Department of Economics Queen s University ECON435/835: Development Economics Professor: Huw Lloyd-Ellis Assignment # Answer Guide Due Date:.30 a.m., Monday October, 202. (48 percent) Let s see the extent to which the results of Maniw, Romer and Weil (992) continue to hold using more recent data. On the ECON 435/835 website, clic on "Assignments". Next to Assignment # you will nd data les containing some growth-relevant data for 97 countries. Of these, 65 have data on investment rates in both physical and human capital. (a) Consider the convergence equation from the basic Solow model: g y t;0 = V ( e t ) ln y 0 + ( e t ) where y 0 denotes the initial GDP per worer, V = ( [ln (s ) ln (n + g + )] ; e t ) ln A 0 + gt represents the technological in uence and = ( )(n+g+). Assuming that = =3; g+ = 0:075 and t = 43, use the average population growth rate for the 65 countries to compute the theoretical rate of convergence. What are the implied theoretical values of the two coe cients = ( e t ) and 2 = =( ) in this equation? The mean value of population growth is n = 0:02. From the assumed values of the parameters the theoretical values are = 0.064, = 0.936 and 2 = 0.468. (b) For the 65 countries, plot g i against ln y60 i + 2 ln s i the values of and 2 computed in (a). ln n i + 0:075 ; using What does the slope of this plot tell you? The line has an (insigni cant) estimated slope much lower than one (about 0.08). This tells us that the model underpredict the variation in per capita income by a wide margin. (c) Using OLS, estimate the regression g i = v + a ln y i 60 + b ln s i + c ln n i + 0:075 + i ;
where v; a, b and c are parameters. errors and the adjusted R-squared. Report your estimated parameters and standard Using your estimate for ^a what is the implied value of. Test the restriction that ^b = ^c. From your restricted regression, what is the implied value of. v -0.048 0.00 a -0.006 0.003 b 0.04 0.04 c -0.035 0.034 Adj. R 2 0.384 v 0.002 0.003 a -0.006 0.00 b = c 0.06 0.002 You should nd that the P-value for the F-test is 0.5, implying that the restriction is rejected at the 95% level (d) = 0:000 ^ = 0:73 Now consider the convergence equation from the augmented Solow model: g y t;0 = V ( e t ) ln y 0 + ( e t ) [ln (s ) ln (n + g + )]+ ( e t ) [ln (s H) ln (n + g + )] ; where = ( )(n + g + ). Assuming = = =3, compute the theoretical rate of convergence and the implied values of the coe cients = ( e t ), 2 = ( e t )=( ) and 3 = ( e t )=( ). The theoretical values are: = 0:032; = 2 = 3 = 0:747 (e) Plot g i against ln y60 i + 2 ln s i using the values of, 2 and 3 computed in (d). tell you? ln n i + 0:075 + 3 [ln (s H ) ln (n + g + )] What does the slope of this plot In the gure below the estimated slope is signi cantly positive and now much closer to one (about 0.6). However, the slope is still signi cantly lower than one. The conclusion is that when considering convergence to steady state the Solow model with human capital is a remarable improvement over the general Solow model, although not a perfect one. (f) Using OLS, estimate the regression g i = v + a ln y i 60 + b ln s i + c ln s i + d ln n i + 0:075 + i ; 2
where v; a, b; c and d are parameters. standard errors and the adjusted R-squared. Report your estimated parameters and Using your estimate for ^a what is the implied value of. Test the restriction that ^b + ^c = ^d. From your restricted regression, what are the implied values of and. v -0.06 0.032 a -0.00 0.002 b 0.009 0.003 c 0.009 0.003 d -0.045 0.03 Adj. R 2 0.476 v 0.004 0.003 a -0.009 0.002 b 0.0 0.003 c 0.008 0.003 d -0.09 0.002 You should nd the P-value for the F-test is 0.046, implying that the restriction cannot be rejected at the 95% level. = :0002 ^ = 0:39 ^ = 0:30 2. (28 percent) (a) Assuming a Cobb-Douglas production function with = = =3 and that g + = 0:075; what is the steady state level of output per worer in the augmented Solow model as a function of n, s ; s H and A: The SS level of output per worer is y = Y L = A s s H (n + g + ) +! y = As s H (n + 0:075) 2 According to Hall and Jones (999), the quality of institutions should be important for the proximate determinants n, s ; s H and A, which in turn should be important for y. In the data set used in question, the variable isi is the index of social infrastructure used by Hall and Jones. 3 Of the 65 countries for which we have
estimates of s and s H ; we have the isi for 64 of them (the missing one is Nepal). Let the USA be the country of comparison. (b) For each of the remaining 63 countries, use your answer to part (a) to decompose the income per worer relative to the US, y i =y US in 2003 into components associated with the relative (residual) technology level, A i =A US ; the relative investment rates, s i =sus and si H =sus H and the relative population growth term, ni + 0:075 = n US + 0:075. (c) Plot y i =y US against your estimate of A i =A US across the 63 countries. Comment with respect to whether technological di erences seem to be important for understanding income di erences. The slope estimate is 0.08 and insigni cant. However, removing the two outliers in the southeastern region (Burundi & Rwanda) yields a slope estimate of 0.37 which is statistically signi cant. It thus appears that technological di erences are somewhat important for understanding income di erences. (d) In four gures plot each of A i =A US ; s i =sus, si H =sus H and ni + 0:075 = n US + 0:075 against isi across the 63 countries. Are these plots consistent with Hall and Jones (999) hypothesis? What do these imply for the regressions ran by Maniw, Romer and Weil (992)? All slope coe cients are statistically signi cant except that for A i =A US. The empirics thus seem not to contradict Hall and Jones (999): a higher level of social infrastructure goes hand in hand with higher propensity to invest and to educate and (maybe) higher factor productivity. With regard to the regressions run by Maniw, Romer and Weil (992), these plots suggest that there is liely to be omitted variable bias in their estimates. 4
3 (24 percent). Consider the following version of the basic Solow model where land is required for production and is in xed supply, X. The aggregate production function is given by Y = (AL) X : The remainder of the model is standard: _ = sy, _ L=L = n and _ A=A = g. (a) Show that the per capita production function and the capital-output ratio, z, are, respectively where y = Y=L, = =L and x = X=L. y = A x z = A x + Divide the production function on both sides by L to nd the per capita production function: y = A x. Insert this into the de nition of the capital output ratio: z = A x = A x + (b) Show that the law of motion for z following from the model is given by the linear di erential equation: where = ( ) + (n + g). _z = ( )s z Now taing logs on both sides and di erentiating wrt. time gives: Since =L, one has Finally, since x = X=L, one has Inserting these gives: _z z _z z = ( = ( ) ) _ " = g ( ) _x x x x = Inserting the capital accumulation equation we nd that: _z sy = ( ) z = ( ) s z n # n n g + ( )n (g + n) ( ) (g + n) 5
Multiplying through by z yields the result. (c) Derive the steady state value for the capital-output ratio, z, and explain why z must converge to this value. Set _z = 0 to nd the steady state level for z: z = ( )s ( ) + (n + g) Whenever z < z, _z is positive and whenever z > z, _z is negative. Hence, z will converge towards z* from any initial level. (d) Derive the steady state growth of per capita output. What e ect does the existence of land in the production function have on the long-run growth rate? Taing logs on both sides of the per capita production function and di erentiating wrt. time gives: _y y _y y = _ + g + ( ) _x x = _ + g ( )n Since the capital-output ratio is constant in steady state, and y must grow at the same rate. Hence: ( ) _y y = g ( )n _y y = g ( )n In the standard Solow model we now that _y y = g. Since > 0 or <, from the above equation it is apparent that the existence of land in the production function lowers the long run growth rate because _y y < g. will be the per capita income growth. Moreover, the higher the population growth, n, the lower 6