Report

ECN741: Urban Economics The Basic Urban Model: Comparative Statics The Basic Urban Model Class Outline The point of comparative statics Open model comparative statics results Closed model comparative statics results Comparative statics tables Comparative statics graphs The Basic Urban Model Why Comparative Statics? Urban models describe urban residential structure when (simplified!) models of 6 markets are combined. A key set of questions involves the way urban residential structure changes when one of the parameters of the model changes. CS involves finding the derivative of the variables in the model with respect to key parameters—accounting for interactions across markets. The Basic Urban Model Why Comparative Statics, 2? Key parameters include Y, t, R , U* (open), and N (closed). With the basic model we can ask what happens to urban residential structure when incomes rise over time; transportation innovation or investment takes place; the cost of non-urban activity (e.g. agriculture) changes; the situation in one city changes (or the opportunities in other cities change); or population increases everywhere (or in one city where out migration does not occur). The Basic Urban Model How to Do Comparative Statics Comparative statics are based on total derivatives, not partial derivatives. CS results must account for all the variables in one of the equations we have derived. For example, many of the equations in the model, including equations for R{u}, P{u}, and D{u}, can be expressed as a function of the parameters and only one variable, namely, u . The Basic Urban Model How to Do Comparative Statics, 2 In these cases, CS derivations are based on equation like this one for R{u}: dR u R u R u du d u d In this equation, δ can be any of the model’s parameters (at least any of the ones that appear in the R{u} equation!). A key to finding many CS derivatives, therefore, is to find the impact of the relevant parameter on u . The Basic Urban Model Open Model Comparative Statics Most CS results are relatively easy to obtain with an open model because of the form taken by the indirect utility function: k Y tu k Y tu * U a R P C or P U Y k u t * a R U Y C k t * The Basic Urban Model Open Model Comparative Statics, 2 In this formulation, u is on the left side of the equation and all the key parameters are on the right. So it is straightforward to derive CS results for the impact of all these parameters on u . These results can then be inserted into the formula given earlier to find CS results for R{u} and other variables. The Basic Urban Model Open Model Comparative Statics, 3 From P U Y k u t * a R U Y C k t * We can see that du 1 0; dY t a du u 0 dt t a 1 du P du aU R 0; * dU kt k t C dR * 0 The Basic Urban Model Open Model Comparative Statics, 4 So the physical size of an urban area: Increases when income rises. Decreases when commuting costs rise. Decreases when opportunities improve elsewhere. Decreases when agriculture becomes more profitable. The Basic Urban Model Open Model Comparative Statics, 5 Now take the expression for R{u} 1 a Y tu R u R Y tu To find the CS derivative for Y, differentiate with respect to Y, recognizing that Y affects u , and plug in the above result for d u/dY. The result: d R u R u 0 dY a Y tu The Basic Urban Model Open Model Comparative Statics, 5 Similarly, we can start with the expression for N R Y b 1 Y tu N u b t b 1 t t b 1Y tu Then we can differentiate with respect to Y, recognizing that Y affects u , and plug in the above result for d u/dY. This yields: d N R Yb 2 1 0 b dY t Y tu The Basic Urban Model Open Model Comparative Statics Table Parameter Variable u R{u} or P{u} or D{u} N Y t R + - - - + - 0 - + - - - U* The Basic Urban Model Closed Model Comparative Statics In a closed model, the derivatives of u with respect to the parameters come from the population equation, now with a bar over the N: R Y b 1 Y tu N u b t b 1 t t b 1Y tu This equation is messier than the one for an open model, but its nonlinearity does not get in the way of CS as it does for solving the model. The Basic Urban Model Closed Model Comparative Statics, 2 For example, after a little algebra one can show that: t b 1 N du 1 0 b 1 2 dR R b Y 1 Y tu Not surprisingly, increasing agricultural rents shrinks the urban area. Recall: b = 1/aα The Basic Urban Model Closed Model Comparative Statics, 3 To find the CS derivative dR{u} / dR , we must substitute this result into: d R u R u tbR du 1 d R R Y tu dR The result (derive as an exercise) is a bit surprising because R{u} decreases at small values of u and increases at large values of u. The Basic Urban Model Closed Model Comparative Statics Table Parameter Variable Y t R N u + - - + R{u} small u:- small u:+ or P{u} or D{u} large u:+ large u: - + + U* - - + - The Basic Urban Model Comparative Statics Intuition We can develop an intuition for these results with some simple graphs for R{u} (or P{u} or D{u}). To interpret these graphs note that Population depends on density and urban size (= u ) A change in Y flattens P{u}and R{u} (remember, Pʹ{u}= -t/H!) A change in t steepens R{u}. The Basic Urban Model CS Result for Y Open Model R{u} Closed Model R can’t drop at u=0 because utility is fixed R{u} With city size increase, density cannot increase at all locations Density declines near center Density increases in suburbs, which grow R R u1 u2 u u1 u2 u The Basic Urban Model CS Result for t Open Model R{u} Closed Model R does not change at u=0 because tu=0. R{u} With city size decrease, density cannot decrease at all locations Density goes up in the center and down in the suburbs, which shrink. R R u2 u1 u u2 u1 u The Basic Urban Model CS Result for R Open Model R{u} Closed Model Utility level (indexed by height of R{u}) cannot change R{u} R2 R2 R1 R1 u2 u1 u City size and density cannot both move in the same direction Implies higher density everywhere u2 u1 u The Basic Urban Model Other CS Results Open Model (U*) R{u} City shrinks and becomes less dense R Closed Model (N) R{u} City grows and becomes more dense R u2 u1 u u1 u2 u The Basic Urban Model Informal Tests of CS Results These results predict that cities will get less dense in the center, more dense in the suburbs, and larger as incomes rise and transportation costs fall. Many estimates of population density functions for cities around the world find this to be true. But the models have only one worksite and many other simplifications. Is this just a lucky coincidence or do the models capture something fundamental?