16. Consider the basic Solow model with population growth where the depreciation rate & is between 0 and 1, and where population evolves according to: Liyi =1 +n)L, Assume the usual Cobb-Douglas production function: Y, = AKFL" (a) Write the accumulation equation for capital per worker. Then, write one expression for capital per person and one for GDP per worker in the steady state, k*and y*, as a function of exogenous parameters only. (5 marks) (b) Using the result just derived, write an expression for consumption per worker in the steady state, *, as a function of exogenous parameters only. Find the consumption- maximizing saving rate. (6 marks) (Continued overleaf) EC2010 (c) How does the growth rate of population impact the optimal saving rate in comparison to an identical economy without population growth? (2 marks) (d) Suppose just for this question that both depreciation rate and growth rate of population are equal to zero. What would happen to the economy in this scenario? (2 marks) Consider now a Solow model where there is full depreciation (6=1) and no population growth (n=0). (e) The economy is in a steady state. Capital per worker is higher than the level of capital per person that maximises consumption per person (k* > k). Would a decline in the depreciation rate lead to a new steady state with higher consumption? Explain briefly. (4 marks) Suppose now that the production function per worker, v, is a function of capital per worker, k, and that there is still full depreciation (6=1) and no population growth (n=0). Assume the following logarithmic production function per worker: v, =A In(1+k,) where A>0. Remember that the derivative of y, = A In(1 + k,) with respect to k; is AJ(1+ky). (f) Write the accumulation equation for capital per worker. (3 marks) (g) Write down the profit function of a representative firm and find the demand function for capital input K; (remember that k; = K; / L;). (5 marks) (h) Does the aggregate production function (i.e., the aggregate version of y, = A In(1+ k.)) have the same properties as the usual Cobb-Douglas aggregate production function? Explain. (3 marks) 16. (@) k sAk;"-i(l}k: kt i () 1-a * A( sA )1_.(. t+1 = 1+n A positive population growth rate implies a higher optimal savings rate compared to an identical economy without population growth, in order to sustain the steady-state level of capital per worker and thus maintain steady-state consumption per worker. (d) Assuming both the depreciation rate and population growth rate are zero, the economy would experience continual capital accumulation since investments would permanently increase the capital stock without depreciation. The constant workforce would mean any capital accumulation directly raises capital per worker. Over time, this would theoretically lead to unbounded growth in output and consumption per worker, as the economy would not reach a steady state. The concept of an optimal savings rate becomes less clear in this scenario, as there's no limit to the returns from savings and investment due to the absence of capital dilution and a need for new workers. (e)When capital per worker is higher than the golden rule level of capital, this implies that the economy is overaccumulating capital: too much output is being invested rather than consumed, which is not optimal for consumption per capita.lf the depreciation rate declines, the steady-state level of capital per worker will increase because less capital is being lost to depreciation each period. As a result, the new steady-state will have more capital per worker than before. The decrease in the depreciation rate would also mean that the economy does not need to invest as much to maintain a given level of capital. Therefore, more of the output can be devoted to consumption, which means that consumption per worker can increase. (f) k]H]_ :sln(l +kt) k-g (9 the profit function II; = p;qu 111(1 + k;) 11K wely the demand function for capital e A K= () (h)Both production functions have some common features such as diminishing marginal returns to capital, they differ fundamentally in elasticity of substitution, returns to scale , distribution of income, and the specific form of the marginal products. The logarithmic production function provided does not have the same flexibility and direct economic interpretations for factor shares as the Cobb-Douglas function