Deriving the short run aggregate supply curve
(turning Economics upside down)

Individual income can grow as fast as productivity rises.

Alex Berenson

**P**_{pc}=α_{1}Y_{pc}^{n}+...+β_{1}Y_{pc}+γ_{1}, **α**_{1}Y_{pc}^{n}+...+β_{1}Y_{pc}+γ_{1}>0, n integer >0, dP_{pc}/dY>0, d^{2}P_{pc}/dY^{2}>0, α_{1}, β_{1}, γ_{1} are numbers to be found by interpolation

Figure 3

A.R.=y

M.R.=x

M.C.=z

m_{c}: slope of the M.C. curve

c_{c}: intersection of the M.C. curve to the y-axis or Price-axis
_{1}= mq_{1}+ c_{1}

y_{2}= mq_{2}+ c_{2}

∴Δy= mΔq+ Δc

_{2}q+ γ'_{2}

To express q units of output produced into Dollars we multiply q by price_{2018}: output produced in Dollars. We multiply every value of q by price_{2018} as we did earlier in perfect competition.

We find β'_{2} and γ'_{2} by interpolation we know one point P=100, p_{2018}q_{2018} output of the monopoly.

P= β'_{2}pq+ γ'_{2}

If the above monopoly_{(1)} is 30% of the output of all monopolies then:

0.3P= 0.3β'_{2}pq+ 0.3γ'_{2}

∴0.3P= 0.3β'_{2}Y_{(1)}+ 0.3γ'_{2}

∴0.3P= 0.3β'_{2}✕0.3Y_{m}+ 0.3γ'_{2}

We add all other monopolies so P_{m}= β_{2}Y_{m}+ γ_{2}

Therefore the point 2018 will be:

P=100, Y_{m}(p_{(1)2018}q_{(1)}+p_{(2)2018}q_{(2)}+...)

p_{(1)2018} is the price of 2018 of monopoly_{(1)}

q_{(1)} is the 2018 output at price_{2018} of the monopoly_{(1)}

**P**_{m}= β_{2}Y_{m}+ γ_{2}, β_{2}>0, γ_{2} is a number

0.5P= 50

_{avc.min}:

Monopoly

P_{m}= 2.1Y_{m} +95.695

P_{m}= 0.21Y +95.695

0.1P= 0.021Y + 9.5695 —✱

Monopolistic competition

P_{mc}= 1.1Y_{mc} +93.235

P_{mc}= 0.33Y +93.235

0.3P= 0.099Y + 27.9705 —✱

Oligopoly

0.5P= 50 —✱

P= 0.00176Y^{4} - 0.0859Y^{3} + 1.571Y^{2} -12.67Y + 128.7278 for Y≥19.704

For Y<19.704:

Monopoly

P_{m}= 2.1Y_{m} +95.695

Pc is not included that is 10% of the economy is not included the remaining is 90% to include the whole economy 100%: the increased share of the monopoly in the economy is: 100/90✕10%= 10/9✕0.1= 0.1:

P_{m}= 0.23Y +95.695

0.1P= 0.0259Y +10.6327 —✱

With the same thinking:

Monopolistic competition

P_{mc}= 1.1Y_{mc} +93.235

P_{mc}= 0.36Y +93.235

0.3P= 0.12Y +31.0783 —✱

Oligopoly

0.5P= 55.5 —✱

P= 0.148Y +97.267 (these numbers are rounded to three decimal places)

Combining the equations for Y≥19.704 and Y<19.704 the graph is as follows:

Figure 8.2 Since the two curves end and begin above each other we connect them with a vertical line to make them one curve. At Y=19.704 pc is included and Price drops slightly because goods and services in pc have lower Price than Monopoly, Monopolistic Competition and Oligopoly at Y<19.704 please see equations.

_{1} is reduced to q_{2}, P_{1} is increased to P_{2}. Please, note that unlike changes in the demand curve when the marginal cost changes price and quantity move in opposite directions the quantity produced is reduced the price is increased. As in monopoly the same happens in monopolistic competition. In perfect competition the supply curve is the marginal cost curve. When the marginal cost curve shifts the supply curve, which is the same, shifts. For example, in perfect competition if the marginal cost curve shifts upwards the quantity produced decreases while the price increases. How can we incorporate changes of marginal cost in this model? These changes can be included as shifts of the s.r.a.s.c. When marginal cost changes the s.r.a.s.c. shifts. For example, when the marginal cost curve shifts upwards the s.r.a.s.c. shifts upwards: equilibrium price is increased equilibrium quantity produced decreases. In the short run we assume the marginal cost curve does not shift because fixed costs, plant and machinery are the same, the demand curve shifts, please, see Technical information 2.

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Individual income can grow as fast as productivity rises.

Alex Berenson

The law of diminishing returns gives us that for every unit of additional output produced, the addition to total cost increases more rapidly. Therefore, in perfect competition the supply curve is a positively sloped curve that as output increases is steeper. When we ask the firms how much they are willing to produce at different prices we derive the supply curve done in an empirical fashion. Perfect competition, monopoly, monopolistic competition, oligopoly comprise the economy and are all added up to define the short run aggregate supply curve (s.r.a.s.c.). In this essay some of the effects of changes in aggregate demand(A.D.) are taken into consideration. When the marginal cost curve shifts the s.r.a.s.c. shifts. In the long run, buildings and machinery increase but so do the wages therefore the long run aggregate supply curve is vertical at national income of full employment Y. At liquidity trap the A.D. is vertical, we prove it as this essay unfolds, resulting in sluggish economic growth or if the s.r.a.s.c. does not meet the A.D. in negative economic growth and negative inflation. It is proved that all short run supply curves are valid at price equal or greater than the average variable cost (A.V.C.). Finally, we tackle economic problems of countries such as liquidity trap, inflationary pressures and try to solve them. The solutions are suggestions.

Marginal cost

Technical information 1

Perfect competition

Monopoly

Technical information 2

Monopolistic competition

Oligopoly

Adding up all markets

An example with figures

When A.D. increases

When marginal cost changes

Shape of the s.r.a.s.c.

Long run aggregate supply curve is vertical

Liquidity trap

Average variable cost ≤ price, for production

When s.r.a.s.c. does not meet A.D.

Liquidity trap

Long run aggregate supply curve shifts

Tight fiscal policy

Deficit financing

Marginal cost

Marginal cost(M.C.) is the cost incurred in producing one additional unit of output. It is the addition to Total cost(T.C.) for producing 1 additional unit of output. The law of diminishing returns gives us that the marginal cost increases at a faster pace. Why?

When we add input to produce more goods, for example, when we add workers (labor) there comes a time that the output does not increase by the same proportion of the additional labor employed. Too many workers collide, their relative space decreases they are more disorganized. The output they produce is not proportional to the input of labor. For example, a factory building with the machines has a capacity of 100 workers if the workers are increased to 200 the output will not double. Workers are not the only variable cost. If the firms' demand for raw materials increases the price of raw materials will increase thus marginal cost progresses. Therefore, to produce one additional unit the cost (wages) is increasing at a faster pace. As output increases the cost is rising at a steeper rate: diseconomies of scale. The following diagram illustrates it:

Figure 1
As quantity produced increases the M.C. is steeper.

From the graph below we conclude the M.C. curve is the supply curve in the short run in perfect competition we explain below.

Figure 2
From figure 2 at output A, M.C. is more than the marginal revenue(M.R.) which in perfect competition is the price. M.C. is greater than the price. The firm loses money so it will produce less. At output C, M.R. is greater than M.C. the price of one unit of output is $150- the M.C.=$33-. The firm gains profit and it will produce more to gain more profit until M.C.=M.R. at B (see figure 2). If it produces more than B M.C.>M.R. the firm loses money. The firm will produce at B where Price=M.C. The firm produces where the price of the product=M.C. it produces that output. Therefore, the M.C. curve is the supply curve in perfect competition(pc). At a given price how much it will produce: supply curve.

Technical information 1

Consider the following example of a pizza firm, in the year 2018, for simplicity values are rounded:

The accountant of the firm can provide the selling price for different values of output and will be like the table above. Recent past data with other values than the current value can be used, adjusted for inflation, to give the values in the table. By an interpolation(given the values the equation is derived) calculator these values give the function:

P= (q^{2})/2,000,000 -q/400 +6

P: price per unit of pizza

q: quantity in units of pizza

Now we take 2018 statistics, for example, the total production of units of pizza in 2018 of the economy is 100,000 and the price is $6- per unit. We take the 2018 prices to be price= 100-. Now if price increases by 50% the new price= 150-. This means that now the price of pizza is $9- (6x1.50= 9). From the table above we find the corresponding output which is 6,000 units of pizza.

x= Total production, at price $9-=

(6,000✕100,000)/5,000= 120,000- units of pizza

To repeat, what we do is if price increases by 50% we find the new price: $9- and the corresponding output from the equation:

P= (q^{2})/2,000,000 -q/400 +6 from an inverse function calculator we find: **Output=q=500(√ 8P-23 +5)**

Output=q=500(√ 8✕9-23 +5)= 120,000- units of pizza

assuming that the market share of pizzas of the firm in the example remains the same and also the supply curve of the pizza firm is the same shape with the other pizza firms which are both acceptable assumptions to make. We do the same with all other goods and services in GDP(national output or national income(Y)) we choose one representative firm and calculate the total production of the industry as it was with pizzas and add them all up.

Then we find:

*the price is $6 not $9 to be able to compare GDP outputs at the same price: Real GDP at price 150.

We plot a graph by finding the points:

(100, GDP_{2018})

(150, as calculated)

(200, calculated)

Consider the following example of a pizza firm, in the year 2018, for simplicity values are rounded:

Output production units of pizza per year |
Price per unit of pizza $ |
---|---|

3,000 | 3 |

4,000 | 4 |

5,000 | 6 |

6,000 | 9 |

7,000 | 13 |

The accountant of the firm can provide the selling price for different values of output and will be like the table above. Recent past data with other values than the current value can be used, adjusted for inflation, to give the values in the table. By an interpolation(given the values the equation is derived) calculator these values give the function:

P= (q

P: price per unit of pizza

q: quantity in units of pizza

Now we take 2018 statistics, for example, the total production of units of pizza in 2018 of the economy is 100,000 and the price is $6- per unit. We take the 2018 prices to be price= 100-. Now if price increases by 50% the new price= 150-. This means that now the price of pizza is $9- (6x1.50= 9). From the table above we find the corresponding output which is 6,000 units of pizza.

Therefore at

price | output | Total production |
---|---|---|

6 | 5,000 | 100,000 |

9 | 6,000 | x |

(6,000✕100,000)/5,000= 120,000- units of pizza

To repeat, what we do is if price increases by 50% we find the new price: $9- and the corresponding output from the equation:

P= (q

Output=q=500(√ 8✕9-23 +5)= 120,000- units of pizza

assuming that the market share of pizzas of the firm in the example remains the same and also the supply curve of the pizza firm is the same shape with the other pizza firms which are both acceptable assumptions to make. We do the same with all other goods and services in GDP(national output or national income(Y)) we choose one representative firm and calculate the total production of the industry as it was with pizzas and add them all up.

Then we find:

Price | N.Y.(Y) |
---|---|

100 | GDP_{2018} |

150 | (6*✕120,000(no. of pizzas) +all other...) |

We plot a graph by finding the points:

(100, GDP

(150, as calculated)

(200, calculated)

Perfect competition

Using the above technique we apply it to perfect competition(pc) and we derive the pc equation and graph by interpolation:

P=αY ^{n}+βY ^{n-1}+...+γ

αY ^{n}+βY ^{n-1}+...+γ>0, n integer >0, dP/dY>0, d^{2}P/dY^{2}>0

∴

Figure 3

Monopoly

In monopoly, there is no unique relation between market price and quantity supplied. There is no distinct supply curve under monopoly. Yet when the demand curve shifts to the right (increases) there is a relation between increase in prices and increase in quantity supplied as the following graph shows under the assumption that the demand curve shifts parallel:

Figure 4
The monopolist produces that output where M.C.=M.R. to maximize profit as it was explained. From the graph above you can see the output determined is at the intersection of the M.C. curve (black line-) and the M.R._{1}, M.R._{2}, M.R._{3} curves (dotted lines). The monopolist sets the price at which consumers are willing to pay at the determined output the demand curves D_{1}, D_{2}, D_{3}. Where the output meets the demand curve we get the price. From the above graph you can see that the change in quantity ΔQ increase is less than the change in price ΔP (increase). The price compared to quantity increases at a larger rate.

Please see figure 5, below.

The demand curve is the average revenue(A.R.) curve.

A.R.=Total revenue/quantity = price✕quantity/quantity = Pq/q = P

Let the equation of the demand curve be P_{D}=mq+c where m: slope, c: intersection to the y-axis or Price-axis

Marginal Revenue= d(T.R.)/dq = d(P_{D}q)/dq = d(mq^{2} + cq)/dq = 2mq+c

The slope of the demand curve is m and is negative. As proved the slope of the M.R. curve is 2m it is twice as steep.

The A.R. curve meets the q axis:

0=mq+c

mq=-c =>q=-c/m

The M.R. curve meets the q-axis:

0=2mq+c =>q=-c/2m

The horizontal distance from(0,0) is half for the marginal curve. When M.R. is zero it intersects the q-axis at q=-c/2m at that q the A.R.= mq+c= m(-c/2m) + c= c/2

When M.R.=0 A.R.= c/2

A.R.=y

M.R.=x

M.C.=z

m

c

x_{1}=2mq_{1}+c_{1}

x_{2}=2mq_{2}+c_{2} => Δx= 2mΔq+ Δc

At the two points of intersection between M.R.s and M.C. we have:
z_{1}=m_{c}q_{1}+c_{c}

z_{2}=m_{c}q_{2}+c_{c} => Δz= m_{c}Δq

∴Δz= Δx

∴m_{c}Δq= 2mΔq+ Δc

∴Δc= m_{c}Δq- 2mΔq —✱

y

∴Δy= mΔq+ Δc

From just the above equation and from ✱: Δy= mΔq+ m_{c}Δq- 2mΔq

∴Δy=m_{c}Δq- mΔq

∴Δy= (m_{c} -m)Δq

∴ΔP= (m_{c} -m)Δq

∴ΔP/Δq= m_{c} -m

∴P= (m_{c} -m)q+ γ_{2}

∴P= β_{2}q+ γ_{2}

β_{2}= m_{c}- m, m_{c}>0 the slope of the marginal cost curve, m<0 the slope of the demand curve which is negative so -m is positive therefore β_{2}>0, γ_{2} is a number

Figure 5
Why is the demand of a monopolist steep, |m| is high?

Because consumers cannot switch to another brand it is monopoly. A change in price brings a small change in quantity sold and thus produced. Therefore we relax on the assumptions that the demand curve shifts and not pivots(elasticity of demand changes dramatically) and also the M.C. is a straight line that does not shift (please see figure 5).

To express q units of output produced into Dollars we multiply q by price

We find β'

P= β'

If the above monopoly

0.3P= 0.3β'

∴0.3P= 0.3β'

∴0.3P= 0.3β'

We add all other monopolies so P

Therefore the point 2018 will be:

P=100, Y

p

q

∴

Technical information 2

Figure 5.1 As explained previously the perfect competitive firm will produce that output where M.R.=M.C. to maximize profits. The M.R. curve is the same as the A.R. curve which is the demand curve and it is parallel to the x-axis and to other demand curves. The points of intersection between M.R. curves and the M.C. curve describe the supply curve for example at P_{A} it will produce q_{A}, at P_{B} it will produce q_{B}, at P_{C} it will produce q_{C} but these points as it is illustrated is the M.C. curve: points A, B, C.

Figure 5.2 Please, see also figure 4. In monopoly the monopolistic firm will produce the output where M.R.=M.C. to maximize profits where M.C. curve intersects the M.R. curve. At that output we find the price we move upwards vertically until we meet the demand curve, D, of the monopolistic firm. There, is the price of the product. Therefore, at output q_{A} the price is P_{A}. At output q_{B} the price is P_{B} and at output q_{C} the price is P_{C}. The line which connects these points A, B, C,: (q_{A}, P_{A}) (q_{B}, P_{B}) (q_{C}, P_{C}) is the supply curve of the monopoly the philosophy is the same as in perfect competition. Of course, we assume that the demand curve shifts parallel to the previous demand curve. If it pivots we connect the points as done above and we will find the supply curve.

Figure 5.1 As explained previously the perfect competitive firm will produce that output where M.R.=M.C. to maximize profits. The M.R. curve is the same as the A.R. curve which is the demand curve and it is parallel to the x-axis and to other demand curves. The points of intersection between M.R. curves and the M.C. curve describe the supply curve for example at P

Figure 5.2 Please, see also figure 4. In monopoly the monopolistic firm will produce the output where M.R.=M.C. to maximize profits where M.C. curve intersects the M.R. curve. At that output we find the price we move upwards vertically until we meet the demand curve, D, of the monopolistic firm. There, is the price of the product. Therefore, at output q

Monopolistic competition

In monopolistic competition there are many firms with freedom of entry and exit (competition). Yet each firm has some power over price because each sells a product that is somehow different from the firm's competitors (monopoly). The demand curve is rather elastic because there are substitute products from other firm competitors: As it was mentioned in monopoly the demand curve does not pivot because the elasticity of demand normally does not change it is a monopoly, consumers cannot switch to other goods. In monopolistic competition, however, the demand curve of a firm pivots because consumers can consume other similar products as the graph below demonstrates.

Figure 6
When the demand curve pivots from D_{1} to D_{2} there is a small change in price ΔP.

We showed in monopolistic competition(mc) that variation in quantity changes the price.

Now let:

A.R.=y

M.R.=x

M.C.=z

m_{c}: slope of the M.C. curve

c: intersection of the A.R. & M.R. curves or y & x curves to the y-axis or Price-axis

c_{c}: intersection of the M.C. curve or z curve to the y-axis or Price-axis

y_{1}= m_{1}q_{1} + c

y_{2}= m_{2}q_{2} + c

∴Δy= m_{1}q_{1} - m_{2}q_{2}

x_{1}=2m_{1}q_{1}+c

x_{2}=2m_{2}q_{2}+c

∴Δx=2m_{1}q_{1}-2m_{2}q_{2}

z_{1}=m_{c}q_{1} + c_{c}

z_{2}=m_{c}q_{2} + c_{c}

∴Δz=m_{c}Δq

But Δx=Δz

∴2m_{1}q_{1} - 2m_{2}q_{2}= m_{c}Δq —✱

But Δy= m_{1}q_{1} - m_{2}q_{2}

From just the above equation and from ✱: Δy= 1/2m_{c}Δq

∴y= 1/2m_{c}q + γ'

P= 1/2m_{c}q + γ'

P_{mc(1)}= β'Y_{mc(1)}+ γ'

∴β'= 1/2m_{c}, γ' is a number

Consider the following more realistic case:

The demand curve moves at random that is it shifts and pivots from D_{1} to D_{2}:

Figure 6.1
What we do is that we split this change in demand from D_{1} to D_{2} in 2 movements: From D_{1} to D_{1}' (a parallel shift) and from D_{1}' to D_{2} a pivot.

From D_{1} to D_{1}' as already proved in monopoly:

P= (m_{c} -m)q + γ'

From D_{1}' to D_{2} as already proved in monopolistic competition:

P= 1/2m_{c}q + γ

In order to find the full effect we add them up:

**P= (1 ^{1}/2m_{c} -m)q + γ_{1}'**

Where m

If the demand curve pivots to the left and not to the right, as we took into consideration just above, we subtract 1/2m

What if the demand curve moves a third time randomly and the new point(quantity, Price) is not on the supply curve. We do either regression analysis (best line through the points) or interpolation (best equation that describes the points) and we find the best possible equation of Price and quantity. We know one point P=100, p

Subsequently, we do the same analysis as done in technical information 1 and monopoly, the principle is the same, and we derive the equation of the monopolistic competition:

**P _{mc}= β_{3}Y_{mc}+ γ_{3}**, β

Oligopoly

In oligopoly few firms compete with each other. Each firm has enough power so that it cannot be a price taker(take price as given), but it is subject to enough inter-firm rivalry that it cannot consider the market demand curve as its own. **This is probably the dominant market structure outside agriculture and basic industrial materials**. In oligopoly the price is sticky:

In the figure above the oligopolistic firm is assumed to be selling q_{1} units at a price of p_{1}. It then considers altering its price and it makes two key assumptions. First, it assumes that if it cuts its price, all of its competitors will match its price cut. Its demand will then expand along the relatively steep curve below a, which indicates the effect of the firm's price cut when its share of the market is unchanged. Second, it assumes that if it raises its price, above p_{1}, none of its competitors will raise theirs. Its demand for prices above p_{1} thus contracts along the relatively flat curve indicating the effects of raising prices with a declining market share. To summarize, above price p_{1} the competitors will not increase the price thus the firm loses a lot of demand: a small increase in price leads to a large decrease in demand the demand curve is flatter. At price below p_{1} the competitors will also reduce the price there will be no change in the firm's demand, competitors follow suit therefore the demand curve is steep a change in price brings a small change in output. Therefore, changes in the (M.C.) do not alter the price: see figure 7 M.C._{1}, M.C._{2}.

Figure 8
As demand increases the demand curve shifts from D_{1} to D_{2} the quantity changes from q_{1} to q_{2} but the equilibrium price remains the same p_{1}

In oligopoly as quantity changes price is constant.

**P _{o}= 100**

Adding up all markets

To conclude, we add up all markets of the economy as displayed above to derive the short run aggregate supply curve (s.r.a.s.c.) of the economy. We assume that perfect competition= 10% of the economy, oligopoly= 50%, monopoly=10% and monopolistic competition= 30%. Therefore, Y_{pc}= 0.1Y=> Y_{pc}^{n}= 0.1^{n}Y^{n} and Y_{m}= 0.1Y. The P_{pc} to express it in terms of Price, P for the whole economy we multiply the whole pc equation by 0.1 because this is the share of pc in the economy to derive the weighted average for the whole economy (P_{pc}=0.1P). We integrate the analysis done separately before and the just previous assumptions and we have:

Perfect competition

0.1P= 0.1α_{1}✕0.1^{n}Y ^{n}+...+0.1✕0.1β_{1}Y+0.1γ_{1}

Monopoly

0.1P= 0.1✕0.1β_{2}Y+ 0.1γ_{2}

Monopolistic competition

0.3P= 0.3✕0.3β_{3}Y+ 0.3γ_{3}

0.5P= 50

P=0.1α_{1}✕0.1^{n}Y ^{n}+...+(0.1✕0.1β_{1}+0.1✕0.1β_{2}+0.3✕0.3β_{3})Y+0.1γ_{1}+0.1γ_{2}+0.3γ_{3}+50

The above equation applies for Y≥Y_{avc.min} because the supply curve for pc is at an output, Y, greater or equal than where the minimum of the average variable cost(avc) curve occurs because at a value less than the minimum avc firms make losses hence they close down, it will be later explained and proved in detail.

Monopoly 11.111%

P_{m}= β_{2}Y_{m}+ γ_{2}

Pc is not included that is 10% of the economy is not included the remaining is 90% to include the whole economy 100%: the increased share of the monopoly in the economy is: 100/90✕10%= 10/9✕0.1= 0.111= 11.111%:

P_{m}= 0.111β_{2}Y+ γ_{2}

0.111P= 0.111✕0.111β_{2}Y+ 0.111γ_{2} —✱

0.556P= 55.556 —✱

Pc is not included that is 10% of the economy is not included the remaining is 90% to include the whole economy 100%: the increased share of the monopoly in the economy is: 100/90✕10%= 10/9✕0.1= 0.111= 11.111%:

P

0.111P= 0.111✕0.111β

Monopolistic competition 33.333%

P_{mc}= β_{3}Y_{mc}+ γ_{3}

P_{mc}= 0.333β_{3}Y+ γ_{3}

0.333P= 0.333✕0.333β_{3}Y+ 0.333γ_{3} —✱

0.556P= 55.556 —✱

P= (0.111✕0.111β_{2} +0.333✕0.333β_{3})Y +0.111γ_{2} +0.333γ_{3} +55.556

∴

Overall the s.r.a.s.c is a positively sloped curve which becomes steeper as national output increases because in the short run firms have a specified capacity leading to a s.r.a.s.c. which has a vertical asymptote at a large national output, please see figure 3.

An example with figures

Below there is an **example** of a s.r.a.s.c.:

The given equations are not the real US GDP they are just an example which resemble reality more realistic equations can be obtained from empirical evidence that is at given prices how much the producers are willing to produce as explained before. The base year, Price=100, is 2018. US GDP_{2018}= $20.5 trillion.

Perfect competition

The equation for perfect competition(pc):

P_{pc}= 176Y_{pc}^{4}-859Y_{pc}^{3}+1571Y_{pc}^{2}-1279Y_{pc}+411.878

Please, note that the supply curve for pc starts where the M.C. meets the Average Variable Cost curve (a.v.c.c.) at minimum. We have to derive the a.v.c.c. We integrate M.C. equation which is above and we get:

T.V.C.= 35.2Y_{tvc}^{5} -214.75Y_{tvc}^{4} +523.6667Y_{tvc}^{3} -639.5Y_{tvc}^{2} +411.878Y_{tvc} +k

In the above equation T.V.C. is the Total Variable Cost(tvc), Y_{tvc} is the output is the variable and k is a constant number.

The input of variable cost, for example, labor when it is 0 total variable cost is 0 and output Y is 0. Consequently, when Y=0, T.V.C.=0 substituting these values in the above equation we have:

k=0

∴T.V.C.= 35.2Y_{tvc}^{5} -214.75Y_{tvc}^{4} +523.6667Y_{tvc}^{3} -639.5Y_{tvc}^{2} +411.878Y_{tvc}

Then we divide by Y because the above equation is the Total Variable Cost to derive Average Variable Cost:

A.V.C.= 35.2Y_{avc}^{4} -214.75Y_{avc}^{3} +523.6667Y_{avc}^{2} -639.5Y_{avc} +411.878

The a.v.c.c. is as in the graph below (red curve). We find the point of intersection of the two curves and indeed is the minimum of a.v.c.c. where the supply curve for pc starts: it is output=1.9704.

Now, because Y_{pc}= 0.1Y, Y= Y_{pc}/0.1= 10Y_{pc}, Y=19.704 from that point pc equation is taken into consideration in deriving the s.r.a.s.c. Of course we later multiply by 0.1 but the a.v.c.c. is also affected, its minimum is the same: 19.704.

According to the above two equations: M.C. and A.V.C. the graph is as follows:

Figure 8.1
Now, as explained before Y_{pc}=0.1Y we substitute 0.1Y for Y_{pc} to express the above equation in terms of Y, national output:

P_{pc}= 0.0176Y^{4}-0.859Y^{3}+15.71Y^{2}-127.9Y +411.878

The above P_{pc} to express it in terms of Price, P for the whole economy we multiply the above equation by 0.1 because this is the share of pc in the economy to derive the weighted average for the whole economy:

0.1P= 0.00176Y^{4}-0.0859Y^{3}+1.571Y^{2}-12.79Y+41.1878 —✱

Monopoly

P

P

0.1P= 0.021Y + 9.5695 —✱

Monopolistic competition

P

P

0.3P= 0.099Y + 27.9705 —✱

Oligopoly

0.5P= 50 —✱

P= 0.00176Y

For Y<19.704:

Monopoly

P

Pc is not included that is 10% of the economy is not included the remaining is 90% to include the whole economy 100%: the increased share of the monopoly in the economy is: 100/90✕10%= 10/9✕0.1= 0.1:

P

0.1P= 0.0259Y +10.6327 —✱

With the same thinking:

Monopolistic competition

P

P

0.3P= 0.12Y +31.0783 —✱

Oligopoly

0.5P= 55.5 —✱

P= 0.148Y +97.267 (these numbers are rounded to three decimal places)

Combining the equations for Y≥19.704 and Y<19.704 the graph is as follows:

Figure 8.2 Since the two curves end and begin above each other we connect them with a vertical line to make them one curve. At Y=19.704 pc is included and Price drops slightly because goods and services in pc have lower Price than Monopoly, Monopolistic Competition and Oligopoly at Y<19.704 please see equations.

When A.D. increases

Please, see the graph below.

When aggregate demand, A.D. increases: shifts to the right, the increase in price is enough at low values of output. For example, in the graph from A.D._{green} to A.D._{blue} the price went from 99 to 105: change 6. Then by the same amount of increase in A.D. from A.D._{blue} to A.D._{red} the price went from 105 to 159: 54. This is because the supply curve as output increases becomes steeper. At full employment output (supply curve is vertical) an increase in A.D. causes high inflation and zero economic growth(annual percentage increase of real National Output, Y). The shift of A.D._{green} to A.D._{blue} Y increased (in trillions) from 12 to 22: 9. From A.D._{blue} to A.D._{red} Y increased from 22 to 26: 4. The above paragraph is merely an example to demonstrate the effects of changes in A.D. This is what is happening to the US economy right now: inflation is increased, economic growth is reduced.

Figure 9

When marginal cost changes

Figure 9.1

When marginal cost changes it is reflected as a shift in the marginal cost curve. In oligopoly a certain change in the marginal cost curve has no effect on either price or quantity produced, please, see figure 7. In monopoly, for example, an increase in marginal cost (the marginal cost curve shifts upwards) reduces the equilibrium quantity produced and increases the equilibrium price, please, see figure above: qShape of the s.r.a.s.c.

As output increases costs are rising at a faster pace. As output increases labor and raw materials become scarce so the wages, costs rise. Costs are also rising from diseconomies of scale and diminishing returns as previously explained. In all markets perfect competition, monopoly, monopolistic competition, oligopoly, the marginal cost curve leads to finding the output produced and the price (where the marginal cost(M.C.) curve intersects the marginal revenue curve). For example, if the M.C. curve bends upwards: the equilibrium output q is restrained while the price increases. We are talking about one particular M.C. curve. Thus, as output increases the price increases at a faster pace giving us a short run aggregate supply curve which is upward sloping. At very high output the economy's potential is reached: full employment, full capacity, the M.C. curve becomes vertical, thus the output remains constant while price escalates.

Long run aggregate supply curve is vertical

The assumption in deriving the s.r.a.s.c. is that the factory buildings, heavy equipment remain the same. In the long run factory buildings, machinery increase. Thus the s.r.a.s.c. shifts to the right:

Figure 10
At equilibrium the economy is at its natural level of output (potential output) Y and the s.r.a.s.c. meets aggregate demand curve at point A. When buildings, machinery increase the s.r.a.s.c. shifts to the right s.r.a.s.c._{2}. At point B the actual output Y_{2} exceeds the potential output workers are scarce thus wages rise and s.r.a.s.c. shifts to the left to s.r.a.s.c._{1}. Therefore, at point A after the shift we have more factories and machinery and higher wages. In reality the economy is dynamic: aggregate demand and aggregate supply are ever-changing. The above example (figure 10) is a simplification, it demonstrates that the long run aggregate supply curve is vertical.

Liquidity trap

Figure 11
At liquidity trap the money demand curve (M_{d}) is flat as money demand changes the rate of interest is the same r_{1}. The money supply curve M_{S} does not change with interest rate changes in fact the rate of interest does not have an effect on money supply it remains the same: 40. Suppose we have an increase in prices the real money supply decreases because real money supply= M/P= 40 where P is the price. If prices double then real money supply= M/2P= 20.

A change in prices does not change real money demand because the higher the prices the more money people will demand canceling each other out. Money demand= (M/P)^{d} after doubling the price: (2M/2P)^{d}= (M/p)^{d} real money demand is the same.

From the previous graph the rate of interest is constant thus the LM curve is flat. Therefore, national income does not change it remains the same Y_{1} as in the graph below.

Figure 12
Because the Y_{1} remains the same we have a vertical national income or aggregate demand curve. As price changes A.D. is the same:

Figure 13

Average variable cost ≤ price, for production

In the short run a firm should produce if and only if total revenue is equal or greater than total variable cost. A firm to make a profit its total revenues must be greater than its total costs: T.R.>T.C. Profit= total revenues- total costs

Total costs= Fixed costs + total variable costs

Fixed costs: interest, rent, depreciation

Total Variable costs: labor, raw materials for the product

The fixed costs are there even if the output is zero the fixed costs exist.

In the short run the firm produces if the total revenues are equal or greater than the total variable costs. Why? Even if: total revenues- total variable costs= $1- the firm subtracts this one dollar from the fixed costs. The fixed costs will be $1- less. The fixed costs always exist. Except when the firm closes down.

Therefore the firm to reduce its fixed costs produces where the total revenue is equal to or greater than the total variable cost. At that point Total revenues= Total variable costs and at values where the total revenues are greater the firm produces and at that point onwards the marginal cost curve is the supply curve in perfect competition. Please bear in mind that

total revenues= price✕quantity produced

total variable costs= average variable cost✕quantity produced

total revenues= total variable costs

Therefore price✕quantity produced= average variable cost✕quantity produced

Therefore price= average variable cost(A.V.C.)

Consequently, in the short run, the firm produces where the price= average variable cost or at a greater price. This applies to all the markets in the economy: perfect competition, monopoly, monopolistic competition, oligopoly.

Figure 14 As it was previously explained the supply curve is the red line-curve (please see figure above). At price-output less than point B (less than A.V.C.) the firms close down. The supply curve is a vertical line along the vertical-axis, price-axis at output 0 until point A. Then the supply curve is the curve from point B onwards.

Figure 15 When wages, costs, prices fall: Supply curve shifts to the right, down (please, see the figure above). There is a vertical movement from A to B the supply curve moves vertically down from A to B the new supply curve is the red line curve. Please, note that at price less than B the supply curve moves from B to C. There is a horizontal movement from B to C.

When s.r.a.s.c. does not meet A.D.

Figure 16
At the above graph there is inconsistency: point A the s.r.a.s.c. does not meet the A.D. curve. There is excess supply. When there is excess supply the prices fall. When prices fall below A the supply curve moves from point A to point B. What that means is that supply moves to zero, firms not having at least their average variable costs covered stop production they are closing down there is a movement: a arrow (please, see figure below). When firms are closing down unemployment increases demand for raw materials decrease. Wages and prices decrease therefore the supply curve falls there is a movement: b arrow. The result is a diagonal movement of the s.r.a.s.c. from point A to point E. At point E prices fell and output fell. This is what happened during the Great Depression in U.S.A. and to a lesser effect in Germany and Japan in 2009.

Figure 17

Liquidity trap

The problem with liquidity trap is that individuals do not deposit their money in the banks. The European Central Bank (ECB) injects money into the economy individuals take the money out of the economy by holding money as cash, money demand is flat please figure 11. Why is this happening? Europeans do not feel safe by depositing their money in the banks because the banks may default. What the ECB, Bank of England and Bank of Japan should do is to guarantee all bank deposits: in case a bank goes bankrupt the central bank will pay for the deposits and be legally bound to keep their word. Central banks have an unlimited supply of money. Secondly, even if all bank deposits are guaranteed individuals are indifferent of depositing their money because the rate of interest is very low. They do not forego any interest so they keep money as cash. Therefore, the interest rates should be increased this can be achieved by increasing the discount rate (the interest rate central banks charge the commercial banks for lending the commercial banks) and by increasing the funds rate (the interest rate banks charge each other). When banks need money they will sell bonds that they hold they will not borrow from the central bank or other banks because the rate of interest is higher. By selling bonds the price of bonds will go down the rate of interest will go up. There is the paradox how interest rates can increase (and they should be increased) and the economy and inflation grow? At a higher interest rate the opportunity cost of holding money increases individuals will deposit their cash injecting thus money into the economy. This will lead to banks using and/or lending money because they will have more cash from depositors and because the lending rate will increase. Banks will trade off between risk of loan and higher interest and they will lend more. I suggest the interest rate on deposits be driven to 3% even if selling bonds is required. As US interest rate rose there is an upward pressure on the interest rates in the rest of the world. Do not fight these upward pressures let the interest rates rise. Thus, evading the liquidity trap. The above hold true for the Eurozone, United Kingdom and Japan. Also, with E.U. citizens leaving the U.K. there has to be a way of increasing the labor force. I think this could be achieved if a tax is imposed on couples who live in the same house and only one is working and they do not have child/ren below 6 years of age. The tax should be an additional tax of 10% on the income of the person who is working. If both people work this tax does not apply. The money received could be given to married couples who both work for example in the form of increased marriage allowance (allowance: a sum to be deducted from gross income in the calculation of taxable income). This is an incentive for people to work and be married. It will increase the British labour force participation rate. British housewives will displace E.U. citizens in U.K. The classic measure in eliminating liquidity trap is a generous fiscal policy please see figure 12 IS curve shifts to the right. It proved to work in USA but not in Japan which ran generous fiscal policies. Also, if the government wants to reduce the expenses to achieve budget surplus the generous fiscal policy measure cannot be used.

To my Japanese brothers

During the liquidity trap monetary policy is ineffective. Your government buys bonds keeping the wealth of households the same. Households give your government the bonds (decrease of wealth) and earn money (increase of wealth). Instead of owning bonds they own the same amount of money.

Fiscal policy which in liquidity trap is very effective your government sells bonds thus households receive the bonds and when the government spends the money from the sale of bonds households own the bonds and the money, their wealth increases.

National Income= GDP= C+I+G+NX

G is the government spending if the government spending increases the national income increases. It increases by the multiplier effect. Multiplier= 1/(1-mpc). Mpc is the marginal propensity to consume: how much of an additional 1 Yen is spent on consumption. By increasing G the GDP increases you have economic growth.

What can be done is a mixed monetary and fiscal policy: The Bank of Japan (BoJ) prints money or the government increases its deposits at the BoJ and increases the salaries of all Japanese people. It does not buy bonds which takes wealth (bonds) from Japanese households. By doing the above the national income will **certainly** grow by the amount of the (total increase of the salaries that is money printed) ✕ 1/(1-mpc). The interest rate remains the same, there is no crowding out. This measure will continue until inflation reaches an acceptable level. When you stop this measure the effect will go on for a period of time. The just above is demand-side economics, how A.D. is derived. How A.D. and A.S. interact please, see Figure 9.

Long run aggregate supply curve shifts

In conclusion, the s.r.a.s.c. is a diagonal curve starting from low price and low national output (Y) and as Y increases it becomes steeper, at large Y there is a vertical asymptote because in the short run firms have a maximum capacity and because of diminishing returns and diseconomies of scale. The s.r.a.s.c is derived empirically and perfect competition, monopoly, monopolistic competition and oligopoly are taken into account. Some effects of changes in A.D. are examined. In the long run the marginal cost curve shifts so the s.r.a.s.c shifts. Also, in the long run the s.r.a.s.c. shifts to the right because more buildings and machinery are used but the s.r.a.s.c. shifts back to the left because wages increase therefore the long run aggregate supply curve is vertical. At liquidity trap the A.D. is vertical and if it is low the s.r.a.s.c. does not intersect it leading to a reduction in both prices and Y. The short run supply curves hold true for price equal or greater than the average variable cost as previously thoroughly explained. Now, if s.r.a.s.c. shifts to the right and long run supply curve shifts to the right because natural level of output increases then prices fall and Y is increased, please see figure below. This can be achieved if the labor is increased if the number of workers increases. The US government can increase labor if it taxes couples living in the same house and only one is working and they do not have child/ren below 6 years of age by decreasing the tax deduction from their adjusted gross income. The US government can increase the tax deduction of couples who both work and are married with a church wedding and are husband and wife. This measure will increase the US labor force participation rate increasing the natural level of output Y decreasing the prices, please see figure below. How can productivity increase? The US government can introduce a measure that when an employer dismisses an employee the salary of the employee is deducted from the net profit before tax of the employer assuming no other person is then employed in a similar job of the dismissed employee. When a new employee takes the similar job this measure stops. This measure cannot be used if the period of the dismissal and employment is less than 3 months. This measure will reduce the number of unproductive employees and compel the employees to work more productively in order not to be dismissed. Also, it will have a positive effect on unemployment.

Figure 18

Tight fiscal policy

Is there a way to reduce inflation and simultaneously reduce interest rates? Yes, there is. If the government runs a tight fiscal policy for example if government expenses are reduced:

Figure 19
A tight fiscal policy shifts the IS curve down because National income Y is reduced. Government spending is one of the components of Y.

Figure 20
Government spending is reduced consequently aggregate demand is reduced, shifts down reducing Y to Y_{3}.

In the above two graphs interest rate is reduced from r_{1} to r_{2} and price level is reduced from P_{1} to P_{2}.

Deficit financing

We all want lower taxes and a generous government. How can we achieve this without causing inflation to rise?

As Professor William H. Branson says in his book Macroeconomic Theory and Policy there are 5 ways for a government to finance its deficit:

1. The US government can sell bonds to the nonbank public. It is rather inflationary.

2. The US government can sell bonds to the commercial banks. Thus, the banks pay the US government and reduce lending to the private sector. It is rather inflationary.

3. The US government reduces its demand deposits(money it has in the commercial banks) at the commercial banks. It is inflationary.

4. The US government issues and sell bonds to the Federal Reserve System which Fed prints money, buys the bonds giving the money to the US government. It is highly inflationary.

5. The US government takes money from the Fed. It is highly inflationary.

What the US government can do is to sell bonds **only** to US people and institutions inside U.S.A. By doing this it takes money out of the economy and when it spends it, it puts the money back into the economy. When the US government sells bonds to foreigners there is an injection of money into the economy from outside. The US government must not sell its bonds to foreigners because the foreigners flood the US economy with US dollars overheating the already hot US economy. Another way to see it is that the US government does not take money out of the US economy but from abroad, which is inflationary.

I mentioned 4 measures to combat US inflation the fifth and most important is please pray that inflation is reduced and you may prosper.

Christmas is a time when kids tell Santa what they want and adults pay for it. Deficits are when adults tell the government what they want and their kids pay for it.

Richard Lamm

Bibliography

Macroeconomics, N. Gregory Mankiw

Macroeconomics, Rudiger Dornbusch, Stanley Fischer, Richard Startz

Macroeconomic Theory and Policy, William H. Branson

Macroeconomics, Paul A. Samuelson, William D. Nordhaus

Macroeconomics, Dernburg, McDougall

Principles of Economics, N. Gregory Mankiw

An introduction to Positive Economics, Richard G. Lipsey

Economics, David Begg, Stanley Fischer, Rudiger Dornbusch

Economics, Paul A. Samuelson, William D. Nordhaus

Fundamental Methods of Mathematical Economics, Alpha C. Chiang

Microeconomic Theory A Mathematical Approach, James M. Henderson, Richard E. Quandt

Price theory and applications, Jack Hirshleifer

January 2020

The author is an economist, a graduate from City, University of London

Lord our God, our all-powerful Father, You who gave birth to wisdom let all the words in this essay be Yours and not mine. In Jesus' name. Amen.

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