on Notes the Economics of Land Use
Susan E. Stratton
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Land Rent
The key concept for land rent is tied to scarcity. Since land is scarce, the owner of the land can claim the economic value of that scarce resource. In our basic model, land is the only scarce resource, so only land can earn positive economic profits. In some of our models all land is identical. In others, land varies by location or productivity. Land values or rents will adjust to consider those changes in either land or productivity. The models we develop are designed to show us how they adjust.
1.1
Approach 1: Marginal and average worker productivity
Consider a landowner who can hire workers at a wage rate w. As more workers are added, their marginal and average productivity will (eventually) decline. The landowner will maximize profits by solving maxn pq (n) - wn (1) where n is the number of workers, p is the price of output and q () gives productivity as a function of the number of workers. Clearly, the solution to this problem is pq (n) - w = 0 (2)
or that marginal productivity is equal to the wage rate. Because we've assumed an infinite supply of workers at a wage rate of w, the workers will all receive w. The landowner will get pq (n) - wn or n pq(n) - w . This is equivalent to the average product per worker minus the n wage rate times the number of workers. We can draw two different graphs showing this.
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1.2
Approach 2: Variation in land productivity
Consider multiple plots of land. Plot A is very productive, plot B less so, and so on. Mathematically, this means that M PA > M PB > M PC and so on. 1.2.1 Fixed supply of workers, two plots
We'll equate the marginal product of labor on the two plots and set the wage rate that way. Clearly, if the marginal product of labor where higher on one plot than the other, we should shift workers toward the more productive plot. Thus the marginal product must be equal in equilibrium.
1.2.2
Infinite elastic supply of workers
All landowners solve a problem like the one in section 1.1, setting the marginal productivity of their land equal to the wage rate. Each plot earns a rent determined by its marginal productivity. The marginal plot of land earns zero economic rent.
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1.2.3
Upward-sloping labor supply curve
In this case, the wage rate is determined by the intersection of the marginal product on the marginal plot with the labor supply curve. How is surplus divided?
1.3
Market structure and land rent
Let's compare social welfare maximization with competition. Remember while we will find as usual that competition is social welfare maximizing, we have to prove this. B () = the benefit of agricultural production q (n) = productivity (in units of output) of n workers on one plot of land Q (N ) = productivity (in units of output) of N workers on all plots of land Assume all plots are identical and the labor supply is infinitely elastic at a wage rate of w 1.3.1 Social welfare maximization
Workers get w regardless so their wages are not part of the analysis. The social planner wants to maximize maxN B (Q (N )) - wN. (3) Make sure you understand why this is the expression for the social surplus i.e. how you get this from adding consumer and producer surplus. Using the chain rule, the first-order condition is B (Q (N )) Q (N ) - w = 0. (4) 1.3.2 Competition with each landowner controlling her own plot maxn pq (n) - wn which has first-order conditions pq (n) - w = 0 3 (6) (5)
which tells us the supply curve of an individual farm. To find a market equilibrium, we will use three conditions. 1. Producers maximize profits the condition we just derived, i.e. equation 6. 2. Consumer utility maximization implies that p = B (Q) where Q is the total quantity sold in the market. This is equal to Q (N ). 3. Notice that since all farms are identical, they'll all employ the same number of workers. This implies that q (n) is constant across all firms in equilibrium, which means that no matter where we put another worker, it will have the same impact on output. This means that Q (N ), the change in total output from adding one more worker total is the same as q (n), the change in a single farm's output from adding another worker to that farm. Using each of these conditions, we can determine the full equilibrium. To do, we'll substitute B (Q (N )) for p and Q (N ) for q (n) into equation 6, giving us exactly the same equation as social welfare maximization (Eqn. 4). 1.3.3 Open-access
If there are many farmers with access to the same land, workers will be added until profits of every farmer are zero. The key assumption is that each farmer with workers on the land receives the same output per worker. Moreover, in open-access, individual farmers considering sending workers out onto the land perceive the output they get per worker as constant. Individual farmers therefore have production functions that look like q = qn. This leads to a maximization problem of the form pqn - wn which has first-order condition pq = w (8) Again, we can substitute in the consumer utility maximization condition which tells us that p = B (Q). Moreover, since all farmers are identical q = Q(N ) . Making both subsitions, we N have Q (N ) B (Q) = w. (9) N Instead of equating marginal value product of labor with the wage rate, we're equating the average value product of labor with the wage rate. If anyone can use the land, someone will try to add workers until are profits driven to zero. This results in more employment and more production but less social surplus. If this story seems a little confusing in terms of workers, think of it instead in terms of cattle. Suppose that the total weight of each cow grazing on a plot of land declines as more cattle are added to the same plot of land. We can therefore think of an average and marginal product of meat as a function of the number of cattle. Further, suppose that cattle can be bought at the beginning of a season at a cost c per cow. Every cow is identical. If each 4 (7)
producer has only one cow, producers will therefore look at the weight of their cow (i.e. the average cattle weight) and not the marginal weight of all cattle when deciding whether to purchase another cow and let it graze on the plot.
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2.1
Competing Uses of Land
Location and value
Basic conclusion land value will decline as we move away from transportation centers because the price farmers receive for their output is adjusted by the cost of getting it to market. Again, the differences in land value accrue to the landowner, because the landowner controls the scarce resource (land close to transportation centers). price of output at port is p cost of transportation is cx where x tells us the distance from the port q = f (l, n) is crop output Farmer at location x will solve maxl,n (p - cx) f (l, n) - wn - r (x) l which gives (p - cx) fl = w and (p - cx) fn = r (x) . (12) The quantities (p - cx) fl and (p - cx) fn are the value marginal product (VMP) or marginal revenue product (MRP) of land and labor. Notice that these conditions tell us that that a profit-maximizing producer will set the ratio of marginal products equal to ratio of factor prices. Under constant returns to scale, this implies that the ratio of factor inputs will be the same at any level of production. This means that the output per unit input is determined only by the relative prices of the two inputs and not by the scale of production. The relative prices will vary by location, x. Therefore, we can write the output per unit input (q/n and q/l) as an (x) and al (x). Finally, since landowners control the land, they will be able to extract all potential profits. Therefore, in equilibrim, we must also have (p - cx) q = wn + r (x) l. Dividing through by q gives us (p - cx) = w l n + r (x) q q (14) (13) (11) (10)
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or p - cx =
w r (x) + . an (x) al (x)
(15)
Solving for r (x), we get the bid-rent function r (x) = (p - cx) al (x) - w al (x) . an (x) (16)
If we focus on Leontieff or fixed-proportion production technology, al (x) and an (x) will be constants, implying that the bid-rent function is a straight line.
2.2
Location and different crops
Basic conclusion crops/products that cost more to transport will be grown/produced close to transportation centers and crops/products that are easier to transport will grown/produced further away. To focus on the basic intuition, suppose both crops/products have fixed proportion technology, so both bid-rent curves are linear. A landowner can derive the highest rent she can change each type of producer. Clearly, a profit-maximizing landowner will set the rent at the highest rent either type of producer is willing to pay. As a result, the producer willing to pay more is the one that will end up renting the land.
2.3
Housing location
Basic conclusion land value adjusts to compensate people who live further away for their commuting costs. Consider a central business district (CBD). Plots of land surrounding the business district are identified by their distance x from the CBD. All workers (one per family for simplicity) work in the CBD and receive a common wage w. The cost of commuting from location x to the CBD is given by t. Household utility has the form U = C - lnA where A is the amount of land consumed and C is disposable income after communting costs and land purchases. maxx,A w - tx - p (x) A + ln (A) which gives us the first-order conditions 6 (17)
-t - p (x) A = 0 and -p (x) + From the first condition, we get A=- Substituting into the second equation, we get p (x) = - or p (x) t t . p (x) = 0. A
(18) (19)
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(21)
t p (x) =- . (22) p (x) This is a differential equation because it relates a function, p (x), to its derivative p (x). To solve it, we need to find a function which satisfies the relationship. This turns out to be a very simple differential equation to solve. We integrate both sides over x. ^ ^ p (x) t dx = - dx (23) p (x) This tells us that ln (p (x)) + c1 = - or that ln (p (x)) = - tx + c2 (24) (25)
tx + c0 where c0 = c1 + c2 . To isolate p (x), we raise e to both sides of the equation or p (x) = e- ec0 .
tx
(26)
Note that we still have the unknown constant c0 in this equation. Any value of c0 we plug in will satisfy the first-order conditions. We need to know the value of p (x) at some location since our model has only told us something about housing prices change over location. Suppose land costs P0 in the CBD.1 This means that p (0) = P0 = e- (0) ec0 . Since anything raised to the 0 power is 1, we have P0 = ec0 and we can rewrite our original function as p (x) = P0 e- .
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tx t
(27)
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(29)
To find P0 we'd need to look at how many people live in the city and find some kind of equilibrium for total land prices. For simplicity, we'll just take P0 as given and look at how land values change as we move away from the center.
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