy must occurs to the left and the optimal level of factor 1 must decrease.
\[y = \frac{\pi}{p} + \frac{w_2}{p}\bar{x}_2 + \frac{w_1}{p}x_1\]
It means that if the price of factor 1 increases, the demand of factor 1 must decrease.
Profit maximization can be written as:
\[max_xpf(x)-wx\]
Take the first derivative:
\[pf'(x^*) = w\]

Second derivative:
\[f''(x^*)\le0\] If the output price decreases the isoprofit line must become steeper. High \(w_1\) = steeper isoquant, low output. low \(w_1\) flatter isoquant, high output. Low \(p\) steeper isoquant, decrease in output. high \(p\) flatter isoquant, increase in output. A reduction in the output price must decrease the supply of output.

Profit Maximization int the Long Run

\[max_{x_1}pf(x_1, x_2)-w_1x_1-w_2x_2\]
The same order condition from the time we were holding one variable constant applies, except we take the derivatives of both variables:
\[p\frac{\partial f(x_1^*, x_2^*)}{\partial x_1} - w_1 = 0\]
\[p\frac{\partial f(x_1^*, x_2^*)}{\partial x_2} - w_2 = 0\]
It can be written as:
\[pMP_1(x_1^*, x_2^*) = w_1\]
\[pMP_2(x_1^*, x_2^*) = w_2\]
The value of the marginal product of each factor should equal its price at the optimal choice.

Example: Maximize. production, output, with Cobb-Douglas production

\[f(x_1,x_2) = x_1^ax_2^b\]
First order condition:
\[w_1=\lambda ax_1^{a-1}x_2^b\]
\[w_2=\lambda bx_1^{a}x_2^{b-1}\]
Multiply the first equation by \(x_1\) and the second equation by \(x_2\) to get:
\[ax_1^{a}x_2^{b} - w_1x_1 = 0\]
\[bx_1^{a}x_2^{b} - w_2x_2 = 0\]
Using \(y = x_1^ax_2^b\) to denote the level of output of this firm we can require these expression ass:
\[pay = w_1x_1\] & \[pby = w_2x_2\]
Solve for \(x_1\) and \(x_2\) we have:
\[x_1^* = \frac{apy}{w_1}\]
\[x_2^* = \frac{bpy}{w_2}\]
This gives the demand function for the two factors as a function of the optimal output choice.

We still have to solve for the optimal choice of output.
\[(\frac{pay}{w_1})^a(\frac{pby}{w_2})^b=y\]
Factoring the \(y\) gives:
\[(\frac{pa}{w_1})^a(\frac{pb}{w_2})^by^{a+b}=y\]
or
\[(\frac{pa}{w_1})^{\frac{a}{1-a-b}}(\frac{pb}{w_2})^{\frac{b}{1-a-b}}=y\]
This gives us the supply function of the Cobb-Douglass firm. along with the factor demand functions derived above.

Inverse Factor Demand Curve

Factor demand curve of a firm measures the relationship between the price of a factor and the profit maximization choice of that factor. To find the profit maximizing choice for any price \((p,w_1,w_2)\) we just find those factor demand \((x_1^*,x_2^*)\) such that the value of the marginal product of each factor equals its price.

The inverse factor demand curve measures the same relationship, but from a different point of view. It measures what the factor price must be for some given quantity of inputs to be demanded. Given the optimal choice of factor for either good 1 or 2 we can draw the relationship between the optimal choice of factor. if we use factor 2 we will get the optimal choice of factor for 1.
\[pMP_1(x_1,x_2^*) = w_1\]
The curve will be downward sloping by the assumption of diminishing marginal product. The above function depicts what the factor price must be in order to induce the firm to demand that level of \(x_1\), holding factor 2 fixed at \(x_2^*\).

Profit maximization and return to scale.

In the long run level of profit for a competitive firm that has constant returns to scale at all levels of output is a zero level of profit.

Summary

1. Profits are the difference between revenues and costs. In this definition it is important that all costs be measure using the appropriate market prices.
   
2. Fixed factors are factors whose amount is independent of the level of output; variables factors are factors whose amount used changes as the level of output changes.
   
3. In the short run, some factors must be used in predetermined amounts. In the long run, all factors are free to vary.
   
4. If the firm is maximizing profits, then the value of the marginal product of each factor that it is free to vary must equal its factor price.
   
5. the logic of profit maximization implies that the supply function of a competitive firm must be an increasing function of the price of output and that each factor demand function must be a decreasing function of its price.
   
6. If a competitive firm exhibits constant returns to scale, then its long run maximum profit must be zero.

Cost Minimization

Suppose that we have two factors of production that have prices \(w_1\) and \(w_2\) and that we want to figure out the cheapest way to produce a given level of output, \(y\). If we let \(x_1\) and \(x_2\) measure the amounts used of the two factors and let \(f(x_1,x_2)\) be the production function for the firm, we can write this problem as:
\[min_{x_1x_2}w_1x_1+w_2x_2\]
s.t.
\[f(x_1,x_2)=y\]
The solution to this cost minimization problem will depend on \(w_1\), \(w_2\) and \(y\) so we write it as \(c(w_1,w_2,y)\). This function is known as the cost function. It measures the minimal costs of producing \(y\) units when factor prices are \((w_1,w_2)\).

The isoquants give’s the technological constraints which are all the combinations of \(x_1\) and \(x_2\) that can produce \(y\).

Suppose that we want to plot all the combinations of inputs that have some given level of costs, \(C\), we can write this as:
\[w_1x_1 + w_2x_2 = C\]
Which can be rearranged to give:
\[x_2=\frac{C}{w_2}-\frac{w_1}{w_2}x_1\]
The slope of the above function is: \(-\frac{w_1}{w_2}\). The vertical intercept is \(\frac{C}{w_2}\). Every point on an isocost curve has the same cost, \(C\) and higher isocost lines are associated with higher costs.

The cost-minimization problem can be phrase as: find the point on the isoquant that has the lowest possible isocost line associated with it. The cost-minimization point will be characterized by a tangentci condition: the technical rate of substitution must equal the factor price ratio:
\[-\frac{MP_1(x_1^*,x_2^*)}{MP_2(x_1^*,x_2^*)}=TRS(x_1^*,x_2^*)=-\frac{w_1}{w_2}\]
Consider any changes in the pattern of production \((\delta{x_1},\delta x_2)\) that keeps output constant. The changes must satisfy:

\[MP_1(x_1^*,x_2^*)\delta{x_1} + MP_2(x_1^*,x_2^*)\delta{x_2} =0\]
Note that \(\delta{x_1}\) and \(\delta x_2\) must be the opposite signs. If you increase the amount used of factor 1 you must decrease the amount used of factor 2 in order to keep output constant.
If we are at the cost minimum, then this changes cannot lower cots, so the final equation is:
\[w_1\delta{x_1} + w_2\delta{x_2} = 0\]
Then solving for \(\frac{\delta{x_2}}{\delta{x_1}}\) in these two equations: \(MP_1(x_1^*,x_2^*)\delta{x_1} + MP_2(x_1^*,x_2^*)\delta{x_2} =0\) and \(w_1\delta{x_1} + w_2\delta{x_2} = 0\). This equal to :
\[\frac{\delta{x_2}}{\delta{x_1}}=\frac{w_1}{w_2}= -\frac{MP_1(x_1^*,x_2^*)}{MP_2(x_1^*,x_2^*)}\]
Which is the condition for cost minimization derived by geometric argument.

Example II

\[min_{x_1,x_2}w_1x_1+w_2x_2\]
Such that:
\[f(x_1,x_2) = y\]
To solve we can use the Lagrange multipliers.
\[L =w_1x_1 + w_2x_2 - \lambda(f(x_1,x_2)-y)\]
Differentiate with respect to \(x_1\), \(x_2\), and \(\lambda\). This will gives us the first order conditions:
\[w_1 - \lambda \frac{\partial f(x_1,x_2)}{\partial {x_1}} = 0\]

\[w_2 - \lambda \frac{\partial f(x_1,x_2)}{\partial {x_2}} = 0\]
\[f(x_1,x_2)-y = 0\]
Rearrange the first two equations and divide the first equation by the second equation to get:
\[\frac{w_1}{w_2} = \frac{\frac{\partial f(x_1x_2)}{\partial x_1}}{\frac{\partial f(x_1,x_2)}{\partial x_2}}\]
The consumer problem and the producer problem they almost look the same but they are not. On the consumer problem the straight line was the budget constraint, and the consumer moved along the budget constraint to find the most preferred position. In the producer problem, the isoquant is the technological constraint and the producer moves along the isoquant to find the optimal position.
The choice of inputs that yield minimal cost a for the firm will in general depend on the input prices and the level of output that the firm wants to produce, so we write these choices as \(x_1(w_1,w_2,y)\) and \(x_2(w_1,w_2,y)\). They are called conditional factor demand functions. They measure the relationship between the prices and output and the optimal factor choice of the firm.

There are differences between conditional factor and profit maximizing factor. The conditional factor demand give the cost minimizing choices for a given level of output. Profit maximizing factor demand five the profit maximizing choices for a given price of output.

Conditional factor demands are not directly observed; they are a hypothetical construct. They answer the equation of how much of each factor would the firm use if it wanted to produced a given level of output in the cheapest way. However they are useful as a way of separating the problem of determining the optimal level of output from the problem of determining the most cost effective method of production.

Example: Minimizing cost for specific technologies

Suppose that we consider a technology where the factor are perfect complement, so that \(f(x_1,x_2) = min[x_1,x_2]\). Then if we want to produce \(y\) units of output, we clearly need \(y\) units of \(x_1\) and \(y\) units of \(x_2\). Thus the minimum cost of production will be:
\[c(w_1,w_2,y) = w_1y + w_2y = (w_1 +w_2)y\]
because the equation has perfect substitutes in technology:
\[f(x_1,x_2) = x_1 +x_2\]
The firm will use whichever is cheaper. Thus, the minimum cost of producing \(y\) units of output will be \(w_1y\) or \(w_2y\), which ever one is less. It could be written as:
\[c(x_1,x_2,y) = min[w_1y,w_2y] = min[w_1,w_2]y\]

Example II, Cobb-Douglas

\[c(x_1,x_2,y) = Kw_1^{\frac{a}{a+b}}w_2^{\frac{b}{a+b}}y^{\frac{1}{a+b}}\]
where \(K\) is constant that depends on \(a\) and \(b\).
The cost minimization problem is:
\[min_{x_1,x_2}w_1x_1 + w_2x_2\]
Such that: \[x_1^ax_2^b = y\]
The above problem can be solve using the substitution method or the Lagrangian method. The Lagrangian method is easier. First find the first order condition: \[w_1=\lambda ax_1^{a-1}x_2^b\]
\[w_2=\lambda bx_1^{a}x_2^{b-1}\]
\[y = x_1^ax_2^b\]
Multiply the first equation by \(x_1\) and the second equation by \(x_2\) to get:
\[w_1x_1 = \lambda ax_1^ax_2^b = \lambda ay\]
\[w_2x_2 = \lambda bx_1^ax_2^b = \lambda by\]
Solve for \(x_1\) and \(x_2\) so that:
\[x_1 = \lambda \frac{ay}{w_1}\]
\[x_2 = \lambda \frac{ay}{w_2}\]
Now use the third equation to solve for \(\lambda\). Substitute the solutions for \(x_1\) and \(x_2\) and we have:
\[(\frac{\lambda ay}{w_1})^a(\frac{\lambda by}{w_2})^b = y\]

Cost Cuve

Perfect competition.

Output function: The economical idea of cost and profit. Cost can be use in production and also profits. We minimize cost by minimizing the cost fuction. We maximize the profit function. the cost and profit function can come in many ways. The production function can also come in many ways. By the fact they are functions; they can come in many different ways. The most common production function is called the cob douglass function. The reason is more common is because it fallows the rules for an indifference curve.

Elasticity:

The formula for price elasticity of of \(x\):
\[\frac{\%\Delta Qdx}{\%\Delta I}\]
Price Elasticity of demand: Measures the responsiveness of consumers of a particular good to a change in the good’s price.
Cross Elasticity of demand: Measures the responsiveness of consumers of one good to a change in the price of a related good. (substitutes or complements).
Income Elasticity of demand: Measures the responsiveness of consumers of a particular good to change in their income.

Luxury goods \(\frac{\%\Delta{Qdx}}{\%\Delta{I}}\) > 1.
Necessity \(\frac{\%\Delta{Qdx}}{\%\Delta{I}}\) < 1.
Normal good: \(\frac{\%\Delta{Qdx}}{\%\Delta{I}}\) > 0.
Inferior good: \(\frac{\%\Delta{Qdx}}{\%\Delta{I}}\) < 0.

Home Work

Example

Consider a consumer with utility function. Consier the prodctin functio: \(q = 600\bar{K}^2L^2-\bar{K}^3L^{3}\). Where \(q\) represents output. \(\bar{K}\) represents capital (fixed) and \(L\) represents labor.

Find \(x_1^*(w_1,w_2,P)\), \(\pi = TB-TC\). \(y = x_{1}^{1/3}x_{2}^{1/3}\). \(\pi = py-w_1x_1-w_2x_2\).
Find \(x_1^*(w_1,w_2,P)\) & \(x_2^*(w_1,w_2,P)\).

Set up th Langrangian. Maximize profits with a constraint. \(p\{x_{1}^{1/3}x_{2}^{1/3}\}-w_1x_1-w_2x_2\).

Find the partal derivitives:
This I th prtia derivitive inrelation to \(x_1\): \(\frac{\partial\pi}{\partial{x_1}} = p\{\frac{1}{3}x_{1}^{-2/3}x_{2}^{1/3}\} - w_1 = 0\)
This is the partial derivitive inrelation to \(x_2\): \(\frac{\partial\pi}{\partial{x_2}} = p\{\frac{1}{3}x_{1}^{1/3}x_{2}^{-2/3}\} - w_2 = 0\)
We then set them equal to one another. One way o doing this is to divive them: \(\frac{p\{\frac{1}{3}x_{1}^{-2/3}x_{2}^{1/3}\} - w_1 = 0}{p\{\frac{1}{3}x_{1}^{1/3}x_{2}^{-2/3}\} - w_2 = 0}\)