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ECON7040 Semester 1, 2022

Tutorial 2: The basic Solow model

1. Suppose that we have a Solow model with one twist. The twist is that there is a government. Each period, the government consumes a fraction of output, sG . Hence, the aggregate resource constraint is: Yt = Ct + It + Gt  .

where Gt = sGYt  . Define private output as Ytp  = Yt − Gt . Suppose that investment is a constant fraction, s, of private output (consumption is then 1 − s times private output). Otherwise the model is the same as in the text.

a) Re-derive the central equation of the Solow model under this setup (the TD equation).

b) Suppose that the economy initially sits in a steady state. Suppose that there is an increase in sG that is expected to last forever. Graphically analyze how this will affect the steady state value of the capital stock per worker. Plot out a graph showing how the capital stock per worker will be affected in a dynamic sense.

2. Suppose that we have a standard Solow model with a Cobb- Douglas production function. The central equation of the model is as follows:

kt+1  = sAkt(a)  + (1 − 6)kt

a)   Solve for steady state consumption per worker.

b)  The Golden rule of optimal savings rate: Use calculus to derive an expression for the s which maximizes steady state consumption per worker.

3. Suppose that we have a standard Solow model with a Cobb- Douglas production function. The central equation of the model is as follows:

kt+1  = sAkt(a)  + (1 − 6)kt

a)   Suppose that A is constant at 1. Solve for an expression for the steady state capital per worker, steady state output per worker, and steady state consumption per worker.

b)  Suppose that α = 1/3 and δ = 0.1. Create an Excel sheet with a grid of values of s ranging from 0.01 to 0.5, with a gap of 0.01 between entries (i.e. you should have a column of values 0.01, 0.02, 0.03, and so on). For each value of s, numerically solve for the steady state values of capital, output, and consumption per worker. Produce a graph plotting these values against the different values of s. Comment on how the steady state values of capital, output, and consumption per worker vary with s.

c)   Approximately, what is the value of s which results in the highest steady state consumption per worker? Does this answer coincide with your analytical result on question 2b above?

4. Use the Solow model’s diagram (TD equation vs. the 45 degree line) to describe how two economies (economy A and economy B) that only differ in terms of their initial capital per- capita (k0A > k0B ) converge to their long-run equilibrium. Discuss the implications for convergence.