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- Optimization Research in the Construction Sector -

Most esteemed readers of Ekonomi Politik,

One of the sectors that was affected the most by the arrival of the new version is without doubt the construction sector. As a result of the careless modification of the production formulae, the profitability and desirability of the housing, hospital and defense system sectors have changed the negative direction. Because of these changes and the uncertainty regarding future changes, the decrease in production in the construction sector has even superceded the overall stagnation in the economy.

One result of this worldwide drop in construction has been the decreasing construction worker wages, which is due to the fact that the builders, carpenters and architects who have distributed their skill points cannot reallocate these points after seeing the demand. However, this also means that the costs of labour for construction has also decreased in comparison to pre-V2 levels. In addition to this, the amount of health replenished by a house per day has been increased by 400% in a recent update; and these events points towards the possibility that the housing sector has once again become profitable. Furthermore, since the construction firms can create twice the amount of jobs for the same amount of capital, this industry is one of the best for the government to invest in, so that unemployment may be reduced.

In the light of the aforementioned changes, I have decided to undertake an optimization research in the housing industry. It is my hope that my findings and advices will prove beneficial to both private investors and government officials who are thinking of opening or operating firms in the housing sector. Since the cost-benefit analysis of defense systems and hospitals would necessitate a thorough investigation of the wa module, I choose to postpone the affiliated research to a later date.



Quality or Quantity?

In the new economy module, the housing firm owners are required to distribute a limited amount of customization points among three properties: Health (W), Happiness (H) and Durability (D). The health and happiness properties linearly determine the amount of health and happiness that is replenished each day, and the amount of customization points that can be allocated to them is allowed to vary between 0 and 100. The durability property defines how long the house can be used, and it is allowed to have values between 10 and 100. The quality of the house determines the total amount of customization points that can be allocated, and is fixed as 40 times the firm quality.

The optimal choice between health and happiness may vary according to personal preferences. However, independent of this choice, the needs to be a choice between the quality and quantity of the benefits a house provides. As a result, we choose to postpone the problem of health or happiness, and we aggregate the points allocated to these properties under a single variable called Quality ( Q = W + H). In this case, the choice that has to be made is between the quality of the house and the duration of receiving the benefits.

Let’s assign the variable F to the quality of the firm. If we rewrite the constraints we have been talking about mathematically, we reach the following optimization problem:

Objective function to be maximize😛 U(W,H,D) = V(Q,D)

Constraints:

Q = W + H
0 ≤ W ≤ 100
0 ≤ H ≤ 100
10 ≤ D ≤ 100
W + H + D ≤ 40F

The main issue here is the functional form of the objective function U or V. If we can determine it, the problem can be solved. In the theory of economics, and in the problem of determining the prices of financial derivatives, the most usual approach is to discount future benefits to the current date by multiplying them with a discount factor depending on how far the benefits will be received, and calculate a single net present value.

There are possible functional forms other than the discount factor method, like hyperbolic functions; but they usually arise much more rarely, and in cases when the choices have more immediate consequences. Since buying a house in eRepublik is a long-term decision, it is harmless to assume our function has the proposed form.

Since the formulas indicate that the mechanicswise daily benefit of a house is linear in the quality, the net discounted value of a house can be calculated in the following manner:

Discount factor: 0 ≤ B ≤ 1

V(Q,D) = (1 * Q) + ( B * Q ) + ( B^2 * Q ) + ... + ( B ^ D * Q )

Since this function is a geometric series, it can be simplified as follows:

V(K,D) = ( ( 1 – B^D) / ( 1 – B ) ) * Q



Now that we have an objective function, we can move on to solving the optimization problem. This part will be somewhat technical: Since the constraints are linear functions, all the points in the feasible set are also a solution to the corresponding linearized optimization problem, and thus satisfy the characterization qualification. Consequently, this means any optimal solution to the problem needs to satisfy the Karush-Kuhn-Tucker conditions. As a result, what we have to do to find the optima is nothing other than taking the derivative of the function most widely known as the Lagrange function with respect to each choice variable, and set it equal to zero, while also taking note of the complementary slackness conditions.

Lagrange Function:

L(Q,D,a,b,c,d,e) = V(Q,D) + a( Q ) + b( D – 10 ) + c( 200 – Q ) + d( 100 – D ) + e(40F – Q – D)

Karush-Kuhn-Tucker Conditions:

( (1 – B^D ) / (1 – B ) ) + a – c – e = 0

( - D / (1 – B ) ) * ( B^(D-1) ) * Q + b – d – e = 0

0 ≤ Q
0 ≤ 200 – Q
0 ≤ D – 10
0 ≤ 100 – D
0 ≤ 40F – Q – D
0 = aQ
0 = b(D – 10)
0 = c(200 – Q)
0 = d(100 – D)
0 = e(40F – Q – D)

a,b,c,d,e nonnegative real numbers.

If we know the discount factor of the consumers in the economy, we can solve this problem numerically. But is this possible? Of course! We have two methods available:

1) Conducting a survey to understand how much value people assign to benefits in the future (this is not something hard; I can conduct the survey if I have the time. In case I don’t, you can calculate your own discount factor by answering the question I state at the end of this article)

2) Calculating the discount factor by using the existing data on interest rates as a natural experiment. Three months ago, the people in Turkey who bought some treasury bills have made this possible. These people have accepted to pay 45.429 TRY to get 60 TRY in 8 weeks with equal weekly payments. If we ignore the risk of late payment by the government, and if we take the expectation of the trend of exchange rate changes to be zero, it is calculated that these people must have a discount factor of 0.990869 or more in order to find this deal profitable (the decimals after the point do matter, since we are taking their 60th power). If we accept that the aggregate discount factor of the consumers in general is this value (that means we are assuming everyone is more or less as patient as these people – and since we also took the risk premium to be zero, this is a pretty conservative assumption), we reach the following results:



Solution for Q1 House:
Q = 21 D = 19

Solution for Q2 House:
Q = 43 D = 37

Solution for Q3 House:
Q = 67 D = 53

Solution for Q4 House:
Q = 93 D = 67

Solution for Q5 House:
Q = 119* D = 81

In the solution of the problem for houses of quality below Q5, none of the constraints other than the customization point feasibility hold with equality. However, in the case of Q5 houses, if a person would like to invest less than 19 points to either happiness or health (which means this person wants to obtain less than 9.5 health or 1.9 happiness from the house per day), it might be more profitable to assign more than 81 to the durability of the house. In the case of all other houses, the discount factor used implies a unique optimal value for the durability of a house.

According to these calculations the ratio of the net present discounted values of each type of house is as follows:

1 Q5 House = 1.46 Q4 House = 2.42 Q3 House = 5.04 Q2 House = 18.58 Q1 House

This means that the price of the best possible Q5 house must be 18 times the price of the best possible Q1 house. So this can be taken as the proof that Q1 and Q2 houses are unprofitable with the current formulas. The investors and government officials should therefore observe this fact and refrain from producing Q1, and most probably Q2 houses.



Health or Happiness?

The answer to this question will be different for each person. The people who fight daily should prefer houses that focus solely on health. The people who never fight should cover all their wellness needs from their house, since it is cheaper than replenishing it by food, and then allocate the rest of the customization points to happiness.

The citizens who do not fight professionally, but join the fights at least once per day have the opportunity to maintain their health at 70 at no monetary cost. I would suggest this technique especially to new players. My personal advice to these players is to invest all their income to food and housing that gives them happiness, so that they can minimize the time they need to allocate to leisure; thus maximizing the speed of their personal development by having more time to spend in the training fields or the library. This would help to reduce the gap between the new and old players at some degree.

TL😉R. What does all this stuff mean?

In this research, I have determined the optimal quality-quantity ratio for houses with respect to a given discount factor obtained through a natural experiment. The private entrepreneurs and government officials who are planning to open, upgrade or downgrade housing firms should take these calculations into consideration when choosing the durability they want to assign to their products. If you would like to obtain the optimal results for a different discount factor, I would be glad to provide assistance.

Discount Factor Determination Test:

Suppose I guarantee to pay you 60 GBP over 8 weeks, with equal weekly payments. In order to purchase this right, how much would you be willing to pay today? You can find your discount factor according to your answer from the table below:

60.0 GBP : 1.000000 (infinitely patient)
57.5 GBP : 0.998643
55.0 GBP : 0.997210
52.5 GBP : 0.995694
50.0 GBP : 0.994986
47.5 GBP : 0.992374
45.0 GBP : 0.990544
42.5 GBP : 0.988581
40.0 GBP : 0.986466
37.5 GBP : 0.984173
35.0 GBP : 0.981674

If you would give a lesser offer, you are too impatient; you don’t need to buy a house 😉 If everyone can write their own choice in the comments, we can determine the average discount factor for the UK.

I hope that my research will be helpful, and thank you for your interest.

Best regards,

Kord. Kemal Ergenekon
3 x Minister of Economics of eTurkey